Rodney Philani

Majozi

The expansion of sin nθ and cos nθ in

Keywords: De Moivre's formula, compound angles, binomial co-efficients Mathematics Subject Classification: Primary 33B10, 00A22. Secondary 05A10, 11B65

The expansion of sin nθ and cos nθ in terms of the powers of sin θ and cos θ respectively

Philani Rodney Majozi

To cite this version:

Philani Rodney Majozi. The expansion of sin nθ and cos nθ in terms of the powers of sin θ and cos θ respectively. 2017.

Introduction

The De Movre's formula and the compound indentities can be used to expand sin nΘ and cos nΘ into the powers of cos Θ and sinΘ, where n is an integer. The expansion of cos nΘ can be written in terms of powers of cosΘ, for all positive values of n. e.g cos 6θ = 32 cos 6 θ -48cos 4 θ + 18cos 2 θ -1. [START_REF] Kreysig | Advanced Engineering Mathematics[END_REF] However sin nΘ can only be written in terms of sinΘ, where n is an odd number.e.g sin 5θ = 5 sinΘ -20sin 3 Θ + 16sin 5 Θ.This is due to the fact that sin 2Θ = 2cos Θ sin Θ, which can cannot be simplified further. Therefore sin nΘ must be expressed in terms of sin 2Θ and sin Θ, for even values of n. [START_REF] Kreysig | Advanced Engineering Mathematics[END_REF] The De Movre's formula can be written as follows [START_REF] Kreysig | Advanced Engineering Mathematics[END_REF]:

(cos Θ + i sin Θ) n = n P =0 n P (cos θ) n (i sin θ) n-P (1)
The expansion of cos nΘ and sin nΘ can written as follows [START_REF] Kreysig | Advanced Engineering Mathematics[END_REF]:

cos nΘ = Re n P =0 n P (cos θ) n (i sin θ) n-P (2)
sin nΘ = Im n P =0 n P (cos θ) n (i sin θ)

n-P (3)
The answers in (2) and (3) will be written in terms of sin θ and cos θ. In this paper, we want to find the general formula that express sin nΘ and cos nΘ into the powers of cos Θ and sinΘ respectively.

(2) and (3) suggest that we can expect the co-efficients of cos Θ and sinΘ to be made of the binomial co-efficients.

1 Development of the formula to determine the co-effiecient of cosΘ in the expansion of cosnΘ

Laws of determing co-efficient of cosΘ

The formulas have laws which govern them. It is a essential to contruct laws to simplify the proceedings of finding a formula. Note: cos nΘ= 2 (n-1) cos n Θ -Acos n-2 Θ + Bcos n-4 Θ -Ccos n-6 Θ + Dcos n-8 Θ -...±n cosΘ/1, where A,B,C,D... are co-efficients. The exponents of cosΘ corresponds to the value of n. i.e If n is an odd number then the exponent of cosΘ will be an odd number. Definition 1.1. [START_REF] Kreysig | Advanced Engineering Mathematics[END_REF] The cosΘ that has an exponent(E) of n is called the First Term. i.e In the second term E=n-2, third term E=n-4,... Definition 1.1. [START_REF] Graham | Concrete Mathematics[END_REF] The first term has the co-efficient of 2 n-1 .The last term has a co-efficient of n,where n is an odd numbers. If n is an even number, the last term is always 1. Since the co-efficient of cosΘ and the value of 1 in the last terms are kwown, We don't consider them as terms. i.e The first term start at cos(2×Θ). So anyterm start to exist at cos(2×T m × Θ), where T m is the term number. i.e In first term T m =1, second term T m =2,... The term number has a relationship with n and E, which can be written as follows:

T m = 1 2 [n -E] + 1
1.2 Simplification of the co-efficients

1.2.

Expansions of cosnΘ

The expansion of cosnΘ in terms of the powers of cosΘ.

cosΘ = cosΘ cos2Θ = 2cos 2 Θ -1 cos3Θ = 4cos 3 Θ -3cosΘ cos4Θ = 8cos 4 Θ -8cos 2 Θ + 1 cos5Θ = 16cos 5 Θ -20cos 3 Θ + 5cosΘ cos6Θ = 32cos 6 Θ -48cos 4 Θ + 18cos 2 Θ -1 cos7Θ = 64cos 7 Θ -112cos 5 Θ + 56cos 3 Θ -7cosΘ cos8Θ = 128cos 8 Θ -256cos 6 Θ + 160cos 4 Θ -32cos 2 Θ + 1 cos9Θ = 256cos 9 Θ -576cos 7 Θ + 432cos 5 Θ -120cos 3 Θ + 9cosΘ cos10Θ = 512cos 10 Θ -1280cos 8 Θ + 1120cos 6 Θ -400cos 4 Θ + 50cos 2 Θ -1
As mentioned above, first term and the last term were only two terms that can be easily determined. So it essential to come with simplifications means. In order to simplify, there were adjustment were done on the co-efficients of the cos Θ.

Let

n 1 =1, n 2 =2,n 3 =3,...n n =n and 2 (n2-1) = 2, 2 (n3-1) = 4, ...2 (nn-1) = 2 n-1 1.2.

Simplification of the expansions of cosnΘ

After simplification the expansion of cosnΘ above will look as follows:

cosΘ = 2 (n1-1) cosΘ cos2Θ = 2 (n2-1) cos 2 Θ -1 cos3Θ = 2 (n3-1) cos 3 Θ -n 3 cosΘ cos4Θ = 2 (n4-1) cos 4 Θ -(2 (n2-1) + 2n 3 )cos 2 Θ + 1 cos5Θ = 2 (n5-1) cos 5 Θ -(2 (n3-1) + 2 n2 + 4n 3 )cos 3 Θ + n 5 cosΘ cos6Θ = 2 (n6-1) cos 6 Θ -(2 (n4-1) +2 n3 +2 (n2+1) +8n 3 )cos 4 Θ + (2 (n2-1) +2n 3 + 2n 5 )cos 2 Θ + 1 cos7Θ = 2 (n7-1) cos 7 Θ -(2 (n5-1) + 2 n4 + 2 (n3+1) + 2 (n2+2) + 16n
3 )cos 5 Θ + (2 (n3-1) + 2.2 n2 + 8n 3 + 4n 5 )cos 3 Θ -n 7 cosΘ cos8Θ = 2 (n8-1) cos 8 Θ -(2 (n6-1) +2 n5 +2 (n4+1) +2 (n3+2) +2 (n2+3) +32n 3 )cos 6 Θ + (2 (n4-1) +2.2 n3 +6.2 n2 +24n 3 + 8n 5 )cos 4 Θ -(2 (n2-1) +2n 3 + 2n 5 + 2n 7 )cos 2 Θ + 1 cos9Θ = 2 (n9-1) cos 9 Θ -(2 (n7-1) +2 n6 +2 (n5+1) +2 (n4+2) +2 (n3+3) +2 (n2+4) + 64n 3 )cos 7 Θ + (2 (n5-1) + 2.2 n4 + 6.2 n3 + 16.2 n2 + 64n 3 + 16n 5 )cos 5 Θ -(2 (n3-1) + 3.2 n2 + 12n 3 + 8n 5 + 4n 7 )cos 3 Θ + n 7 cosΘ cos10Θ = 2 (n10-1) cos 10 Θ -(2 (n8-1) + 2 n7 + 2 (n6+1) + 2 (n5+2) + 2 (n4+3) + 2 (n3+4) + 2 (n2+5) + 128n 3 )cos 8 Θ + (2 (n6-1) + 2.2 n5 + 6.2 n4 + 16.2 n3 + 40.2 n2 + 160n 3 + 32n 5 )cos 6 Θ -(2 (n4-1) + 3.2 n3 + 12.2 n2 + 48n 3 + 24n 5 + 8n 7 )cos 4 Θ + (2 (n2-1) + 2n 3 + 2n 5 + 2n 7 + 2n 9 )cos 2 Θ -1

We have established the co-efficients written in terms of 2 (nn-1) and n n . Each term can be now written in a more simplified form called {L : R} form. The L side is occupied by elements which has 2 (nn-1) and the R side is occupied by elements which has structure of n n .

The pattern of the co-efficients Of The Terms

Let us look at each term closely and study their pattern.

First Term

By the definition the first term start at cos(2Θ), where S=1.

cos1Θ

-→ 2 (n1-1) : 0 = 2 0 {1 : 0} cos2Θ -→ 2 (n2-1) : 0 = 2 1 {1 : 0} cos3Θ -→ 2 (n3-1) : 0 = 2 2 {1 : 0} cos4Θ -→ 2 (n4-1) : 0 = 2 3 {1 : 0} cos5Θ -→ 2 (n5-1) : 0 = 2 4 {1 : 0} cos6Θ -→ 2 (n6-1) : 0 = 2 5 {1 : 0} 1.3.

Second Term

The Second term start at cos(4Θ), where S=1. cos4Θ -→ 2 (n2-1) :

2n 3 = 2 1 {1 : 1n 3 } cos5Θ -→ 2 (n3-1) , 2 n2 : 4n 3 = 2 2 {1, 1 : 1n 3 } cos6Θ -→ 2 (n4-1) , 2 n3 , 2 (n2+1) : 8n 3 = 2 3 {1, 1, 1 : 1n 3 } cos7Θ -→ 2 (n5-1) , 2 n4 , 2 (n3+1) , 2 (n2+2) : 16n 3 = 2 4 {1, 1, 1, 1 : 1n 3 } cos8Θ -→ 2 (n6-1) , 2 n5 , 2 (n4+1) , 2 (n3+2) , 2 (n2+3) : 32n 3 = 2 5 {1, 1, 1, 1, 1 : 1n 3 } cos9Θ -→ 2 (n7-1) , 2 n6 , 2 (n5+1) , 2 (n4+2) , 2 (n3+3) , 2 (n2+4) : 64n 3 = 2 6 {1, 1 , 1, 1, 1, 1 : 1n 3
} Note: There are S(subscript number) number of terms in the L side and there are (T m -1) terms in the R side. The common factor in the {L : R} f orm is 2 S .

Third Term

The Third term start at cos(6Θ), where S=1. cos6Θ -→ 2 (n2-1) : 2n 3 ,

2n 5 = 2 1 {1 : 1n 3 , 1n 5 } cos7Θ -→ 2 (n3-1) , 2 n2 : 8n 3 , 4n 5 = 2 2 {1, 2 : 2n 3 , 1n 5 } cos8Θ -→ 2 (n4-1) , 2.2 n3 , 6.2 (n2+1) : 24n 3 , 8n 5 = 2 3 {1, 2, 3 : 3n 3 , 1n 5 } cos9Θ -→ 2 (n5-1) , 2.2 n4 , 6.2 n3 , 16.2 n2 : 64n 3 , 16n 5 = 2 4 {1, 2, 3, 4 : 4n 3 , 1n 5 } cos10Θ -→ 2 (n6-1) , 2.2 n5 , 6.2 n4 , 16.2 n3 , 40.2 n2 : 160n 3 , 32n 5 = 2 5 {1, 2, 3, 4, 5 : 5n 3 , 1n 5 } cos11Θ -→ 2 (n7-1) , 2.2 n6 , 6.2 n5 , 16.2 n4 , 40.2 n3 , 96.2 n2 : 384n 3 = 2 6
{1, 2, 3, 4, 5, 6 : 6n 3 , 1n 5 } Note: There are S(subscript number) number of terms in the L side and there are (T m -1) terms in the R side. The common factor in the {L : R} f orm is 2 S .

