Maria Bas
email: bas@pse.ens.fr

Ivan Ledezma
email: ledezma@pse.ens.fr

Hugo Hopenhayn

Philippe Martin

Thierry Mayer

James R Tybout

Trade liberalization and heterogeneous within-firm productivity improvements

Keywords: trade liberalization, firm heterogeneity, endogenous productivity gains, extensive margin of trade trade liberalization, …rm heterogeneity, endogenous productivity gains, extensive margin of trade. JEL Classi…cation: F10, F12 and F41

This paper develops an intra-industry model of trade with heterogeneous …rms to investigate the impact of trade on the evolution of within …rm productivity. The main contribution is to incorporate endogenous labor productivity gains. Heterogeneous …rms have di¤erent incentives to invest in foreign technology which in turns enhances e¢ ciency heterogeneously. Trade liberalization reduces the price of imported capital equipment and increases factor demands. These mechanisms introduce two novel results. First, aggregate productivity increases due to within-…rm productivity improvements. Second, tari¤s reduction has little impact on the extensive margin of trade in countries already highly open.

Introduction

Empirical works at the …rm level have shown that trade policy shapes the evolution of …rm productivity. This "within-…rm" e¤ect of trade, however, can not be reproduced by the wave of the recent heterogeneous …rms'models. In these models, trade induces aggregate productivity improvements as a consequence of the exit of the least productive …rms and the reallocation of market shares towards the most productive ones. This "between-…rm" e¤ect of trade is in line with stylized facts provided by …rm-level data. Nonetheless, the understanding of the e¤ect of trade on productivity calls for further analysis on the mechanisms explaining …rm's productivity improvements.

This paper investigates the relationship between trade liberalization, technology investment and productivity gains at the …rm level. We focus on developing economies, usually characterized as highly dependent on foreign technology. We propose an extension of [START_REF] Melitz | The impact of Trade on Intra-Industry Reallocations and Agregate Industry Productivity[END_REF] incorporating endogenous labor productivity gains determined by an initial investment in technologies embodied in imported capital goods. While we assume an exogenous initial distribution of productivity levels, we allow for further modi…cations of the initial level as a consequence of …rms' decisions. The main contributions of this paper to the existing literature can be resumed as follows. Firstly, the model allows for endogenous productivity gains. Thereby, it explains how trade policy shapes the incentives of …rms to undertake investments to enhance their productivity gains, a topic that has received fewer attention in the existing theoretical literature of trade with heterogeneous …rms. Secondly, the model sheds some new light on the impact of trade liberalization on the intensive (volume of exports) and the extensive margin of trade (number of new exporters). Actually, …rms' productivity improvements are themselves heterogeneous.

Initial productivity matters and self-selection mechanisms are emphasized. The result is that gains of trade are concentrated on a reduced number of exporters.

Several empirical works have studied the impact of trade integration on industry productivity evolution. One of their contributions is to understand whether trade liberalization in ‡uences aggregate productivity and by which mechanisms. By decomposing aggregate productivity (see [START_REF] Olley | The dynamics of productivity in the telecommunication equipment industry[END_REF][START_REF] Foster | Aggregate Productivity Growth : Lessons from Microeconomic Evidence[END_REF], these studies have stressed three main channels : [START_REF] Aw | Productivity and Turnover in the Export market: Micro Evidence from taiwan and south Korea[END_REF] Resources reallocation towards most productive …rms (between-…rm channel), (2) Net entry and (3) Improvements of …rms' e¢ ciency (within-…rm channel). In general, these works use plant panel data to carry out study cases of countries which have experienced trade reforms. Results vary across countries and industries. [START_REF] Pavcnik | Trade liberalization, exit and productivity improvement: evidence from chilean plants[END_REF] investigates the impact of trade liberalization on …rm productivity in Chile (1979Chile ( -1986)). In her aggregate productivity decomposition, a major role is played by the reallocation process of inputs and production towards most productive …rms. [START_REF] Bernard | Who dies? International Trade, market structure and industrial restructuring[END_REF] estimate the determinants of aggregate productivity at the industry level in the US (1983)(1984)(1985)(1986)(1987)(1988)(1989)(1990)(1991)(1992) and …nd similar results. In both studies, productivity improvements at the industry level are mainly explained by the between-…rm channel.

On the other hand, other empirical works highlight the explanatory power of within-…rm productivity improvements and the net-entry of more productive …rms: Aw, Chung and Roberts (2000) on Taiwan (1986)(1987)(1988)(1989)(1990)(1991); [START_REF] ‡er | The Long and Short of the Canada-U.S. Free Trade Agreement[END_REF] on Canada, De [START_REF] Loecker | Creative Destruction and Productivity Growth in an Emerging Economy : Evidence from Slovenian Manufacturing[END_REF] on Slovenia (1994)(1995)(1996)(1997)(1998)(1999)(2000), [START_REF] Bergoeing | Market reforms and e…ciency gains in chile[END_REF] on Chile (1979Chile ( -2001) ) and [START_REF] Bas | Market access and the Evolution of within Plant Productivity in Chile[END_REF] on Chile . These studies con…rm that …rm productivity evolves over the time and that this evolution is a key factor to explain aggregate levels.