Fourth Term

The Fourth term start at cos(8Θ), where S=1.

cos8Θ

-→ 2 (n2-1) : 2n 3 , 2n 5 , 2n 7 = 2 1 {1 : 1n 3 , 1n 5 , 1n 7 } cos9Θ -→ 2 (n3-1) , 3.2 n2 : 12n 3 , 8n 5 , 4n 7 = 2 2 {1, 3 : 3n 3 , 2n 5 , 1n 7 } cos10Θ -→ 2 (n4-1) , 3.2 n3 , 12.2 n2 : 48n 3 , 24n 5 , 8n 7 = 2 3 {1, 3, 6 : 6n 3 , 3n 5 , 1n 7 } cos11Θ -→ 2 (n5-1) , 3.2 n4 , 12.2 n3 , 40.2 n2 : 160n 3 , 64n 5 , 16n 5 = 2 4 {1, 3, 6, 10 : 10n 3 , 4n 5 , 1n 7 } cos12Θ -→ 2 (n6-1) , 3.2 n5 , 12.2 n4 , 40.2 n3
, 120.2 n2 : 480n 3 , 160n 5 32n 7 , = 2 5 {1, 3, 6, 10, 15 : 15n 3 , 5n 5 , 1n 7 } cos13Θ -→ 2 (n7-1) , 3.2 n6 , 12.2 n5 , 40.2 n4 , 120.2 n3 , 336.2 n2 : 1344n 3 , 384n 5 , 64n 7 = 2 6 {1, 3, 6, 10, 15, 21 : 21n 3 , 6n 5 , 1n 7 } Note: There are S(subscript number) number of terms in the L side and there are (T m -1) terms in the R side. The common factor in the {L : R} f orm is 2 S .

Fifth Term

The Fifth term start at cos(10Θ), where S=1.

cos10Θ Note: There are S(subscript number) number of terms in the L side and there are (T m -1) terms in the R side. The common factor in the {L : R} f orm is 2 S .

-→ 2 (n2-1) : 2n 3 , 2n 5 , 2n 7 , 2n 9 = 2 1 {1 : 1n 3 , 1n 5 , 1n 7 , 1n 9 } cos11Θ -→ 2 (n3-1) , 4.2 n2 : 16n 3 , 12n 5 , 8n 7 , 4n 9 = 2 2 {1, 4 : 4n 3 , 3n 5 , 2n 7 , 1n 9 } cos12Θ -→ 2 (n4-1) , 4.2 n3 , 20.2 n2 : 80n 3 , 48n 5 , 24n 7 , 8n 9 = 2 3 {1, 4, 10 : 10n 3 , 6n 5 , 3n 7 , 1n 9 } cos13Θ -→ 2 (n5-1) , 3.2 n4 ,

Sixth Term

The Sixth term start at cos(10Θ), where S=1. cos12Θ -→ 2 (n2-1) : 2n It can be also noted that the co-efficients are made of the binomial coefficients of the Pascal triangle [START_REF] Graham | Concrete Mathematics[END_REF]. If the L side of the all the terms above is combined and give the following results:

The construction of the formula

In the above array of numbers; We can notice that the terms are made of the binomial co-efficient, which is inline with the formula in ( 2) and (3). We can now be able to formulate the general formula for the co-efficients of cosΘ.Firstly, we have to write Table 1 using binomial notations [START_REF] Graham | Concrete Mathematics[END_REF] as follows: Let's make the co-efficient of cos E Θ in the expansion of cosnΘ be CcosΘ E n . e.g The co-efficient of cos 16 Θ in the expansion of cos20Θ is CcosΘ 16 20 . CcosΘ E n can written in the binomial notation as follows: We have seen that there are S(subscript number) number of terms in the L side and there are (T m -1) terms in the R side. The common factor in the {L : R} f orm is 2 S .

CcosΘ 6 8 = 2 5 0 0 , 1 1 , 2 2 , 3 3 ,
Based on the findings above, we can write the general formula to determine the co-efficients of cosΘ in the expansion of cosnΘ. The General Formula is given by :

CcosΘ E n = 2 S S-1 P =0 R P + Tm-2 P =0 M S -1 (2P + 3)
where R = (P + T m -2) and M = (T m + S -3 -P ).

The sign of anyterm is given by the formula (-1) (Tm+1) . 2 Development of the formula to determine the co-efficient of sinΘ in the expansion of sinnΘ

Laws of determing co-efficient of sinΘ

The expansion of sinnΘ have laws which are slightly different from that of cosnΘ.

Remark 2.1.1 Recall that for the n of even numbers, sin nΘ must be expressed in terms of sin Θ and sin 2Θ. For n of even numbers; sin nΘ=[ n n 2 -Asin 2 Θ + Bsin 4 Θ -Csin 6 Θ +...±2 (n-2) sin (n-2) Θ]sin2Θ,where A,B,C,... are coefficients.

If n is an odd number; sin nΘ=nsin Θ + Asin 3 Θ -Bsin 5 Θ + Csin 7 Θ -Dsin 9 Θ +...±2 (n-1) sin n Θ, where A,B,C,D... are co-efficients. Definition 2.1.2 For n of odd numbers, the sinΘ that has an exponent(E) of n is called the First Term. i.e In the second term E=n-2, third term E=n-4,...

For n of even numbers, the sinΘ that has an exponent(E) of (n-2) is called the First Term. i.e In the second term E=n-4, third term E=n-6,... Definition 2.1.3 For n of odd numbers, The first term has the co-efficient of 2 n-1 .The last term has a co-efficients of n,where n is an odd numbers. Since the co-efficient of sinΘ is kwown, We don't consider it as terms. i.e The first term start at sin(2×Θ). So anyterm start to exist at sin(2×T m × Θ), where T m is the term number. i.e In first term T m =1, second term T m =2,...

If n is an even number;The first term has the co-efficient of 2 n-2 .The last term has a co-efficient of n n 2 . Since the co-efficients of the first term and the last terms are kwown, We don't consider the first term and the last term as terms. i.e The first term start at sin(4×Θ). So anyterm start to exist at sin((2×T m + 2) × Θ), where T m is the term number. i.e In first term T m =1, second term T m =2,...

Remark 2.1.4

The first term and last term does exist and are ignored for simplification purposes. Definition 2.1.5 For n of odd numbers; the number of times that the term existed since appeared for the first time is called the subscript(S), where S= (E -1).

If n is an even number; the number of times that the term existed since appeared for the first time is called the subscript(S), where S= n 2 -2 = E 2 .

Definition 2.1.6 For n of odd numbers; the term number has a relationship with n and E, which can be written as follows:

T m = 1 2 [n -E] + 1
If n is an even number; the term number has a relationship with n and E, which can be written as follows:

T m = 1 2 [n -E]
2.2 Simplification of the co-efficients

Expansions of sinnΘ

The expansion of sinnΘ in terms of the powers of sinΘ.

sinΘ = sinΘ sin2Θ = sin2Θ sin3Θ = 3sinΘ -4sin 3 Θ sin4Θ = [2 -4sin 2 Θ]sin2Θ sin5Θ = 5sinΘ -20sin 3 Θ + 16sin 5 Θ sin6Θ = [3 -16sin 2 Θ + 16sin 4 Θ]sin2Θ
Note: There are S(subscript number) number of terms in the L side and there are (T m -1) terms in the R side. The common factor in the {L : R} f orm is 2 S .

Third Term

sin7Θ -→ 2 2 {1, 2 : 2n 3 , 1n 5 } sin9Θ -→ 2 4 {1, 2, 3, 4 : 4n 3 , 1n 5 } sin11Θ -→ 2 6 {1, 2, 3, 4, 5, 6 : 6n 3 , 1n 5 } Note: There are S(subscript number) number of terms in the L side and there are (T m -1) terms in the R side. The common factor in the {L : R} f orm is 2 S .

The construction of the General Formula

Let's make the co-efficient of sin E Θ in the expansion of sinnΘ be CsinΘ E n . e.g The co-efficient of sin 16 Θ in the expansion of sin20Θ is CsinΘ 16 20 . Based on the three terms above; It can said that terms in the of the expansion of sin nΘ, where n is an odd number are the same as that of cos nΘ of the same kind. i.e They have the same formula, which is written as follows:

CsinΘ E n odd = 2 S S-1 P =0 R P + Tm-2 P =0 M S -1 (2P + 3)
where R = (P + T m -2) and M = (T m + S -3 -P ).

2.4

The formula of the co-efficients of sinΘ, where n is an even number

In this subsection, we will study the pattern of the terms where n is an even number. Then later,we will write the general formula of the expansion.

First Term

By definition, the second term start to exist at sin 4Θ. sin2Θ

-→ 2 0 {1 : 0} sin4Θ -→ 2 2 {1 : 0} sin6Θ -→ 2 4 {1 : 0} sin8Θ -→ 2 6 {1 : 0}

Second Term

The second term start to exist at sin 6Θ. sin6Θ -→ 2 2 {0, 1 :

1n 3 } sin8Θ -→ 2 4 {0, 1, 1, 1 : 1n 3 } sin10Θ -→ 2 6 {0, 1, 1, 1, 1, 1 : 1n 3 } sin12Θ -→ 2 8 {0, 1, 1, 1, 1, 1, 1, 1 : 1n 3 } Note:
There are E(exponent) number of terms in the L side and there are (T m -1) terms in the R side. The common factor in the {L : R} f orm is 2 E .

Third Term

The Third term start to exist at sin 8Θ. sin8Θ -→ 2 2 {1, 1 : 1n 3 , 1n 5 } sin10Θ -→ 2 4 {1, 1, 2, 3 : 3n 3 , 1n 5 } sin12Θ -→ 2 6 {1, 1, 2, 3, 4, 5 : 5n 3 , 1n 5 } sin14Θ -→ 2 8 {1, 1, 2, 3, 4, 5, 6, 7 : 7n 3 , 1n 5 } Note: There are E(exponent) number of terms in the L side and there are (T m -1) terms in the R side. The common factor in the {L : R} f orm is 2 E .

Fourth Term

The Fourth term start to exist at sin 10Θ.

sin10Θ -→ 2 2 {0, 2 : 2n 3 , 1n 5 , 1n 7 } sin12Θ -→ 2 4 {0, 2, 4, 7 : 7n 3 , 3n 5 , 1n 7 } sin14Θ -→ 2 6 {0, 2, 4, 7, 11, 16 : 16n 3 , 5n 5 , 1n 7 } sin16Θ -→ 2 8 {0, 2, 4, 7, 11, 16, 22, 29 : 29n 3 , 7n 5 , 1n 7 } Note: There are E(exponent) number of terms in the L side and there are (T m -1) terms in the R side. The common factor in the {L : R} f orm is 2 E .

Fifth Term

The Fifth term start to exist at sin 12Θ. sin12Θ -→ 2 2 {1, 2 : 2n 3 , 2n 5 , 1n 7 , 1n 9 } sin14Θ -→ 2 4 {1, 2, 6, 13 : 13n 3 , 7n 5 , 3n 7 , 1n 9 } sin16Θ -→ 2 6 {1, 2, 6, 13, 24, 40 : 40n 3 , 16n 5 , 5n 7 , 1n 9 } sin18Θ -→ 2 8 {1, 2, 6, 13, 24, 40, 62, 91 : 91n 3 , 29n 5 , 7n 7 , 1n 9 } Note: There are E(exponent) number of terms in the L side and there are (T m -1) terms in the R side. The common factor in the {L : R} f orm is 2 E .

Sixth Term

The Sixth term start to exist at sin 14Θ. sin14Θ -→ 2 2 {0, 3 : 3n There are E(exponent) number of terms in the L side and there are (T m -1) terms in the R side. The common factor in the {L :

R} f orm is 2 E . b (4)
where a and b are integers greater or equal to 0.

The first element will be written as ¯ a 0 . i.e The final answer of the first element must be converted.

The correct array can be represented in a new notation as follows: ¯ 0 0 The sum of the entries of the sine array in each row can result to a single element of the following row. These is inline with binomial co-efficient array. The theorems above can be applied to the unsimplified part of the general formula of the sine expansion to understand why it equal to 2 E Tm+E E + 1 . We will use (7) to simplify the first element of the express below.

E-1 P = 0 Tm+P -2 P + Tm-2 P =0 Tm+E-3-P E -1 (2P + 3) = Tm-2 P =0 Tm+E-3-P E -1 (2P + 3) + Tm+E-2 E -1 = Tm+E-2 E -1 + 3 Tm+E-3 E -1 + 5 Tm+E-4 E -1 + 7 Tm+E-5 E -1 + 9 Tm+E-6 E -1 + ...... + E E -1 (2T m -3) + E-1 E -1 (2T m -1)
Then split the terms such that we can use (5).

=

Tm+E-2 E -1 + 3 Tm+E-3 E -1 + 3 Tm+E-4 E -1 + 2 Tm+E-4 E -1 + 5 Tm+E-5 E -1 + 2 Tm+E-5 E -1 +5 Tm+E-6 E -1 +4 Tm+E-6 E -1 +......+ E E -1 (2T m -5)+ E E -1 (2T m + 2)+ E-1 E -1 (2T m -1) = Tm+E-2 E -1 +3 Tm+E-3 E -1 + Tm+E-4 E -1 +2 Tm+E-4 E -1 + Tm+E-5 E -1 + 5 Tm+E-5 E -1 + Tm+E-6 E -1 + ...... + E-1 E -1 (2T m -1)
When we apply (4)and (5) we get the following results.

=

Tm+E-2 E -1 +3 Tm+E-3 E -1 +2 Tm+E-4 E -1 +5 Tm+E-5 E -1 +......+ E-1 E -1 (2T m -1) = Tm+E-2 E -1 - Tm+E-3 E -1 + Tm+E-4 E -1 - Tm+E-5 E -1 + ...... ± E-1 E -1 + 3 Tm+E-3 E -1 + 2 Tm+E-4 E -1 + 5 Tm+E-5 E -1 + ...... + E-1 E -1 (2T m -1) = Tm+E-2 E -1 +2 Tm+E-3 E -1 +3 Tm+E-4 E -1 +4 Tm+E-5 E -1 +......+ E-1 E -1 (2T m -1 ± 1) = Tm+E E + 1
We have seen how CsinΘ E neven is simplified. So we can write the identity that is obtain on this simplification. The co-efficient of sin E Θ in the expansion of sin nΘ in terms of powers of sin Θ can now be written as follows:

CsinΘ E neven = 2 E E-2 P =0
Tm+E-2-P E -1 (P + 1)

= 2 E Tm+E E -1

Cosine Expansion

Theorem 3.2.1 If a,k and r are integers then the parallel summation of the binomial co-efficient can be written as follows [START_REF] Graham | Concrete Mathematics[END_REF]:

k≤a r+k k = r+a+1 a (11)
CsinΘ E neven can be simplied further by simplifying the two sigmas of the co-efficient. The co-efficient of cos E Θ in the expansion of cos nΘ in terms of powers of cos Θ can now be written as follows: 4 The general formula of the expansion of cos nΘ and sin nΘ in the powers of cos Θ and sin Θ respectively

CcosΘ E n =
The general formula for the co-efficients can be used to find the general formula of cos nΘ and sin nΘ. We have seen that the expansions have different formulas for n odd and n even .So we will use n odd and n even to represent expansions where n is an odd and even respectively.

Remark 1 . 1 . 3

 113 Note: The first term and last term does exist and are ignored for simplification purposes. Definition 1.1.4 The number of times that the term existed since appeared for the first time is called the subscript(S). e.g. First term cos(2×Θ)⇒ S=1, cos(3×Θ)⇒ S=2, cos(10×Θ)⇒ S=9, cos(n×Θ)⇒ S=E-1. Definition 1.1.5

1.4. 1

 1 An example of determining the co-efficients using the general formula Example1: Determine CcosΘ 7 15 the co-efficient of cos 7 Θ in the expansion of cos 15Θ in terms of powers of cos Θ. Given: n = 15 and E = 7. Step1: Determine the Term number(T m ). T m = 1 2 [15 -7] + 1 = 5. Step2: Determine the subsctript number(S) S = E -1 = 6. Step3: Calculate the co-efficient. Substitute Step1 and 2 into the general formula.

Theorem 3 . 1 . 1

 311 If a and b are positive integers such that a,b = 0 then a

Theorem 3 . 1 . 2 Remark 3 . 1 . 3 1

 3123131 The sum of the elements in a column(C) and in a row(R) of the sine array is given by: C,R and n are positive integers ≥ 0. Recall: If the value of P in a sigma does not have a cap on the top of 0, the answer of the first element of an array does not change. Theorem 3.1.4 If a is a positive integer such that a = 0: The binomial coefficient has the same value with sine array element where all the numbers in both notations are the same. Simplification of the general formula

Theorem 3 . 1 . 5 b

 315 If a,b are positive integer such that a,b = 0 then: + ...... + (a -b + 1)

1 S - 1 (T m - 1 )

 111 Then we can apply (9) and get the following results:

n - 2 - 2 -2 and neven 2 - 2 Example3:

 2222 2P cos (n-2-2P ) θ + 2 n-1 cos n Θ + (-1) A+2 n odd cosΘ/1 evenwhere A= n odd -1 Expand cos 10Θ in terms of powers of cos Θ. Sustitute the value of n into A and the general formula. A= 10 2 -2 = 3.

  12.2 n3 , 40.2 n2 : 320n 3 , 160n 5 , 64n 7 , 16n 9 = 2 4 {1, 4, 10, 20 : 20n 3 , 10n 5 , 4n 7 , 1n 9 } cos14Θ -→ 2 (n6-1) , 4.2 n5 , 20.2 n4 , 80.2 n3 , 280.2 n2 : 1120n 3 , 480n 5 160n 7 , 32n 9 = 2 5 {1, 4, 10, 20, 35 : 35n 3 , 15n 5 , 5n 7 , 1n 9 } cos15Θ -→ 2 (n7-1) , 4.2 n6 , 20.2 n5 , 80.2 n4 , 280.2 n3 , 896.2 n2 : 3584n 3 , 1344n 5 , 384n 7 , 64n 9 = 2 6 {1, 4, 10, 20, 35, 56 : 56n 3 , 21n 5 , 6n 7 , 1n 9 }

  n4 , 30.2 n3 , 140.2 n2 : 560n 3 , 320n 5 , 160n 7 , 64n 9 , 16n 11 = 2 4 {1, 5, 15, 35 : 35n 3 , 20n 5 , 10n 7 , 4n 9 , 1n 11 } cos16Θ -→ 2 (n6-1) , 5.2 n5 , 30.2 n4 , 140.2 n3 , 560.2 n2 : 2240n 3 , 1120n 5 480n 7 , 160n 9 , 32n 11 = 2 5 {1, 5, 15, 35, 70 : 70n 3 , 35n 5 , 15n 7 , 5n 9 , 1n 11 } cos17Θ -→ 2 (n7-1) , 5.2 n6 , 30.2 n5 , 140.2 n4 , 560.2 n3 , 2016.2 n2 : 8064n 3 , 3584n 5 , 1344n 7 , 384n 9 ,64n 11 = 2 6 {1, 5, 15, 35, 70, 126 : 126n 3 , 56n 5 , 21n 7 , 6n 9 , 1n 11 } Note: There are S(subscript number) number of terms in the L side and there are (T m -1) terms in the R side. The common factor in the {L : R} f orm is 2 S .

[START_REF] Kleitz | Digital and Microprocessor Fundamentals[END_REF]

, 2n 5 , 2n 7 , 2n 9 , 2n 11 = 2 1 {1 : 1n 3 , 1n 5 , 1n 7 , 1n 9 , 1n 11 } cos13Θ -→ 2 (n3-1) , 5.2 n2 : 20n 3 , 16n 5 , 12n 7 , 8n 9 , 4n 11 = 2 2 {1, 5 : 5n 3 , 4n 5 , 3n 7 , 2n 9 , 1n 11 } cos14Θ -→ 2 (n4-1) , 5.2 n3 , 30.2 n2 : 120n 3 , 80n 5 , 48n 7 , 24n 9 , 8n 11 = 2 3 {1, 5, 15 : 15n 3 , 10n 5 , 6n 7 , 3n 9 , 1n 11 } cos15Θ -→ 2 (n5-1) , 5.2

  3 , 2n 5 , 2n 7 , 1n 9 , 1n 11 } sin16Θ -→ 2 4 {0, 3, 9, 22 : 22n 3 , 13n 5 , 7n 7 , 3n 9 , 1n 11 } sin18Θ -→ 2 6 {0, 3, 9, 22, 46, 86 : 86n 3 , 40n 5 , 16n 7 , 5n 9 , 1n 11 } sin20Θ -→ 2 8 {0, 3, 9, 22, 46, 86, 148, 239 : 239n 3 , 91n 5 , 29n 7 , 7n 9 , 1n 11 } Note:

Acknowledgment

I would like to thank all my teachers at Sanzwili Primary School, Edendale Technical High School and Protec Pietermaritzburg for making mathematics an interesting subject. I would also like to express deep gratitude to Mr Sibusiso Ntshangase for his support during the course of this paper. Once again, I would like to thank all other members of the staff at the University of the Witwatersrand that helped and encourage me during the compilation of this paper.

sin7Θ = 7sinΘ -56sin 3 Θ + 112sin 5 Θ -64sin 7 Θ sin8Θ = [4 -40sin 2 Θ + 96sin 4 Θ -64sin 6 Θ]sin2Θ sin9Θ = 9sinΘ -120sin 3 Θ + 432sin 5 Θ -576sin 7 Θ + 256sin 9 Θ sin10Θ = [5 -80sin 2 Θ + 336sin 4 Θ -512sin 6 Θ + 256sin 8 Θ]sin2Θ Let n 1 =1, n 2 =2,n 3 =3,...n n =n and 2 (n2-1) = 2, 2 (n3-1) = 4, ...2 (nn-1) = 2 n- 1 We have seen that the {L : R} form of the expansion of cosnΘ is simplification of the terms written in a form n n = n and 2 (n-1) .

Simplification of the expansions of sinnΘ

After simplification the expansion of sinnΘ above will look as follows: sin1Θ = 2 0 {1 : 0} sin1Θ sin2Θ = 2 0 {1 : 0} sin2Θ sin3Θ = n 3 sinΘ -2 2 {1 : 0} sin 3 Θ sin4Θ = [n 2 -2 2 {1 : 0} sin 2 Θ]sin2Θ sin5Θ = n 5 sinΘ -2 2 {1, 1 : 1n

, 4 : 4n 3 , 3n 5 , 2n 7 , 1n 9 } sin 3 Θ + 2 4 {1, 3, 6, 10 : 10n 3 , 4n 5 , 1n 7 } sin 5 Θ -2 6 {1, 2, 3, 4, 5, 6 : 6n 3 , 1n 5 } sin 7 Θ + 2 8 {1 : 0} sin 9 Θ sin12Θ = [n 6 -2 2 {1, 2 : 2n 3 , 2n 5 , 1n 7 , 1n 9 } sin 2 Θ + 2 4 {0, 2, 4, 7 : 7n 3 , 3n 5 , 1n 7 } sin 4 Θ -2 6 {1, 1, 2, 3, 4, 5 : 5n 3 , 1n 5 } sin 6 Θ + 2 6 {0, 1, 1, 1, 1, 1 : n 3 } sin 8 Θ + 2 10 {1 : 0} sin 10 Θ]sin2Θ 2.3 The formula of the co-efficients of sinΘ, where n is an odd number

In this subsection, we will study the pattern of the terms where n is an odd number. Then later,we will write the general formula of these expansion.