Interestingly, concerning recent empirical works on Chile, one notes that the sample periods are larger than the one used by [START_REF] Pavcnik | Trade liberalization, exit and productivity improvement: evidence from chilean plants[END_REF]. Using di¤erent estimates of …rm productivity, both [START_REF] Bergoeing | Market reforms and e…ciency gains in chile[END_REF] and [START_REF] Bas | Market access and the Evolution of within Plant Productivity in Chile[END_REF] …nd that between 1979 and 1986 aggregate productivity is mostly explained by the reallocation process. Nevertheless, these studies show that the evolution of within-…rm productivity plays an important role from 1986 to 1998, the same period in which Chile experienced stable macroeconomic growth. This result is illustrated by Figure 1.a. It depicts the ratio of the weighted average labor productivity to the simple average (unweighted). While the former re ‡ects average productivity gains arising from reallocation of resources, the latter relates to within-…rm productivity improvements. In Figure 1.b., we overlap the histogram of plant productivity of 1987 with the one of 1995, a period of within-…rm productivity gains. First, we observe that distribution remains highly asymmetric. Since we only observe those …rms that remain in the market, the reduction in the percentage at the lowest productivity levels re ‡ects the exit of least productive ones (both histograms start at zero). Second, it is important to note that productivity improvements concern a much reduced number of plants (histogram for 1995 in gray). Thus, productivity improvements at the …rm level are heterogeneous and concern just a few …rms. Labor productivity evolution (weighted/unweighted); 1979=100 A key assumption of these models is that productivity at the …rm level is exogenous.

Therefore, they are not able to explain the evolution of industry productivity related to within-…rm channels. Firm productivity is exogenously determined and it remains unchanged. This is a key issue motivating this paper. While several theoretical works have explained why only the most productive producers can export and how trade induces a market share reallocation process, the determinants of these productivity di¤erentials have not received enough attention. We contribute to this issue by introducing an endogenous mechanism of productivity divergence across …rms which is reinforced by trade liberalization.

In that sense our model is related to [START_REF] Yeaple | A Simple Model of Firm Heterogeneity, International Trade, and Wages[END_REF] Hence, both "between" and "within" mechanisms are present in this model to explain how the trade frictions reduction contributes positively to aggregate productivity. The between-…rm e¤ect works through the standard selection channel in the domestic market. The novel within-…rm channel is interesting because productivity improvements are heterogenous across …rms. Indeed, the model predicts that only a small fraction of …rms will become more e¢ cient after a reduction of trade costs. Hence, the initial distribution of productivity is modi…ed. Initially high-productive …rms capture most of technological productivity improvements. This result explains why there is a minor change in the extensive margin of trade since gains from trade are concentrated in the most productive …rms, those that more likely already export.

The rest of the paper is organized as follows. In Section 2 the set-up of the model is presented. Section 3 develops the main results. Section 4 concludes.

2 The model

Households consumption

There are two countries: home h and foreign f: They represent two small open economies.

Households allocate consumption between the set of available domestic varieties h and the imported one x f . Both sets are endogenously determined by the entry and the exit of …rms. Whether domestic varieties are exported or not depends on their pro…tability.

Consumers'preferences across varieties are given by a standard CES utility function. All variables with an circle ( ) represent the foreign market and all round brackets () are reserved to specify the arguments of functions. Let C h be the aggregate CES index in the home country. Domestic preferences are then summarized by:

C h = " Z i2 h d h (i) 1 di + Z i 0 2 x f d f (i 0 ) 1 di 0 # 1 (1)
Where d h (i) and d f (i 0 ) are the consumption of home and foreign varieties, respectively.

The elasticity of substitution, > 1; is the same in both countries. Denoting p x (i) the price of variety (i) produced in country x and P h the aggregate price in the home country, this formulation implies the following optimal inverse demand functions:

d h (i) = P h p h (i) C h (2) d f (i 0 ) = " P h p f (i 0 ) [1 + ] e # C h (3)
Where e is the nominal exchange rate quoted in the home currency relative to the foreign one and the variable trade costs, modeled as "iceberg costs". Consequently, the CES price index is given by:

P h = " Z i2 h p h (i) 1 di + Z i 0 2 x f p f (i 0 ) [1 + ] e 1 di 0 # 1 1 (4)

Producers

Each …rm i faces the following schedule. First it enters the market. To do so, before knowing its initial productivity level, h (i) ; the …rm has to pay a start-up cost f e . After paying this sunk entry cost, h (i) is revealed from a common distribution density g( h );

with support [0; 1] and cumulative distribution G( h ). Second, once the …rm knows its initial productivity level, but still before production, it decides its investment in technology embodied in imported capital goods. This technology choice allows it to improve its initial level of productivity. Finally, due to the presence of a …xed production cost f paid at every period, the …rm decides whether it stays or exits the market. It might be the case that, even if a …rm invests to enhance its e¢ ciency, the expected pro…tability is not high enough to produce. In that case, the …rm exits the market. Otherwise it stays and produces with an improved productivity level until economic conditions change and its revenues become insu¢ cient to pay the permanent …xed cost.

Firms enter the market if their expected value allows, at least, to pay the entry cost.