The construction of the General Formula

Let us look at the L side of the expansion of the terms above.We will write their pattern from the second term up to sixth term. 0 1 1 1 1 1 1 1 1 1 1 1 1 2 3 4 5 6 7 8 We can notice that the terms are made of a strange array of numbers, which is not binomial notation. It is essential to study this array in details. Let us examine the pattern of this array in terms of the binomial co-efficients.

We have seen from the second array(pattern of the first array) above that the co-efficients are made of binomial co-efficients in a strange form. However the first elements of the second array contradict with the first elements of the first array. In order to illustrate this contradiction, let's represent the second array in the format of the first array. The difference between the two arrays is the replacement of 1 and 0 of the first array by 0 and 1. However we should recall that the first array is the correct array for finding the co-efficients in the expansion of sin nΘ. i.e We have to find the notation of replacing the value 1 and 0 of the second array by 0 and 1.

Remark 2.4.1 The Boolean notation is made for inputs of 1's and 0's in the logic circuits, where 1 represent switch on and 0 represent switch off. The Boolean algebra uses the symbol Ā for inverting the value of A. e.g. If the input A is 1 then output will be 1 = 0. [START_REF] Kleitz | Digital and Microprocessor Fundamentals[END_REF] The second array can now be written in the correct form, which is same as the first array using the Boolean equation.

We can now write the general formula for each element in an array. The general formula for the elements which made up the co-efficients is shown below:

At this point, we are in a position of writing the general formula for the expansion of sin nΘ, where n is an even number. Let's examine some couple of terms then write the general formula. We have seen that there are E(Exponent value) number of terms in the L side and there are (T m -1) terms in the R side. The common factor in the {L : R} f orm is 2 S .

Based on the findings above, we can write the general formula to determine the co-effiecient of sinΘ in the expansion of sinnΘ. We will use the cap on the general formula to remind us to invert the first entry on the co-efficient. The General Formula is given by :

where R = (P + T m -2) and M = (T m + E -3 -P ).

An example of determining the co-efficients using the general formula

Example2: Determine CsinΘ 6 16 the co-efficient of sin 6 Θ in the expansion of sin 16Θ in terms of powers of sin Θ.

Given: n = 16 and E = 6.

Step1: Determine the Term number(T m ). T m = 1 2 [16 -6] = 5. Step2: Determine the subsctript number(S) S = E 2 = 3. Step3: Calculate the co-efficient. Substitute Step1 and 2 into the general formula. It have been shown above that the expansion of sin nθ for the n of odd numbers share the same general formula with cos nθ of the same kind. However the signs of the co-efficients are not same the between these two expansion. (-1) (3-P ) .2


Bruno Jetin
email: bruno.jetin@ubd.edu.bn

Functional income distribution and growth in Thailand: A Post Keynesian econometric analysis

Keywords: Thailand, functional distribution of income, Post Keynesian economics, growth, development

. Several measures of the labor income share are calculated to take into account the fact that wage labor represents only half of the total labor force and check the robustness of our results. We also introduce a new treatment of external trade to better integrate the price competitiveness of Thailand. The econometric investigation shows that the growth regime is profit-led over the period 1970-2011 which shows that rebalancing the Thai economy will be difficult and requires an overall change of strategy going beyond a simple pro-labor policy.

stronger role for the domestic market implies that an increase of household consumption offsets at least in part a decline in net exports. This article explores this possibility by looking at the relation between the functional income distribution and demand.Rebalancing growth to lessen the dependence on exports will be much easier if an increase in the labor income share has a sufficient positive effect on consumption to offset a negative effect on investment and exports. If it is the case, the demand is wage-led; otherwise it is profit-led. We apply the Post Keynesian model developed by [START_REF] Stockhammer | Functional Income Distribution and Aggregate Demand in the Euro-area[END_REF]which precisely addresses this question. This is the first application of such a model on Thailand, most of previous publications being based on advanced economies or large Asian economies like China and India which have different structural features. Our second contribution is the use of an original approach to calculate the labor income share in the context of a developing country. The objective is to include non-wage labor income in the labor share so that the full impact of a change in income distribution on growth is taken into account. The third contribution is the deepening of [START_REF] Stockhammer | Functional Income Distribution and Aggregate Demand in the Euro-area[END_REF] model with a new treatment of international trade, which employsexport prices of competitor countries to explain exports and export prices of the country under investigation. This had not been done, in previous publications.

The article is organized as follows. Section 1 reviewsthe main characteristics of Thailand"s macroeconomic performance. In Section 2 we compute various estimates of theThai labor income share, which for the major part includes non-wage labor, to test the robustness of our overall results. In Section 3, we present the theoretical model and the changes we introduce in the analysis of the foreign trade. In section 4, we proceed to the econometric investigation and present the results.We find that Thailand is a profit-led economy because an increase in the labor income share has a strong negative effect on net exports which is not offset by a positive effect of the domestic demand. An increase in the labor income share has either a positive but small effect on household"s consumption or a perverse (negative) effect, while in all cases it has no significant effect on investment.

Section 5 concludes and draws some policy implications. A redistribution of income in favor of labor would have short-term negative effect on demand. This does not mean that Thailand should not rebalance its growth but that a mere income policy is not the best suited one to achieve this goal.

Thailand's macroeconomic performance and structural change

Along with the founders of ASEAN (Association of South East Asian Nations), Thailand belongs to the second generation of the newly industrialized countries which emerged at the turn of the nineties. Theyfollow the first generation of the four "dragons" (South Korea, Taiwan, Hong Kong and Singapore) which appeared at the turn of the eighties. Thailand was a poor country after the Second World War, but thanks to a 5.8% average growth rate over the period 1960-2013, it is now part of the upper middle-income group of countries according to the World Bank with a gross national income per capita of $ 5,370 in 2013below China ($6,560) and above Indonesia $3,580 1 .

After an initial phase of import-substitution policy from 1950 to 1977, policy makers adopted a policy of export-promotion growth based on an active support of industry.This led to a high rate of growth achieved thanks to a strong industrial push during a boom phase (1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994)(1995)(1996) characterized by a strong domestic investment rate which at the time induced a trade deficit of 4.4% of the GDP. Despite a solid increase of industrial exports, imports of intermediate and capital products were still higher. After the Asian crisis of 1997-98, which deeply affected Thailand with a 3.2% recession of GDP in real terms, the recovery was prompted by a strong export growth.The trade balance became this time positive amounting to 5.3% of GDP over the period 2000-2007. After the international crisis of 2008-09, the trade surplus still accounted for 3.4% over the period 2010-2014 while investment never recovered its pre-Asian crisis level 2 . For this reason, Thailand has been characterized as an export-led economy since the Asian crisis (Tharnpanich and McCombie, 2013, p 572;Jetin, 2012, p 15, Felipe andLim, 2005, p 22).

Thailand has recently become a very open economy with itssum of exports and imports amounting to around 80% of GDP. Industry in the broad sense (ISIC C-E) now makes up 36% of GDP to be compared with a world average of 22%, which places Thailand as an industrialized country 3 . This fast industrializationwas based on rapid capital accumulation initialized and managed by the State which later on triggered a cumulative growth cycle whereby demand growth promotes productivity gains via the Kaldor-Verdoon relation, and labor productivity gains in turn leads to an increase in demand [START_REF] Naastepad | Strategic factors in economic development: East Asian industrialization 1950-2003[END_REF]. However, except for limited periods of time, Thai workers have never fully benefited from this growth cycle because their real income has lagged behind productivity gainscontributing toward achieving a high profit rate in the Thai economy as a whole and in particular in manufacturing [START_REF] Jetin | Distribution of income, labour productivity and competitiveness: is the Thai labour regime sustainable?[END_REF].

Nonetheless, the industrializationand the high growth rate it promoted induceda profound structural change of the Thai society, which is reflected in the reduction of poverty incidence,the change in employment structure, and the increase of income inequality.In 2013, absolute poverty headcount was close to 0% of the population according to the PPP$ 1.25 international poverty line of the World Bank, and 10% according to the national poverty line in 2013, down from respectively 17.2% and 65.2% in 1988 when the boom period kicked off 4 . The Gini coefficient of income inequality rose from 0.423 in 1962 to 0.527 in 2006 [START_REF] Warr | Poverty Reduction through Long-Term Growth: the Thai Experience[END_REF]. Private sector employees,who accounted for 10% of total employment in 1969 5 add up to 40% in 2014. Adding government employees puts the share of employees to about 50% of total employment. This is still low in comparison with rich countries where employees represent around 80% or more of employment. This increase in the wage-earners" share was prompted by a sharp decline of family helpers in the farm sector to 13% in 2014 down from 50% in 1969 and to a lesser extent by a decline of own-account farmers from 26% in 1969 to 18% in 2014.On the contrary, non-farm own-account workers" share grew from 5% to 15% of total employment over the period mostly because part of rural workers could not find a paid employmentand turned self-employed in cities instead 6 . The major part of these urban own-account workers provides services to wage workers so that their income is very dependent on the overall level of wages.A minor part is composed of high skilledindependent professionals (lawyers, physiciansetc.) like in rich capitalist countries.

To summarize, these deep and durable changes in the employment structure show that wage labor, if not dominant, already impulses the dynamics of the national demand and justifies the application of a Post-Kaleckian theoretical model developed for the rich capitalist countries.To this aim, our article analyzes the impact of a change in labor income share on growth in a broad sense; that is whatever the source of the labor income, be it wage or non-wage. The objective is to observe the effect of a change in the labor share on growth over the long period , which saw an intense structural change. We will also restrict the period under investigation to 1980-2011 when industrialization took off and a new working class of around 5 million blue collar workers in manufacturing and 12 million in the whole economy emerged 7 . These industrial workers, mostly concentrated in the Eastern seaboard of Thailand, have all the characteristics that Kalecki had in mind when he conceived a theoretical model where wages are the result of class struggle and bargaining. In the next section, we explain how we measure the labor share in various ways to either capture labor income in a comprehensive way or restrict it to wage earners only.

Measuring the labor share: the Thai case

Measuring the labor share always involves a degree of arbitrariness because the labor income stems from two categories of workers namely employees who earn a wage and own-account workers who earn a "mixed income" including both the returns on labor inputs and on capital investment. It is therefore necessary to estimate an adjusted labor share of income (ALS) that takes mixed income into account. In rich countries, where own-account work usually amounts to less than 10%, attributing the average compensation of employees to own-account workers (ALS 1) is common practice. To be as comprehensive as possible,we test all these possible adjustments to check the sensitivity of our econometric results on the choice of the measure of the labor"sshare. ALS 4 , is our preferred estimate because it avoids attributing all mixed income to the labor"s share or hypothesizing that own-accounts workers have the same capital intensity than registered companies.Figure 1 presents the evolution of these labor shares in Thailand from 1969 to 2013. The next section presents our theoretical model and tests the impact of a variation of these alternative labor sharesof income on growth.

Theoretical Approach

The theoretical model employed for the analysis of the growth regime in Thailand is a new version of the model by [START_REF] Stockhammer | Functional Income Distribution and Aggregate Demand in the Euro-area[END_REF], which is based on the model of [START_REF] Bhaduri | Unemployment and the real wage: the economic basis for contesting political ideologies[END_REF]. [START_REF] Stockhammer | Functional Income Distribution and Aggregate Demand in the Euro-area[END_REF] aim to find the effect of income distribution on consumption (C), investment (I), and net exports (NX) and thus the total effect on aggregate demand (Y). Those demand components are functions of income (Y), labor share (LS) and some control variables (denoted by z)which are assumed to be exogenous. Government expenditures (G) depend only on income. Then the aggregate demand can bewritten as:

AD= C(Y,LS) + I(Y, LS, z I ) + NX (Y, P, z NX ) + G(Y, z G ) (1)
where,P is the (domestic) price level. The price level is also a function of wage share and other control variables. The prices considered in the model are domestic, export (P x ) and import (P m ) prices. Differentiation of the equation (1) with respect toLS and goods market equilibrium condition (Y*=AD) gives:

d𝑌 * d𝐿𝑆 = ℎ 2 1-ℎ 1 (2)
where,h 1 = ∂C/∂Y+∂I/∂Y+∂NX/∂Y+∂G/∂Y and h 2 = ∂C/∂LS +∂I/∂LS +∂NX/∂LS.For the stability of the model, the multiplier 1/(1-h 1 ) should be positive. Then the sign of h 2, called private excess demand, determines the sign of the total derivative. It gives the change in demand due to a change in income distribution at a fixed level of income. Its sign cannot be determined without an empirical investigation since the partial derivatives have different signs. ∂NX/∂Y is equal to ∂X/∂Y-∂M/∂Y; however, since there is no effect of domestic income on the exports, the term reduces to -∂M/∂Y. Since the government sector is excluded from the model the term ∂G/∂Y disappears. Eventually, h 1 becomes ∂C/∂Y+∂I/∂Y-∂M/∂Y. If dY*/dLS turns out to be positive then the demand regimeis wageled; otherwise it is profit-led.For a given level of Y, the total effect of income distribution on the equilibrium demand can also be written as:

d𝑌 * /𝑌 d𝐿𝑆 = ℎ 2 1-ℎ 1 (3)
which givesthe percentage change in the demand due to a one percent increase in the LS as h 2 = ∂C/Y /∂LS +∂I/Y /∂LS +∂NX/Y /∂LS. 9

In this framework an important distinction is the one between domestic and open economy. In the former, if the consumption is more sensitive to a variation of the labor share than the investment, then the economy is wage-led, otherwise profit-led. Here it is assumed that an increase in labor share promotes total private consumption and discourages investment. The increasing labor share is supposed to stimulate the consumption of the workers more than that of the capitalists. 10 If the economy is domestically wage-led, the entire economy might or not be wage-led depending on the elasticity of exports to the unit labor costs and hence the labor share. However, if the domestic sector is profit-led, the entire economy is necessarily profit-led.

In order to find the effect of the labor income share on the national income, the GDP components and the prices (domestic and export) are modeled as separate equations by [START_REF] Stockhammer | Functional Income Distribution and Aggregate Demand in the Euro-area[END_REF]. While maintaining their analytical framework for the domestic economy,for the analysis of tradewe opt for the approach by [START_REF] Mathis | La competitivité industrielle[END_REF].

These authors proposea framework of international competition which discriminates between price and non-price factors. We integrate some elements related to the international price competition. [START_REF] Thirlwall | The Nature of Economic Growth[END_REF], based on [START_REF] Kaldor | The case for regional policies[END_REF], also employs a similar exports function taking the price competitiveness into account (Hein, 2014, p. 148) Regarding the domestic economy, [START_REF] Stockhammer | Functional Income Distribution and Aggregate Demand in the Euro-area[END_REF] propose a consumption function of the form C= f (W, R), where W is the labor income (or adjusted labor income) and R is the capital income (or adjusted operating surplus). It is expected that workers have a higher propensity to consume than profit earners. The investment is modeled as I=f (Y, R) in order to concurrently take into account the scale of the economy and the profitability. We employ the same functions in our model.

As for the trade equations, firstly exports and imports are estimated as separate equations but they are also treated together as a single net exports equation. The exports function proposed by [START_REF] Stockhammer | Functional Income Distribution and Aggregate Demand in the Euro-area[END_REF] is of the form X = f (Y for , E, P x /P m ),where Y for is the main foreign trade partners" weighted GDP, E is the exchange rate and P x /P m is the ratio of export prices to import prices. The imports are modeled as M = f (Y, E, P/P m ).

The combination of the two functions is NX/Y = f (Y, Y for , E, RULC)where RULC is the real unit labor costs and ULC the unit labor costs.

We employ the same imports function but we exclude the exchange rate variable in the trade equations since all the macroeconomic magnitudes are in constant dollars in our model. In other words, the exchange rate is implicitly present in the model. The exports function is different in two points: firstly, it takes into account the export prices P x and that of the competitors P x comp , hence their ratio, which is the relative export prices P x rel . 11 This assumption is more appropriate since the importing countries compare the export prices of the exporting country by that of its competitors, not by the exporter"s import prices.

However, the impact of the import prices on the export prices is not neglected but captured via the domestic prices, which is also a function of unit labor costs. The domestic prices and P x comp determine the P x .The second difference is that exports are a function of the total (weighted) imports 12 of the world, but not of the global GDP of the rest of the world. We do not use a net exports function.

As for the price equations, we employed the same domestic prices equation proposed by [START_REF] Stockhammer | Functional Income Distribution and Aggregate Demand in the Euro-area[END_REF], which is modeled as P = f (P m , ULC). The export prices are endogenized in our model in a price-maker-price-taker setting, as proposed by [START_REF] Mathis | La competitivité industrielle[END_REF].Export prices are a function of the domestic prices and the weighted export prices of thecompetitors. The comparison of our model with that of [START_REF] Stockhammer | Functional Income Distribution and Aggregate Demand in the Euro-area[END_REF] is in Table1. It should be mentioned that, although subsistence farming plays an important role for poor farmers, especially those living from upland agriculture, cash crop is dominant in the plain and has allowed farmers to also engage increasingly into wage labor. Less than a quarter of rural people live exclusively from farming activities and around two thirds of their income is derived from off-farm activities [START_REF] Walker | Thailand"s Political Peasants. Power in the Modern Rural Economy[END_REF]. In 2013, the share of agriculture, industry and services in value added was respectively 9.5%, 43% and 47.5% 15 .

All these elements indicate that Keynesian policies have become more applicable in Thailand in the course of time.In this respect, the analysis of the subsample that covers the last three decades is expected to give more reliable results in terms of applicability of our post-Keynesian model to Thailand.

The second reason is to detect if the growth regime has changed after 1980 when the neoliberal hegemony began to dominate and the income distribution to deteriorate on a global scale. However, this type of a subsample study does not econometrically correspond to testing regime shift (from profit-led to wage-led, or vice-versa). We resort to such an econometric analysis since the number of the observations is limited and division of the sample into twowould not be sufficient to find robust results.A priori, we expect a more wage-led or less profit-led growth regime in the subsample of 1980-2011 compared to the full sample.

The third reason behind analyzing the subsample starting with 1980 is econometric. The Chow test suggests a structural break in some equations with some measures of the labor"s income share at this year. This point will be explained in the section 4, which is dedicated to the econometric estimations of the model.

A review of empirical literature

There are two approaches to testing Bhaduri-Marglin type of wage-led-profit-led models.

The first type analyzes consumption, investment, exports and imports (or net exports)

separately through single equations. [START_REF] Bowles | Wages, aggregate demand, and employment in an open economy: an empirical investigation[END_REF] are the first to apply such a methodology to some developed countries. There is a vast literature on developed countries in this fashion. However, theoretical framework and equations vary. Among those, Naastepad and Storm (2006-7) examine eight OECD countries and conclude that the demand is wage-led in France, Germany, Italy, the Netherlands, Spain, and the United Kingdom, whereas profit-led in Japan and the USA. [START_REF] Hein | Distribution and growth reconsidered: empirical results for six OECD countries[END_REF], analyze Austria, France, Germany, the Netherlands, the UK and the USA. [START_REF] Stockhammer | Functional Income Distribution and Aggregate Demand in the Euro-area[END_REF] apply the aforementioned model on the Euro area, considering it as a single country, and find that the whole area is wage-led.

The second type of models takes into account the interactions among the variables with the use of Vector Autoregressive Models (VAR). Gordon (1995aGordon ( , 1995b) applies such a model, along with single equation estimations, to USA. Stockhammer and Onaran (2004) employ a structural VAR modelin their analysis of the USA, the UK, and France where they test a set of hypotheses. [START_REF] Onaran | Two different export-oriented growth strategies:accumulation and distribution in Turkey and in South Korea[END_REF] use a similar approach for South Korea and Turkey and findsome evidence for wage-led regimes. More recently, [START_REF] Nishi | Structural VAR analysis of debt, capital accumulation, and income distribution in the Japanese economy: a Post Keynesian perspective[END_REF] examines the relationship among income distribution, debt ratio, and capital accumulation in Japan using a similar technique but a different modelthat distinguishes between debt-led and debt-burdened regimes. [START_REF] Hein | Distribution and Growth after Keynes: A Post-Keynesian Guide[END_REF] provides an extensive literature review on both types of models along with the results of the analyses.

While most of the early literature focused on developed countries, a number of empirical studies have addressed the case of LDC recently. A pioneer workis that of [START_REF] Naastepad | Strategic factors in economic development: East Asian industrialization 1950-2003[END_REF] who analyzed the growth pattern regime of the East and the Southeast Asian countries. They build a different theoretical model derived from the Kaldor-Verdoon relation and found that these countries are profit-led.In the case of Thailand, the profit-led character is due to a positive effect of a rise in profitability on investment while the effect on net exports is not relevant. Onaran and Galanis (2014)have also realized a comprehensive analysis ofboth rich countries and LDCs over the period 1960-2007. 16 They use the theoretical model and methodology of Stockhammer et al. effect of an increase of the profit share on consumption higher than the positive effect on investment. Over the whole period, the Chinese domestic economy is thus wage-led but when the positive effect of decreasing labor costs on net exports is taken into account the growth regime turns profit-led although of a lesser magnitude than in Onaran and Galanis" study.We will compare our results with these but before the estimation procedure of the model equations is explained in the next section.

Estimation procedures and the Unit Root Tests

We firstly examined the stationarity properties of the series, in logarithm, via Augmented Dickey Fuller (ADF) test. The highest order of integration is found to be 1 (i.e. I( 1)).

When all the series in the same equation are I(1) we tested for cointegration among them via Engle-Granger (EG) method. 17 When there was no cointegration, we ran an OLS in difference form. When all the variables in the same equation are stationary we applied the OLS method in differences, since we look for the short-run effects. We added autoregressive terms via Prais-Winsten (PW) method and pertinent dummy variables for the outliers when necessary. When the variables in the same equation are not of the same order, we employed Pesaran"s (2001) Bounds test. Thus we can test cointegrationamong variables of different orders among which the maximum order of integration is 1 18 . When cointegration tests gave no sign of cointegration or were inconclusive, we applied the OLS method to the differences of the variables. As the model aims to capture the short-run dynamics between income distribution and the demand growth, lack of cointegration among the variables will not undermine the model. The ADF tests did not give straightforward results for the estimation of the equations. The unit labor costs and the price variables (except the domestic prices) turn out to be stationary and all the other variables are I(1) for the full sample at 5 % significance level.

In the next section we present the estimation results based on the procedure that we applied.

Estimation of the model equations and results

Consumption

We firstly tested cointegration among consumption and labor and capital income series via EGmethod since the lnC, lnW i andlnR i series are all found to be I(1) according to the ADF tests.As we detected no cointegrationamong the series, with four alternatives of labor and capital income, we estimated the consumption equation in first differencesvia OLS as in the form ∆lnC= constant + e CW ∆lnW i +e CR ∆lnR i , where i is from 1 to 4. e CW and e CR represent the labor income and profit elasticities of consumption, respectively.A priori,

we expect e CW the propensity to consume out of labor income, to be higher than e CR, the propensityto consume out of profits, since workers spend a greater part of their income than profit earners. A shift of the income distribution in favor of labor will rise consumption expenditures. The higher the difference between e CW and e CR the higher the probability of the economy to be wage-led and vice-versa. Empirical research confirms that it is the casewith a difference between the marginal propensities around 0.4, in accordance with the findings of the previous evidence as mentioned byLavoie and Stockhammer,(2013, p 22-24),(see Table 2).