Thus, they anticipate their expected discounted of pro…ts. This also includes the expected gains from investment. We then analyze each step of the schedule and solve the model in backward induction.

Technology

Producers compete within a monopolistic competition framework. When a …rm is active in the market, it produces a speci…c variety by using labor with constant returns to scale.

Focusing on the home country, the production function Y h (i) of producer i is given by:

Y h (i) = A h (i) l h (i) (5)
Labor productivity A h (i) depends on an initial investment in foreign technology I h0 (i).

This investment has a di¤erent impact depending on the initial productivity level of each …rm h (i).

A h (i) = h (i) [I h0 (i)] (6)
Where 0 < < 1. The technology choice of …rm i is made once it knows its initial productivity level h (i) and before it starts producing. This initial investment represents a speci…c …xed technological cost incurred by the …rm in order to improve its e¢ ciency when it enters the market. If it decides to stay, the …rm produces using only labor with constant returns to scale, but with an improved productivity level.

Thus, the …rm's investment I h0 is a decision, which is endogenously determined in the model. Since this decision depends on the heterogeneity h , the level of investment will be heterogeneous across …rms. Firms endowed with a higher initial productivity level, will be able to reinforce even more their e¢ ciency through an "investment channel".

As we focus on developing economies, we think in technology as embodied in imported capital goods. The elasticity ; which is homogeneous across …rms, captures the extent to which labor productivity reacts to this type of technology in the industry. If = 0 one …nds a Melitz-type model. If > 0 further decisions take place and, as we will see, the initial distribution of productivity is modi…ed. In this sense, the model seeks to understand the evolution of …rm productivity in capital intensive industries.

One key assumption is that capital goods are supplied to both countries by the rest of the world in perfect competition. Implicitly, the model considers two small economies and a third country that represents the rest of the world. Since the investment is paid with pro…ts, the trade balance condition between the two small economies does not take into account imported capital goods. Thus, we solve the model for the partial equilibrium between these two small economies.

Price setting and pro…ts

As we mentioned, during production, …rms behave as if they have constant returns on labor with a given level of productivity. Hence, …rst order conditions imply that …rm set prices as a mark-up over marginal costs (wages over labor productivity):

p h (i) = [ 1] W h A h (i) (7)
Where W h is the wage rate in the home country. The price of home goods sold in the foreign markets p h is higher due to variable trade costs (represented by ).

p h (i) = [ 1] W h A h (i) [1 + ] e (8)
Assumption 1: Countries h and f are symmetric.

We can now simplify our notation. Assumption 1 ensures equal wage rates (normalized to 1: W h = W f = 1) and equal aggregate prices (P h = eP f = P ). Then, hereafter we drop country subscripts. Since heterogeneity is totally captured by ; we also drop …rm subscripts and identify …rms by .

Firms' revenues can be divided into those earned from domestic sales r d and those earned from export sales r x . Using ( 7) and ( 6) the former can be written as:

r d (I 0 ) = P p (I 0 ) 1 R ( 9
)
Where R is the aggregate revenue of the country (R = P C). We write revenues as a function of the initial investment in order to highlight the decision schedule. Using [START_REF] Clerides | Is learning by exporting important? microevidence from Colombia, Mexico and Morocco[END_REF], export revenues are given by:

r x (I 0 ) = r d (I 0 ) [1 + ] 1 (10)
Total revenue r (I 0 ) of a …rm with initial productivity depends on its export status:

r (I 0 ) = r d (I 0 ) if the …rm does not export r (I 0 ) = r d (I 0 ) + r x (I 0 ) = r d (I 0 ) h 1 + (1 + ) 1 i if the …rm exports
Similarly, pro…ts can also be divided into domestic pro…ts d (I 0 ) and export pro…ts

x (I 0 ):

d (I 0 ) = r d (I 0 ) ( ) I 0 f (11)
Where f represents …xed production costs that are paid in every period. We assume that technology investment is paid with pro…ts to the country supplying capital goods (the rest of the world).

( ) I 0 is the amortized investment.

( ) ( 1
+ ) p k e w is
the …nal price of imported capital in home currency. e w is the nominal exchange rate between any of the two symmetric small economies and the rest of the world. The price of imported capital goods p k is taken as given since both economies are supposed to be small enough to have any impact on world prices. Export pro…ts x (I 0 ) = rx(I 0 ) f x are then:

x (I 0 ) = r d (I 0 ) (1 + ) 1 f x (12)
Firms having a higher productivity, which depends on the initial technology investment, will charge a lower price, have a higher demand and earn higher pro…ts than less productive ones.

Technology choice: Initial investment

Using backward induction, …rms set optimal prices taking I 0 as given (equations ( 7) and ( 8) ) and decide the level of I 0 that maximizes the present value v of their domestic pro…ts:

Using as the time discounting factor.1

v (I 0 ) = 1 r d (I 0 ) I 0 ( ) f (13)
The …rst order condition implies an optimal investment which depends on the initial productivity:

I 0 ( ) = [P ] 1 ( R ( ) 1 ) 1 (14)
Where 1 ( 1). We assume that 1 ( 1) > in order to ensure non-explosive returns of investment. The power 1 gives the concavity of the e¤ect of on …rm's investment. This term comes from the fact that, when maximizing the discounted value of pro…ts ‡ows, the e¤ect of demand is taken into account twice. First, when setting their price, …rms know that their demand is a decreasing function of their price relative to the aggregate one. This leads to the mark-up price rule. Second, when entering the market they also know that their demand can be enhanced by decreasing their marginal costs through the investment channel. As we will see this anticipation mechanism implies that the e¤ect of the initial heterogeneity is not linear.