Table 2:Consumption Equation Estimations Here

e CW is higher than e CR for the four measures of income distribution ranging from 0.49 to 0.65, whereas those of e CR are between 0.16 and 0.24 22 . ALS 1 , registers the highest difference between the two estimated elasticities(0.49) while for the three adjusted labor shares that include own-account workers,it is between 0.25 and 0.38. One reason may be that own-account workers have a lower propensity to consume [START_REF] Tobin | Essays in Economics: Consumption and Econometrics[END_REF], p 84) either for precautionary motives due to the risky nature of their business or because they earn higher revenues and have a higher propensity to save like rich households and entrepreneurs. In order to estimate the impact of a change in income distribution on consumption for the period under consideration, we transform the elasticities into marginal effects and take the difference of them with the formula:∂C/Y/ ∂LS =e CW C/We CR C/R,where C/W and C/R are sample averages or the values of the corresponding ratios at a certain point of the time period considered.We obtain different resultsat the sample mean, at the beginning and the end of the period of the sample as seen in Table 3. When the whole sample is considered, the effect is positive for ALS 1 , which imputes the average compensation to own-account workers, and negative for the three other adjustments which include the income of own-account workers.It is suggested that a 1% increase of the labor income share leads to a 0.24% rise of consumption when ALS 1 is considered and a decrease of consumption ranging from 0.07 to 0.42% for the other adjusted labor shares. The reason is that in the last three cases, the labor income hike leads to a decline of capitalists" consumption of a greater magnitude than the rise of workers" consumption.Conversely, an increase of the profit share would induce a rise of consumption. This confirms that as far as consumption is concerned, demand is profit-led in Thailand when own-account workers" income is taken into consideration.We observe that a redistribution of income in favor of labor is more likely to raise consumption at the end of the period which means that consumption is getting more wage-led.

The possibility of sucha phenomenon, though in contradiction with the a priori hypothesis ∂C/∂LS>0, has recently been discussed within the Post Keynesian framework by [START_REF] Stockhammer | Goodwin or Kalecki in Demand? Functional Income Distribution and Aggregate Demand in the Short Run[END_REF] and Kapeller and Schütz (2015). The former authors, in their empirical analysis based on the Goodwin and Kalecki models, name occurrence of a such effect as "perverse consumption differential" and observe it in theUK, Ireland, Australia and the USA. They state that their framework does not explain this phenomenon but add that missing variables might be the underlying reason. They suggest that e.g.

strong wealth effects in consumption might engender such a case. The latter authors develop the "perverse effects" on theoretical grounds drawing from the "conspicuous consumption" concept of Veblen. They show, in two different cases, that, a decrease in the labor"s income share might have different impacts within the labor class if the class is composed of heterogeneous workers. In their model, in which the labor class is composed of two types of workers, a decrease in the labor"s income share is only passed on one type of the workers (type 2), while the other"s share stays constant (type 1). In such a setting, the type 2 workers, facing a reduction in their wages, try to keep up with the consumption levels of the type 1 workers through increasing their consumptions with or without debt burden. Thus the relative consumption concerns of the type 2 workers might lead to two cases of consumption-driven profit-led regimes.

Some features of the income distribution in Thailand fit well with these arguments.Ourfindings might be a sign of such a segregation within the labor class for instance between those working in the formal economy which enjoy usually higher income and those working in the informal economy. More broadly, the income inequality is high with a Gini coefficient of income around 0.5 (Jitsuchon, 2014, p. 5) and households are highly indebted to maintain or improve their level of consumption in particular of durable goods such as housing and cars. For the richer households, a wealth effect of 0.02 to 0.03 was estimated for 2010 which was of the same magnitude as countries like the USA, Italy or Spain [START_REF] Rungcharoenkitkul | Wealth Effect and Consumption in Thailand, Bank of Thailand Discussion Paper[END_REF]. Thiscould explain the perverse effect in consumption once mixed income is included in the estimate of the labor"s share.These characteristics are possible candidates to explain the observed profitled pattern in consumption but further investigation of the personal distribution of income in Thailand is required to confirm their macroeconomic effects on demand.

Investment

The lnI, lnY, andlnR i series are I(1) according to the ADF tests. Since the cointegration tests did not give significant results, we estimated a difference equation instead,as ∆lnI= constant + e IY ∆lnY +e IR ∆lnR i where i is from 1 to 4. e IY and e IR represent the income and profit elasticities of investment, respectively. We estimatedthe equations with a dummy variable controlling for the Asian financial crisis of 1997 23 . This variable captures the sharp fall in investment after the 1997 crisis, which constitutes an outlier of the growth rate of the lnIseries. The results of the regressions for the four labor income shares are shown in Table 4.

Table 4: Investment Equation Estimation here

The results demonstrate that e IY lies in the range 2.12-2.40. However, e IR is insignificant for the four alternative measures of profits. The a priori hypothesis on the positive impact of profits on investment is not validated.This result isnot surprising when compared to those foundon developing countries in the literature. Onaran and Galanis (2014) find no significant impact of profit share onprivateinvestment in several developing countries like China, India, and Turkey. As for China, they note that Molero-Simarro (2015) and [START_REF] Wang | Three Essays on Monetary Policy and Economic Growth in China[END_REF] find a significant positive effect of profits on total investment. Itis mentioned that the objectives of the state and private enterprises are different, although an increase in profits would provide more funds for financing investment. It should also be noted that financialization, which is not taken into account in our model, plays an important role in the relation between profits and investment. [START_REF] Onaran | Financialisation, income distribution and aggregate demand in the USA[END_REF], in their article on the USA, finda positive significant effect of the former on the latter, after deducting the interest and dividends from the gross operating surplus.The insignificance of the profits in the investment functionis in line with the statement of Bhaduri and Marglin that "investment, particularly in long-lived fixed capital equipment, is likely to respond more cautiously to a change in profit margin/share compared to consumption " (1990, p. 385).

Exports

The ADF tests show that lnX, lnM w1 and lnM w2 series are I(1); however, lnP xrel turns out to be stationary. Since the variables are integrated of different orderswe appliedBounds test.

This test rejected cointegrationamongthe exports, the total world imports (of the two alternative measures) and the relative export prices series. Consequently, we estimated a difference equation instead of the form∆lnX= constant + β ∆lnM wj +κ∆lnPx rel ,where jis the index for the two measures of the world imports. βand κrepresent the world imports and the price elasticities of the exports, respectively. The results of the regressions are in Table 5.

Table 5:Exports Equation Estimations here

We obtain identical results with the two alternative measures of the world imports.

The coefficients have almost the same magnitudes and they are highly significant. The growth of the world demand, measured by a simple or weighted sum of the exports of Thailand, by 1% leads to 0.3 % growth in the exports of Thailand. The impact of international price competition is more substantial. A 1% increase of the relative prices of the exported products shrinks the exports by around 0.7 %.

Imports

The imports equation has similar characteristics with the exports in terms of econometric properties of the variables. lnM and lnYare I(1)but lnP/Pm is stationary. Pesaran"s (2001)Bounds test shows no sign of cointegration. We estimated the equation in difference form as ∆lnM= constant + ρ∆lnY +γ ∆lnP/P m where ρandγ are the income and (relative)

price elasticities of imports, respectively.The results of the regression is displayed in Table 6.

Table 6: Imports Equation Estimations here

Both coefficients are highly significant. A 1% increase in domestic demand raises imports by 1.67%.The imports are even more sensitive than the exports to the relative prices:an increase of P/P m by 1% tends to increasethe imports by around 0.9%. This high sensitivity to the price of imported goods can be explained by the share of import content of exports which has increased recently. Whereas the import content of export was 12% in 1995, it more than doubled reaching 30.1% in 2008 because of the growing importance of electronics and automobile components which are part of global value chains (Poonpatpibul et al. 2009, p 17-18). This is a strong constraint on a stimulus policy of domestic demand. However, thisdependence of a part of the imports to exports, is not taken into consideration by our model. Stockhammer et al. (2011) employ an income distribution-growth model, for Germany, into which the import content of the exports isintegrated. Whenever the exports decline,e.g. due to an increase of the export prices, imports of some goods, majorly the semi-finished products, are expected to decline accordingly. Integrating this dimension into the income distribution-growth models prevents the overestimation of the impact of the labor income share on net exports. 24

Domestic Prices

The ADF test indicates thatlnP series are non-stationary, but the two explanatory variables lnP m and the four measures of lnULC are stationary. We estimated a difference equation as ∆lnP= constant + η ∆lnULC i +μ ∆lnP m, wherei takes values from 1 to 4 according to the alternative measures of income distribution.ηand μ are the elasticities of unit labor costs and import prices of the domestic prices, respectively. We included two dummy variables in the regressions in order to control for the 1973 petrol crisis 25 and the 1997 Asian financial crisis, which had unprecedented inflationary and deflationary effects on the evolution of the domestic prices in Thailand, respectively.Addition of the dummy variables also enabled correction of heteroscedasticity that arose due to the two crises. We present the estimation results in Table 7.

Table 7: Domestic Prices Equation Estimations here

The estimations display the variability of the significance and magnitude of the coefficients depending on the unit labor cost measure, which depends on the type of adjustment. The coefficient ηis in the range 0.55-0.67, while ηis not significant with respect the first type of labor income adjustment and lies between 0.17-0.19 26 . The dummy variables reflect the impact of the drastic price increases and decreases, while in some specifications they are not significant at 5%. 27 Chow test suggests that there is a structural break at 1980.

Export Prices

As already stated, ADF test indicates thatlnP series are non-stationary; however, the lnP x and lnP x comp series are stationary. We estimated a difference equation as

∆lnP x = constant + λ∆lnP +τ ∆lnP x comp
, where λ andτ are the relevant elasticities. The estimations are reported in Table 8.

Table 8: Export Prices Equation Estimations here

Since the 1973 petrol crisis ceiled the export prices, we added the dummy variable, which is already used in the domestic price regressions. Addition of this variable not only solved the heteroscedasticity problem, but also being highly significant,it displays the inflationary impact of the petrol shock on the export prices. 28 . The results show that a 1 % increase of the domestic prices leads to a 0.31 % increment of the export prices while the effect of the foreign export prices on the latter is 0.41 %. These empirical finding suggest that Thailand is a price-taker country since τ is higher than λ. As in the domestic prices equation, a structural break is present at 1980.

In the next section we combine the theoretical model with the estimations of the coefficients obtained from the regressions in order to characterize the growth regime of Thailand.

Calculation of the Effect of Income Distribution on Growth

The overall effect on an increase of the labor income share on economic growth is by the multiplier in the equation 3, as mentioned in the section on the theoretical framework. We used the coefficients (elasticities) extracted from the regressions and the sample mean of the related ratios for the period considered in the calculations. The model implies that the sign of the private excess demand (h 2 ) characterizes the growth regime and h 1 determines its magnitude.

The first element of the h 1 is the marginal effect∂C/∂Y, which is equal to the multiplication ofe CY , the elasticity of C with respect to Y, and the sample average of the ratio of C to Y. e CY is not directly derived from the regressions. [START_REF] Stockhammer | Functional Income Distribution and Aggregate Demand in the Euro-area[END_REF] simply use the sum of e CR and e CW for e CY , since the consumption function is of type Cobb-Douglas; however, Onaran and Galanis (2014) calculate a weighted sume CR (R/Y) + e CW (W/Y), where R/Yand W/Y are sample averages of shares of capital and labor income, respectively. Since the weighted sum, which underestimates the income elasticity, is always lower than e CW , we opt for the former approach. 29 The second and the third elements ∂I/∂Y and ∂M/∂Y are equal to the multiplication of the elasticities that are directly obtained from the regressions and the sample averages of the relevant ratios.

As for the calculation of 30 In the same manner,∂M/Y/∂LS is calculated as [γη C /(1-η)](M/Y)/LS. 31 The effect of a 1% increase in the labor income share on the net exports is shown in Table9

h 2 , ∂I/Y /∂LS is equal to -e IR (I/R)(Y f /Y). ∂X/Y/∂LS is calculated by the formula[κλη/(1-η)](X/Y)/LS.
Table9:Total Effect of a 1% increase of the labor income share on demand for the full sample (1970-2011)

The results clearly indicate that the economy is profit-led with respect to four different measures of the labor income share. The effect of a 1% increase in the labor income share shrinks the GDP by around 2 % for most of the adjusted labor income shares. In the case of ALS 2 , the decline in GDP approaches to 3%. Our final results show that the total effect of a change in the distribution of income is the highest found until now by the literature. In the case of China, which is also an example of an export-led economy, Galanis and Onaran (2014, op cit) using what we called ALS 1 in our article found that an increase of 1% of the profit share has a total effect of 1.57% on growth, the highest total effect of their sample while Molero-Simarro(2015, op cit) found a small total effect of 0.011%.Our resultsare robust to various estimates of the labor income shareand reveals the strong dependency of Thailand on foreign markets due to its high export (45%) and import (52%) to GDP ratios 32 and its sensitivity to price competitiveness in the past decades.These characteristics of a small open economy combine to explain that the profit-led character of Thailand depends above all on the strong negative effect of a change in the labor income share on net exports (for instance -1.29, with respect to ALS 1 ) and in particular on imports

(1.07, again with respect to ALS 1 ) which reflects the limit of the development model of Thailand. Future research should explore further this topic and bring more insights to give a comprehensive explanation of the dependence of Thailand on imports.

A Sub Period Analysis: 1980-2011

As mentioned above, analyzing the sub-period 1980-2011 enables cross checking the results obtained for the entire sample. Due to space limitations, we skip the intermediate stages of the econometric investigation and present the results in Table10. As for the consumption function, we obtain similar results as in the analysis of the entire sample. However, globally, the negative effects are smaller in absolute value for ALS 2 to ALS 4 and the positive effects are higher for ALS 1 . This reflects the greater influence of wage labor in the economy and confirms that demand is becoming less profit-led.Still, we cannot claim a regime shift after 1980. The results of the regressions are almost identical for the investment functions. The trade equations, except fort the calculations based on ALS 1 , point that the export-led pattern is still present but weaker. The overall character of the growth regime persists to be profıt-led, however, a one percent increase of labor income share resultsin a 1 to 1.5 % decrease of the GDP at market prices, except for ALS 1 which gives an identical result as in the entire sample. Overall, the results validate our expectations of a weaker profit-led regime.

Conclusion and Policy Implications

This article explored the possibility to rebalance the Thai growth with a policy of domestic market stimulus based on an increase of the labor income share. Our investigation shows that,other things being equal, such a policy would lead to a short-term decreaseof the GDP of around 2% because the demand regime is profit-led: an increase in the labor share has a strong and negative impact on net exports which is not offset by a positive impact on consumption while it has no effect at all on investment. This result is in accordance with the conclusions of other researches on East Asian countries and more generally on small open economies [START_REF] Onaran | Is Aggregate Demand Wage-led or Profit-led? A Global Model[END_REF]. In our case, it can be explained by the features of the Thai economy, which combined a relatively small domestic market and a large export sector. The Thai industrial apparatus is geared towards exports, especially the industrial companies that are part of global value chains. For these companies, which include many subsidiaries of multinationals, wages are most of all viewed as a cost and not primarily as a source of domestic demand. A wage hike will not increase the sales of computers, cell phones or TV screens,by much in Thailand and therefore investment in these sectors.However, it will impact more significantly the cost of production and hence the competitiveness of Thai exports. This illustrates a point made by Lavoie and Stockhammer (2013, 19-20): the income distributional policy must be consistent with the economic regime of the economy. A single policy of labor share increase in a profit-led economic regime is bound to fail.This does not mean that the Thai growth regime can be maintained without any change. The results of the econometric analysis must be interpreted with caution as regards the policy implications. The profit-led character reflects the structure of the Thai economy and can be seen as a historical resume of the coevolution of the labor share and the output level of Thailand. The econometric results cannot be extrapolated by suggesting that a continuous decline of the labor"s share due to income suppression will have the same positive impact on net exports in the short or long run. Moreover, partial dependence of the imports to exports, which is not integrated into our model, overestimates the profit-led character of the economy. In a future research, this aspect should be taken into account.

The investment function deserves to be discussed in depth from a theoretical point of view. The model does not take into account the accumulation of capital, which is employed in a typical Kaleckian investment function, and its long-term effects. However, since our research is restrained to short-term demand growth, using such an investment function is not inconsistent. It might be argued that the series span a too long period for estimation of a short-term model. If monthly or quarterly data series were available, the equations could have been estimated for much shorter periods, in terms of years, since the number of observations would increase. Then the econometric estimations would have been much more consistent with the theoretical model. Another point to be addressed is financialization, especially issues concerning household debts and rentiers,who receive dividends and interest payments. These elements are also important for determination of consumption. Integration of rentier class in the model would be more complete for consumption and investment functions, as in the study by [START_REF] Onaran | Financialisation, income distribution and aggregate demand in the USA[END_REF]. However a problem arises: such data is hard to come by for developing countries like Thailand.Last but not least, the structural change is not captured by the investment function. The industrial policies led by the state, e.g. industry-specific credits, might have short and long term impacts on investment decisions. While not ignorable, these issues should be addressed and integrated in a more in-depth research within the post-Keynesian framework.

Despite these limitations, our article points to the need for change. The export-led growth pattern of Thailand may have been coherent with the global economy in the past, but the uncertain recovery of the 2008-09 international crisis, the emergence of new lowcost competitors of Thailand, and the slowdown of China"s growth show that the global economy is no more supportive of ever increasing export sales. In these new circumstances, an export-led growth economy like Thailand must consider rebalancing its growth because one cannot imagine that an ever decreasing labor income share will be the solution.

The real solution is to rebalancethe Thai economy away from export and towards the domestic market by implementing a broad set of economic and social policies that will change the overall structure of the economy. The objective is to shift from a profit-led to a wage-led economy based on the domestic market. However, our results show that a mere wage-policy is not sufficient to achieve this goal and should be accompanied by policies aiming structural change in favor of promoting domestic market.

competitors are from the PWT 8.0 database. P x comp is the weighted price index of the trade partners of Thailand, where the weights are the shares of exports of Thailand to the countries in its total exports. P x rel is simply the ratio of P x to P x comp .

M w : We used two alternatives for the world demand. , is used in the estimations.

Unit labor costs

Unit labor costs (ULC) is the ratio of the nominal compensation of workers to real output, i.e. real GDP at market prices.We obtain the real unit labor cost (RULC) if the labor compensation is also in real terms. The RULC and the LS are slightly different. While the former is the ratio of the real labor compensation to real GDP at market prices (Y), the latter is the share of the (real) labor compensation in (real) GDP at factor costs (Y f ).

Price deflator

The price deflator P used in the estimations is not the one calculated in local currency division by E, the exchange rate. Then it is normalized such that its value in 2005 is equal to 1. This deflator is then used to calculate the unit labor costs.

Derivation of Multipliers

The impact of LS on exports is found indirectly through combining the domestic price, export price and exports equations. At the first stage, replacing ULC by its definition P.LS. Y f /Y in the domestic price equation, we obtain the following reduced equation:

lnP= constant + η C /(1-η C ).lnLS + η/(1-η) Y f /Y + μ /(1-η) lnP m
Secondly, we replace this reduced equation in the export prices equation and obtain another reduced equation:

lnP x = constant +λη C /(1-η)lnLS+λη/(1-η) Y f /Y+ λμ/(1-η) lnP m + ….
Finally replacing this equation in the exports equation and taking the partial derivative of X with respect to LS gives us the multiplier formula [κλη /(1-η)]X/LS. Dividing this multiplier by Y we obtain ∂X/Y/∂LS. The imports multiplier ∂M/Y/ ∂LS is derived in a similar way. We report the highest VIF of the regressors of a given regression. Values higher than 10 point at multicollinearity. 22 These estimations are comparable with those of Onaran and Galanis (2014) who get elasticities in the range of 0.316 for Turkey to 0.845 for South Korea with a labor share calculated like ALS1. 23 This variable takes the value 1 for the years 1997 and 1998, 0 otherwise. We also added the year 1998 since the impact of the crisis persisted. 24 Since we do not have enough data for the import content of the exports of Thailand, we cannot address this issue in our current work. 25 This variable takes the value 1 for the years 1973 and 1974, 0 otherwise. We included the year 1974 since the effect of the petrol crisis remained. 26 ePULC is 0.35 (Turkey)-0.77 (China) for the developing countries in the estimations of Onaran and Galanis (2014) ., while ePPm is not significant for India, Mexico and China and for the others ranges between 0.12 (South Africa)-0.36 (Argentina). 27 We excluded D1973-1974 from the estimations with ULC2 since it was not significant. 28 We also tried a third specification by including the dummy variable for the Asian crisis, but this variable wasnot significant. We opt for the second estimation for our calculations 29 The classical Keynesian equation ∆lnC= constant + e CY ∆lnYgives an income elasticity of 0.88. Although this coefficient cannot be directly used in our model, it gives an idea about the magnitude of the coefficient. Our estimates lie between 0.72 and 0.81. 30 See the appendix on the derivation of multipliers. 31 Here it is implicitly assumed that domestic and imported goods are perfectly substitutable. 32 In constant 2005 dollars. Authors' calculations based on World Development Indicators, the World Bank.

Tables and Figures

  ALS 1 = [(Compensation of employees/Number of employees) * (Number of employees+ Own-account workers)] / GDP at factor cost In Thailand where it amounts to 33% in 2014, this adjustment leads to a strong bias because the vast majority of own-account workers are farmers or owners of small shops or street vendors in cities who usually earn less than the average wage. Most of them are non-registered or in other words are part of the informal economy. For these reasons, following Golin (2002) we calculated alternative ALS 8 : ALS 2 = (Compensation of employees + Mixed income of own-account workers) / GDP at factor cost ALS 3 = Compensation of employees / (GDP at factor cost -Mixed income) ALS 4 = Average of ALS 2 and ALS 3

Figure 1 :

 1 Figure 1: Alternative measures of the labor"s income share in Thailand, 1969 -

(

  2009) but carry it one step further. They conduct an empirical research on 16 of the G20 countries to analyze the interactions among countries calculating global multipliers. Their model enables the examination of the impact of a simultaneous change of the labor income share at a global level on individual countries.For LDCs, their results show that they are profit-led contrary to most of rich countries which are wage-led with the exception of Australia and Canada.The profit-led countries have in common to be relatively small economies and thus very open to foreign trade. China is a special case. Although it is the second largest economy of the worldit is "… very strongly profit-led due to strong effect on exports and imports" (2012, op cit p 25). They conclude that the global economy is wage-led and in case of a worldwide race to the bottom in the labor share the global GDP contracts. Australia, China and South Africa are the only countries which maintain a positive but smaller growth. More recently, Molero-Simarro (2015) applies the Stockhammer et al. (2009) model on China for the period 1978-2007 and finds a negative

  After applying the relevant estimation technique, we check the robustness of the estimated equations by using the standard tests on residuals and coefficients. As for the residuals, we report DW test statistic and the probability values for White"s heteroscedasticity test, the Autoregressive Conditional Heteroscedasticity (ARCH) test with one lag and the Doornik-Hansen normality test 19 . For the overall significance of the model and the parameters, probability values of the F-test, Chow 20 , RESET test and CUSUM tests are presented. In order to check for the multicollinearity among the regressors, we report the VIF (Variance Inflation Factor) 21 .