Thresholds of production and export status

Since there is a …xed production cost (paid in units of labor), there exists a marginal …rm, ; whose domestic pro…ts are equal to zero: d ( ) = 0: This is equivalent to state:

r I d ( ) = f (15)
Where r I d ( ) r d ( ) I 0 ( ) ( ) are domestic revenues net of amortization of initial investment in technology. The value is the production cuto¤. It de…nes the threshold corresponding to the minimum level of productivity that allows to produce. Some …rms decide to exit the market because, even after investment, they are not pro…table enough to pay the …xed production costs.

Similarly, the tractability condition implies that only those …rms with operating pro…ts that counterweight the …xed export costs f x , also paid in units of labor, will be able to export. Again, this de…nes a marginal …rm, x , whose export pro…ts are zero:

x ( x ) = 0 r d ( x ) (1 + ) 1 = f x (16)
From this condition we can derive the export cuto¤ x : the threshold corresponding to the minimum level of productivity which ensures just enough revenues to pay f x .

Aggregation

After applying trade balance condition for symmetric countries, the index price over the support of leads to:

P 1 = Z 1 N p ( ) 1 g( ) [1 G( )] d (17)
+ Z 1 x N x [p ( ) [1 + ]] 1 g( ) [1 G( x )] d
From the left to the right, the integrals represent domestic and imported varieties, respectively. The assumption of symmetric countries implies that the characteristics of imported varieties are identical to those of exported ones. Thus, the number of exporters in both countries is the same N hx = N f x = N x . The total number of varieties available for consumption in a country is then

N T = N + N x : It is composed of N domestic varieties,
including exported and non-exported goods, and N x imported varieties.

The price index P takes into account that prices are a function of the random variable : Consequently, the domestic component considers the distribution of conditional on having entered the market g( ) 1 G( ) ; and the import component the one conditional on having the export status g( ) 1 G( x ) : Trade balance accounting concerns two components: consumption goods and capital goods. The former are considered in the standard export-import balance accounting between the two symmetric countries and the latter in the amortization of the initial investment in capital, imported from the rest of the world and paid by revenues coming from sales. Since both components are supplied and paid independently their accounting can also be done independently. Hence, the index price takes only into account consumption goods. Using the assumption of symmetry and the standard results of CES demand formulation, one obtains directly the aboved-presented expression.

Plugging the optimal prices set by the …rm into the price index we obtain:

P = 1 h R 1 N A( ) 1 g( ) [1 G( )] d + (1 + ) 1 R 1 x NxA( ) 1 g( ) [1 G( x )] d i 1 1 De…ning f A d 1 R 1 A( ) 1 g( ) [1 G( )] g( )d and f A x 1 R 1 x A( ) 1 g( ) [1 G( x )
] d we can express the average productivity and the price index as

f A T 1 = 1 N T h N f A d 1 + N x (1 + ) 1 f A x 1 i (18)
P = N 1 1 T 1 1 f A T (19)
This is the Melitz's (2003) aggregate price summarized by the average productivity f A T : In our framework, productivity is determined by the optimal technology choice [START_REF] Kraay | Exports and economic performance: evidence from a panel of Chinese …rms[END_REF], which depends on the index price. This leads to an externality of investment.

Thanks to investment, the economy becomes more productive and the aggregate price P is reduced. As in this Dixit-Stiglitz framework each …rm competes with the whole economy, the average productivity improvement induces …rms to invest more in order to set a competitive price. However, …rms do not take into account that their behavior determines the average productivity of the economy.

After plugging ( 19) into ( 14) and the result into ( 6) and ( 18); we obtain:

f A T = ( ; ) N T b (20)
Where the following de…nitions apply:

( ; ) R ( ) 1 b 1 1 N T N b 1 d + [1 + ] 1 N x b 1 x (21) b 1 d 1 [1 G( )] Z 1 1 g( )d b 1 x 1 [1 G( x )] Z 1 x 1 g( )d
The average de…ned by b aggregates heterogeneity after taking into account optimal decisions of investment. Similarly, b d and b x are the domestic and export counterparts of b . This average gives a measure of the reaction of labor productivity in the industry to the technology choice.

Proposition 1: Firm's investment is a function of its (exogenous) initial heterogeneity relative to the (endogenous) aggregate of the industry (b ) : The investment function is given by:

I ( ) = ( ; ) N T h b i 1 (22)
Proof: Plugging the global average productivity (20) into the optimal investment [START_REF] Kraay | Exports and economic performance: evidence from a panel of Chinese …rms[END_REF] gives [START_REF] Tybout | Plant-and Firm-level Evidence on the 'New'Trade Theories[END_REF].

Equation [START_REF] Tybout | Plant-and Firm-level Evidence on the 'New'Trade Theories[END_REF] gives further insights concerning the above-mentioned externality of investment. Since b d and b x are endogenously determined by productivity cuto¤s, b is modi…ed by …rms'decisions. Producers drawing a high initial productivity level will bias the initial since they will concentrate most of investment gains. This occurs because …rms anticipate the impact of their investment decisions on their demand. The decision of high productive …rms will deter the least productive ones to undertake a large amount of investment because they compete with the average …rm which has become more productive. As a consequence, …rms'decisions are particularly sensitive to the expected relative advantages. As the average productivity gains are reinforced, the e¤ectiveness of investment is reduced for …rms with a low . This induces the exit of the least productive …rms and as a consequence an increase in b :

Proposition 2: Global productivity f A T can be summarized as the productivity of a representative …rm which its initial productivity can be improved through the "investment channel" by a factor of b : f A T = A(b ): More generally, a …rm with an initial level of heterogeneity will obtain after investment decision a productivity level of :

A ( ) = I ( ) = ( ) N T h b i 1 (23)
Proof: Substitution of ( 22) into …rm productivity (6) leads to equation [START_REF] Yeaple | A Simple Model of Firm Heterogeneity, International Trade, and Wages[END_REF]. Evaluating ( 23) for = b gives (20)

The optimal investment de…nes pro…ts and revenues as functions of the exogenous initial productivity : Plugging ( 22) into (9) gives:

r d ( ) = 1 N T h b i 1 R (24)
Export revenues and pro…ts are pin down by r d ( ) (see equations ( 10) and ( 12)). The

Where the probability of exporting, x = 1 G( x ) 1 G( ) is the probability of having an higher than the export cuto¤ x ; conditional on having entered the market. Applying [START_REF] Tybout | Plant-and Firm-level Evidence on the 'New'Trade Theories[END_REF], domestic revenues net of amortized investment r I d ( ) = r d ( ) I 0 ( ) can be expressed as:

r I d ( ) = r d ( ) (27

Macro Balance

In this subsection we analyze global conditions of stability and macroeconomic balance to close the model.

Entry-Exit

The …rst group of conditions relates to the entry-exit process. The mass of prospective entrants is unbounded, they decide to enter depending on the …rm's value and before knowing their initial productivity level . Hence, in order to decide whether they enter the market, …rms calculate the expected value of the average pro…t ‡ows e

v = [1 G( )]e
and compare it to the sunk entry cost f e . As usual, …rms enter the market until the expected value of …rms e v is equalized to the sunk entry cost. This equality (e v = f e ) states the free entry condition (FE):

e = f e [1 G( )]
(FE)

Among …rms that enter the market N e , only a fraction = 1 G( ) will decide to stay. These …rms are those whose technical conditions allow enough revenues to pay the …xed costs of production. On the other hand, among active …rms N , some of them will exit the market with an exogenous probability (the death shock of [START_REF] Melitz | The impact of Trade on Intra-Industry Reallocations and Agregate Industry Productivity[END_REF]. The stability condition of entry and exit implies:

N = N e (28)

Labor Market and Global Accounting

The labor factor is inelastically supplied in perfect competition. Total labor L T is composed of production workers L p (including labor used to pay the …xed production and export costs) and also workers allocated to pay the sunk cost to enter the market L e :

L T = L p + L e (29)
The N e …rms that enter the market incur a labor cost of start-up equal to

L e = N e f e ( 30
)
The labor market clearing condition is ensured by the global accounting condition.

Recalling that wage rate is normalized to 1:

L p + N e = R (31)
Using (30), the free entry condition (FE) and the stability condition (??) one easily obtains L e = N e : Using (29) yields:

L T = R (32)

Number of Domestic Firms

Starting from the previous conditions we can express average revenues e r as

e r = R N (33)
We can write average pro…ts as e = e r f x f x : After multiplying both sides by N and rearranging terms, we obtain the aggregate revenue:

R = N e r = N [e + f + x f x ] (34)
Replacing R in labor market clearing condition (32) and applying the free entry condition (FE) one gets the number of active domestic …rms:

N = L T h fe + f + x f x i (35)
After considering revenues net of investment, the expression de…ning the number of …rms looks similar to the standard Melitz's framework. However, the outcome is di¤erent.

In this model, the probabilities de…ned by and x are determined by the cuto¤s x and

; which at equilibrium are in ‡uenced by the endogenous investment decision.

Equilibrium

The equilibrium can be solved using the Free Entry condition (FE) once we have determined its left-hand-side: the average pro…ts e . The latter can be related to the minimum level of initial productivity that allows enough revenues to stay in the market (the production cuto¤ de…ned by equation ( 15)). This relationship is what [START_REF] Melitz | The impact of Trade on Intra-Industry Reallocations and Agregate Industry Productivity[END_REF] calls the Zero Cuto¤ Pro…t condition (ZCP). Hence, the equilibrium is jointly determined by both the FE condition (FE) and the ZCP condition. The intersection of both curves, gives at equilibrium, which will then pin down the rest of endogenous variables of the model.