(

  bahts) but its conversion into constant 2005 dollars. It is equivalent to multiplication of the domestic price deflator by PPP 2005 ,the Purchasing Power Index for 2005 and its

Figure 1 :

 1 Figure 1: Alternative measures of the Labor Share of Income in Thailand, 1969 -2013

Table 1 :Comparison of our model with that of Stockhammer et al. (2009) here

 1 We estimated our model over the period 1970-2011 and the sub-period 1980-2011.Detailed information on the variables and the data sources is in the appendice. The

motivation behind using a subsample after 1980s is threefold. The first reason is the impact of what

[START_REF] Dutt | The Role of Keynesian Policies in Semi-industrialized Countries: theory and evidence from India[END_REF]

calls the agriculture constraint. Less developed countries (LDCs) are constrained by a low-productivity agriculture, whichdespite being overstaffed finds it hard to supply enough food for the population.According to Dutt, the more the agriculture constraint is relaxed and the country industrializes, the more Keynesian policies become relevant for LDCs. In Thailand, employment in agriculture represented around 80% of total employmentin 1970, around 70% in 1980 and was down to 40% in 2011

13

. Among those working in agriculture, unpaid family helpers amounted to around 50% of employment in agriculture in 1970, 42% in 1980, and was down to 17% in 2011 as many have left for jobs in industry or services which now accounts respectively for 19% and 40% of total employment. This tremendous process of structural change implies strong gains of labor productivity which was multiplied by 2.7 over the period 1970-2013 in agriculture, almost as much as in industry (2.8) and much more than in services (1.6) 14 .

Table 3 : Change in Consumption due to a 1% increase of the Labor Income Share here

 3

Table 10 : Total Effect of a 1% increase of the labor income share on demandfor the subsample (1980-2011)

 10

  The first one, denoted as𝑀 𝑤1 = 𝛼 𝑖 𝑀 𝑖 𝑁 𝑖, where 𝛼 𝑖 is the share of exports of Thailand towards the country i, i.e. 𝛼 𝑖 =

	𝑋 𝑖 𝑋 𝑖 𝑁 𝑖	, and 𝑀 𝑖 is the total imports of the country i. Alternatively, a simple sum of the
	imports of the rest of the world, denoted as 𝑀 𝑤2 =	𝑁 𝑖	𝑀 𝑖

Table 2 : Consumption

 2

	Dependent variable: ∆lnC	i=1		i=2		i=3
	Regressors	Coefficient	t-ratio Coefficient	t-ratio	Coefficient	t-ratio
	Constant	0	0.69	0.01	1.11	0.01	1.19
	∆lnW i	0.65***	9.05	0.55***	6.66	0.49***	6.27
	∆lnR i	0.16***	7.92	0.17***	5.96	0.24***	6.85
	Diagnostics
	R 2	0.76		0.69		0.69
	Adjusted R 2	0.74		0.67		0.67
	DW statistic	1.87		1.76		1.75
	Heteroscedasticity test	0.72		0.05		0.71
	ARCH test (1 lag)	0.94		0.17		0.22
	Normality test	0.61		0.52		0.89
	F-test	0.00		0.00		0.00
	Chow test (Break at 1980)	0.05		0.22		0.27
	RESET test (with squares)	0.22		0.07		0.05
	CUSUM test	0.82		0.89		0.99
	Highest VIF	1.05		1.00		1.00
	Notes: ***, ** and * represent 1, 5 and 10 % of significance levels, respectively.

Table 3 : Change in Consumption due to a 1% increase in the Labor Income Share

 3

		ALS1 ALS2 ALS3 ALS4
	1970 -0.03 -1.23 -0.22 -0.51
	Mean	0.24 -0.42 -0.07 -0.20
	2011	0.22 -0.02	0.06	0.01

Table 4 : Investment Equation

 4

	Dependent variable:∆lnI	i=1		i=2		i=3
	Regressors	Coefficient	t-ratio	Coefficient	t-ratio	Coefficient	t-ratio
	Constant	-0.07	-4.45	-0.07	-4.47	-0.07***	-4.31
	∆lnY	2.40***	9.44	2.12**	8.32	2.14***	7.89
	∆lnRi	-0.07	-1.23	0.12	1.54	0.11	1.01
	D1997-1998	-0.20***	-4.56	-0.18***	-3.89	-0.18***	-3.90
	ρ a
	Diagnostics
	R 2	0.89		0.89		0.90
	Adjusted R 2	0.88		0.88		0.88
	DW statistic	1.72		1.72		1.75
	Heteroscedasticity test b	0.21		0.51		0.02
	ARCH test b (1 lag)	0.51		0.25		0.29
	Normality test b	0.28		0.07		0.10
	F-test b	0.00		0.00		0.00
	Chow test b (Break at 1980)	0.53		0.43		0.46
	RESET test b (with squares)	0.13		0.12		0.11
	CUSUM test b	0.09		0.07		0.08
	Highest VIF	2.08		2.13		2.34

Notes: ***, ** and * represent 1, 5 and 10 % of significance levels, respectively. a. ρ is the parameter of PW method.

b. Probability values of the corresponding tests

Table 5 : Exports Equation

 5

	Dependent variable: ∆lnX	j=1		j=2
	Regressors	Coefficient t-ratio Coefficient t-ratio
	Constant	0.06***	4.41	0.06***	3.35
	∆lnMw j	0.30***	2.9	0.30***	3.15
	∆lnP x rel	-0.67***	-2.93	-0.66***	-3.28
	AR2			0.35**	2.26
	Diagnostics
	R 2	0.26		0.32
	Adjusted R 2	0.23		0.28
	DW statistic	1.91		1.80
	Heteroscedasticity test a	0.38
	ARCH test a (1 lag)	0.45		0.25
	Normality test a	0.28		0.77
	F-test a	0.00		0.00
	Chow test a (Break at 1980)	0.12
	RESET test a (with squares)	0.07
	CUSUM test a	0.96
	Highest VIF	1.06		1.04

Notes: ***, ** and * represent 1, 5 and 10 % of significance levels, respectively. a. Probability values of the corresponding tests

Table 6 : Imports Equation

 6 **, ** and * represent 1, 5 and 10 % of significance levels, respectively. a. Probability values of the corresponding tests

	Dependent variable: ∆lnM
	Regressors	Coefficient t-ratio
	Constant	-0.01	-0.58
	∆lnY	1.67***	4.51
	∆lnP/P m	0.91***	3.91
	Diagnostics
	R 2	0.63
	Adjusted R 2	0.61
	DW statistic	2.25
	Heteroscedasticity test a	0.62
	ARCH test a (1 lag)	0.89
	Normality test a	0.59
	F-test a	0.00
	Chow test a (Break at 1980)	0.94
	RESET test a (with squares)	0.23
	CUSUM test a	0.58
	Highest VIF	1.25
	Notes: *

Table 7 : Domestic Prices Equation

 7 : ***, ** and * represent 1, 5 and 10 % of significance levels, respectively. a. ρ is the parameter of PW method.

	Dependent variable:∆lnP	i=1		i=2		i=3		i=4
	Regressors	Coefficient t-ratio Coefficient t-ratio Coefficient t-ratio Coefficient t-ratio
	Constant	0.01	1.58	0	1.3	0.01*	1.76	0.01	1.59
	∆lnULC i	0.63***	6.13	0.67***	10.6	0.55***	7.77	0.60***	9.03
	∆lnP m	0.11	1.12	0.17**	3.07	0.19***	2.3	0.18**	2.31
	D1973-1974	0.09***	3.42			0.06**	2.13	0.05*	1.91
	D1997-1998	-0.09***	-3.16	-0.07***	-2.77	-0.11***	-4.01	-0.09***	-3.48
	ρ a			0.30		0.23		0.30
	Diagnostics
	R 2	0.84		0.91		0.88		0.90
	Adjusted R 2	0.83		0.90		0.87		0.89
	DW statistic	2.11		1.80		1.95		1.93
	Heteroscedasticity test b	0.38
	ARCH test b (1 lag)	0.70		0.78		0.35		0.57
	Normality test b	0.20		0.71		0.78		0.77
	F-test b	0.00		0.00		0.00		0.00
	Chow test b (Break at 1980)	0.03
	RESET test b (with squares)	0.00
	CUSUM test b	0.86
	Highest VIF	2.09		2.04		1.75		1.91

Notesb. Probability values of the corresponding tests

Table 8 :

 8 Export Prices

	Dependent variable: ∆lnP x
	Regressors	Coefficient	t-ratio Coefficient	t-ratio	Coefficient
	Constant	0.01	0.65	0.01	1.06	0.00
	∆lnP	0.42***	3.32	0.31***	2.89	0.42***
	∆lnP x comp	0.51***	3.16	0.41***	3.1	0.39***
	D1973-1974			0.13***	4.39	0.12***
	D1997-1998					0.05
	Diagnostics
	R 2	0.65		0.77		0.78
	Adjusted R 2	0.63		0.75		0.75
	DW statistic	1.74		1.64		1.72
	Heteroscedasticity test a	0.03		0.71		0.74
	ARCH test a (1 lag)	0.98		0.72		0.56
	Normality test a	0.05		0.29		0.39
	F-test a	0.00		0.00		0.00
	Chow test a (Break at 1980)	0.00		0.00		0.00
	RESET test a (with squares)	0.00		0.02		0.03
	CUSUM test a	0.40		0.10		0.08
	Highest VIF	1.97		2.09		3.67

Notes: ***, ** and * represent 1, 5 and 10 % of significance levels, respectively. a. Probability values of the corresponding test 21

Appendix

Data sources

All data used in the article are annual over the 1970-2011 period. The empirical calculations will be based on this version of the private excess demand.

10 Our empirical findings in the next section show that this is not the case for all but one measures of the labor"s share that we calculated.

11 Thirlwall (2002, chap 4) employs a similar exports function. He uses domestic prices foreign prices ratio instead of relative export prices. [START_REF] Ederer | Competition-oriented Wage Policy and Its Effects on Aggregate Demand in the Netherlands[END_REF] employs the same exports function as ours, however, export prices are a function of domestic and imports prices in his model, while in our model the export prices depend on along with domestic prices, (weighted) export prices of competitor countries instead of import prices.

12 Razmi (2005) also uses a weighted sum of world imports instead of world income in the estimation of the exports equation in order to test the Balance of Payments Constrained Growth Model for India.

13 Authors' estimations with data for employment by industry taken from the National Statistical Office of Thailand. We use data from the third quarter of each year for consistency over the period and to avoid seasonal fluctuations. 14 Labor productivity is calculated by the authors as value added in the relevant sector, in constant US$ 2005(source: World Development Indicators, 2014) divided by data for employment by industry published by the National Statistical Office of Thailand, third quarter. 15 Source: World Development Indicators. 16 In their study, LDCs are Argentina, China, India, Korea, Mexico, South Africa and Turkey and for these countries the period covered is 1970-2007. 17 The results of the ADF, EG and which are not presented due to space limitations, are available on request. 18 . Three outcomes are possible due to the nature of the test. The series might be cointegrated, they might be all stationary or no conclusion can be drawn from the test. 19 Although normality is not a must for the residuals we report the probability values of the Doornik-Hansen test. Gelman and Hill (2007) underline that '[t]he regression assumption that is generally least important is that the errors are normally distributed. In fact, for the purpose of estimating the regression line (as compared to predicting individual data points), the assumption of normality is barely important at all.'

20 Since we hypothesize a structural break starting from 1980s we applied the Chow test to that year. The same test is also conducted for the mid-sample date 1990, however, no signal of a break was detected.