Starting from equation (26) we need to derive the ZCP in order to express e as a function of the cuto¤ : A convenient treatment is to exploit the aggregation properties of the model. To obtain the domestic average pro…t in (26), we start from the domestic average revenue (24). In order to simplify b ; we express the ratio of the revenue of the representative domestic …rm b d over the one earned by the cuto¤ …rm . Applying this to equation ( 27) gives:

r I d (b d ) = b d 1 r I d ( )
From equation [START_REF] Liu | Productivity Growth in Colombia and Chile: Panel-Based Evidence on the Role of Entry, Exit and Learning[END_REF] we know that operating pro…ts of the production cuto¤ …rm equals the …xed production cost: f can be written now as a function of the production cuto¤ and the average b d ; which only depends on : This leads to:

d (b d ) = ( b d 1 1
) f

In the case of the average export revenues we proceed in a similar way. This time, we divide the export revenue of a representative exporter b x by the one earned by the export cuto¤ …rm x . We also know that the export operating pro…ts of the export cuto¤ …rm equals the …xed cost to reach the foreign market: rx( x ) = f x : Then, export pro…ts of the representative exporter are give by:

x (b x ) = ( b x x 1 1 ) f x
Therefore, the ZCP condition de…nes the average pro…t per …rm as:

e = ( b d 1 1 ) f + x ( b x x 1 1 ) f x (ZCP)
This condition is entirely determined by the production cuto¤ level. We know that

b d = b d ( ) and b x = b x ( x )
: Thus, we just need to …nd x = x ( ). In order to …nd this relationship, we plug the optimal investment (22) into equations ( 15) and ( 16).

Then, we take the ratio of the resulting equations and we …nd:

x = ( ) (36) If ( ) [1 + ] h fx f
i 1 > 1 there will be exported varieties at the equilibrium ( < x )2 : At the end, the ZCP condition depends only on and the exogenous parameters. In order to get a closed solution we solve the model using a Pareto distribution for the initial productivity draws.

Analytical solution

b d = (37) b x = x (38)
Where

h k 1 k i 1
Proposition 3: Under Assumption 2, there exists a unique equilibrium production cuto¤ determined by the ZCP and FE conditions. This cuto¤ is given by:

= 2 4 h 1 1 i h f + f x ( ) k i f e 3 5 1 k (39)
Proof. Equalizing the equations of average pro…t stated by the ZP C and the F E leads to the equilibrium production cuto¤ (39).

Proposition 4: Under assumption 2, the production cuto¤ is a decreasing function of variable trade costs @ @ < 0 :

Proof. From (39) we obtain:

@ @ = 1 k 1 1 fe fx ( ) k [1+ ]
. Since > 0, as long as 1 > 1; we verify @ @ < 0. Note that

1 = 1 1 1 k [ 1 ]
. Thus, if k > 1 ; clearly 1 > 1 . This is exactly what the condition restricting k states in order to get …nite means. [START_REF] Melitz | The impact of Trade on Intra-Industry Reallocations and Agregate Industry Productivity[END_REF] explains this result as a general equilibrium consequence of the increase in the number of potential entrants. After a reduction of variable trade costs, export demand increases. The value of …rms is higher in the new equilibrium, which implies a higher number of entrants. This in turn increases labor demands and also real wages 1 P .

To be able to pay the …xed production cost, the marginal …rm needs to be more productive than before.

In our model, the reduction of variable trade costs also enhances investment demand.

As we saw, investment is more signi…cant when the initial heterogeneity induces higher productivity gains from technology. The externality of investment reinforces the selection process. After a reduction of trade costs, investment remains low for initially-lowproductive …rms. Consequently, they end-up with a lower productivity relative to the economy. Therefore, these …rms will be forced to exit the market after a reduction of trade frictions.

Proposition 5: Under assumption 2, the export cuto¤ is an increasing function of variable trade costs: @ x @ > 0 :

Proof. Applying (39) to the export cuto¤ equation (36) gives the cuto¤ x which veri…es: @ x @ = h @ @ + (1+ ) i : Since @ @ < 0; we need to prove @ @ < (1+ ) , @ @ < (1+ ) : After using @ @ obtained in the proof of proposition 4, this condition is similar to state

1 h f fx k +1 i < 1. Since f fx k > 0 ,
this proposition is unambiguously veri…ed.

Since productivity A ( ) increases monotonically with , a reduction of variable trade costs will decrease the export cuto¤ productivity level. Hence, more …rms are able to acquire the export status. On the demand side, variable trade costs reduction leads to a decrease in the price of home goods sold in the foreign market. This price reduction accounts for an increase in foreign demand, which in turn raises export pro…ts. In this new equilibrium …rms need a lower level of productivity to pay the …xed export costs and to sell in the foreign market.

Note that these properties @ @ < 0; @ x @ > 0 imply directly that higher variable trade costs increases the ex-ante probability of staying in the market @ @ > 0 and reduces the probability of exporting @ x @ < 0 . Intuitively, if the reduction of trade costs increases the minimum level of productivity required to stay in the market, the probability of a successful entry is reduced. Similarly, if less costly trade induces a lower threshold of productivity to export, it also implies a higher probability to reach the foreign market.

The reduction of trade variable costs 4.1 Within-…rm productivity

The standard results presented in the previous section are reinforced by the investment channel introduced in this model. The impact of trade liberalization can be separated into two channels whether one observes the supply or the demand e¤ects of trade variable costs. The former is related to the reduction of the cost of capital equipment, while the latter is based on the anticipation of an increase in …nal goods and factor demands which induces capital-labor substitution. In this section we analyze these channels in more detail.

Consider …rm productivity at equilibrium. Investment [START_REF] Tybout | Plant-and Firm-level Evidence on the 'New'Trade Theories[END_REF] applied to productivity [START_REF] Yeaple | A Simple Model of Firm Heterogeneity, International Trade, and Wages[END_REF], after global accounting states:

A ( ) = 1 L T N T ( ) ( ) b ( ) 1 (40)
N T ( ) highlights the dependency of the number of varieties on trade costs. The latter a¤ects the probabilities of staying in the market and of exporting (see equation ( 35)).

Applying the aggregation of heterogeneity for domestic (37) and exported (38) varieties to the global one [START_REF] ‡er | The Long and Short of the Canada-U.S. Free Trade Agreement[END_REF] gives b as function of : The reason is that this aggregation depends on the cuto¤s of production and export, which are in turn determined by : On the supply side, the e¤ect of trade costs on the …nal capital price is captured by the …nal price of imported capital in home currency ( ) = (1 + ) p k e w . Clearly A ( ) increases when decreases and its impact (in absolute value) is higher for high values of initial productivity gains . On the demand side, foreign consumption increases after a reduction of variable trade costs. The intuition is that …rms anticipate a greater demand and, as a consequence, more requirement of labor. Hence, labor productivity gains become more pro…table, which leads to a raise in investment demand. While the supply channel is homogeneous across …rms, the demand channel depends on . Thus, the latter generates heterogeneous …rms'productivity improvements.

We …nd these results with the help of numerical simulations. The impact of on productivity gains is non-linear and it is more signi…cant when is higher.

We illustrate this mechanism in Figure 2. It shows the plot of A ( ) on for a high ( = 2) and a low ( = 1) levels of initial draws. In both cases a reduction of variable trade costs enhances labor productivity, but this improvement depends positively on the initial productivity level. Moreover, in the graph one clearly observes a steeper slope for the higher .

More interestingly, there is a change in the distribution of initial productivity levels.

This result is illustrated in Figure 3. We plot both the initial productivity draws (dashed line) and the equilibirum-level of productivity after investment (solid line). Firms can decide to modify their initial productivity level by the means of technological investments, but these decisions depend on their initial pro…tability. Thus, technological improvements are biased towards initially high-productive …rms. In an heterogeneous …rms model with …xed-exogenous productivity draws, the levels of …rm productivity will be simply represented by : This is the 45 dashed line 3 . The heterogeneous e¤ect of investment is captured by the concavity of the productivity level at equilibrium (solid line). It might be the case that, as in Figure 3, the productivity level is even reduced. The …xed-exogenous productivity model can be seen as a model in which initial investment is homogenous and normalized to 1. Allowing for a continuous technology choice with investment externalities reinforces the e¤ect of market selection. Firms drawing a low will be deterred to undertake a signi…cant level of investment.

The intensive and the extensive margin of trade

One implication of the previous results is that trade liberalization, in countries highly dependant on imported technologies, improves mainly the volume of exports of initially high-productive …rms. Namely those that already export before trade reform (intensive margin). Foreign demand (3), using the properties of aggregation e A T = A(b ( )), can be expressed at equilibrium as:

d f = 1 b ( ) 1 L T N T ( ) (1 + ) A(b ( ))
We know that a fall of trade variable costs forces the least productive …rms to exit the market. Thus, the average b ( ) increases. We observe again di¤erentiated e¤ects of trade policy depending on the impact of foreign technology on labor productivity. A reduction of variable trade costs induces a positive income e¤ect through an increase in the average productivity of the economy, which boosts up global consumption. However, the demand of each …rm (in monopolistic competition) depends on their technology relative to the average of the economy the by b ( ) , the latter being endogenously determined. The initially high-productive …rms will become even more e¢ cient after trade liberalization and thereby they will enlarge their export demand.

Result 2: A reduction of trade barriers has a higher impact on the intensive margin of trade (volume of exports) relative to the extensive one (number of new exporters).

To illustrate this point, we simulate the relationship between foreign demand and initial productivity of …rms, for two di¤erent levels of variable trade costs. Figure 4 plots foreign demand d f ( ) on . In the plot, the economy changes from a iceberg cost level of 80% (solid line) to a one of 20% (dashed line). Foreign demand curve increases more for high levels of , where the slope is steeper. While there is a huge increase in the intensive margin of trade, there is only a minor one in the extensive margin. Indeed, the reduction of the export threshold x is small.

It will only allow for few …rms to become exporters. Therefore, gains from trade are concentrated on continuing exporters.

Conclusion

This paper addresses the impact of trade on the evolution of within …rm productivity gains in countries highly dependant on technologies embodied in imported capital equipment goods. We proposed an intra-industry trade model of heterogeneous …rms which are able to change their initial e¢ ciency level. The main contribution of this study is to introduce endogenous productivity gains determined by an initial investment in foreign technology.

The model proposed in this paper introduces several results consistent with the empirical …ndings of …rm-level studies. Firstly, trade liberalization induces a pro…tability bias towards the initially more productive …rms. These …rms are able to improve easily their e¢ ciency by foreign technology adoption after a reduction of trade variable costs.

Secondly, contrary to the standard vision that supports the idea of a tari¤s reduction

Figure 1 .Figure 1

 11 Figure 1.a. Source: Bas and Ledezma (2007)

  industry can be aggregated using the weighted averages b ; b d and b x : Average revenue e r (net of investment), steam from average revenues earned from domestic sales r I d (b d ) and from export sales r x (b x ) e r = r I d (b d ) + x r x (b x ) (25) A similar statement applies for average pro…ts e , which can be decomposed into domestic d (b d ) and export x (b x ) average pro…ts. e = d (b d ) + x x (b x )

  ) Hence, r I d (b d ) = r d (b d ) :

  = f: Hence, the domestic pro…ts of the representative domestic …rm d (b d ) = r I d (b d )

FollowingAssumption 2 : 1 .

 21 Ghironi and Melitz (2005) and Melitz and Ottaviano (2005), we parametrize the productivity draws to get tractable solutions of the model. Productivity draws are distributed according to a Pareto distribution g (') = k' k min ' k+1 with a lower bound ' min = 1 and a shape parameter k > When this shape parameter increases there is a reduction of the technological dispersion, which will be concentrated towards the lower bound. As usual, this distribution density requires k > 1 in order to ensure …nite means. The parameter min = 1 implies that the corresponding cumulative distribution function is given by G( ) = 1 1 k . We can easily verify that = 1 G( ) = k and x = 1 G( x ) 1 G( ) = h x i k : After solving the integrals de…ning b d ; b x we obtain:

Result 1 :

 1 The parameters used are: f e = 15; = 10%; f x = 20; f = 1; = 1; = 2=3; k = 4: This setting veri…es both k > 1 and the following condition ( Firm productivity increases with the reduction of trade variable costs .

Figure 4 .

 4 Figure 4.

As in[START_REF] Melitz | The impact of Trade on Intra-Industry Reallocations and Agregate Industry Productivity[END_REF], the time discounting parametter ; represents also the exogenous probaility of exit.

This occurs whenf x ( 2 ) (1 + ) 1 > f

Scales of axis are assymetric.

in order to foster export diversi…cation, this work argues that, in the case of a country highly dependant on foreign technology, trade variable costs reduction will have a minor impact on the extensive margin of trade.

Finally, the model is able to reproduce two important channels through which trade liberalization a¤ects aggregate productivity growth. Besides the standard mechanism of selection and reallocation of resources (between-…rm channel), the model also reproduces aggregate productivity gains related to improvements inside the …rm (within-…rm).

Further research should be oriented to analyze the speci…c pattern of the distribution of productivity levels at equilibrium and to compare it to …rm level data. This can be made by the means of stochastic simulation. It seems clear for us that the link between trade and productivity asks for further research on stochastic dynamic issues.


Andrew E Clark
email: andrew.clark@ens.fr.

Dick Easterlin

David Halpern

Felicia Huppert

Orsolya Lelkes

Andrew Oswald

Claudia Senik

Rainer Winkelmann

BORN TO BE MILD? COHORT EFFECTS DON'T EXPLAIN WHY WELL-BEING IS U-SHAPED IN AGE

Keywords: Well-being, aging, cohort effects, panel analysis C23, I3, J11

The statistical analysis of cross-section data very often reveals a U-shaped relationship between subjective well-being and age. This paper uses fourteen waves of British panel data to distinguish between a pure life-cycle or aging effect, and a fixed cohort effect that depends on year of birth. Panel analysis controlling for fixed effects continues to produce a U-shaped relationship between well-being and age, although this U-shape is flatter for life satisfaction than for the GHQ measure of mental well-being. The pattern of the estimated cohort effects differs between the two well-being measures and, to an extent, by demographic group. In particular, those born earlier report more positive GHQ scores, controlling for their current age; this phenomenon is especially found for women.

Introduction

Interest in subjective well-being across the social sciences has developed in parallel with both the greater availability of panel data, where the same individuals are followed over time, and the wider use of statistical tools to better model individual fixed effects. These statistical techniques consist of the use of panel data, or cross-section analysis with very careful controls (for example, Atwin@ studies, where the initial distribution of the genetic pack of cards can be controlled for: see [START_REF] Bouchard | Sources of Human Psychological Differences: The Minnesota Study of Twins Reared Apart[END_REF][START_REF] Kohler | Partner + Children = Happiness? The Effects of Partnerships and Fertility on Well-Being[END_REF][START_REF] Tellegen | Personality Similarity in Twins Reared Apart and Together[END_REF]. The application of these techniques allows subjective well-being to be split up into a permanent or fixed part, and a transitory component that depends on life events. Recent contributions in this spirit include [START_REF] Lucas | Re-Examining Adaptation and the Setpoint Model of Happiness: Reaction to Changes in Status[END_REF][START_REF] Senik | When Information Dominates Comparison: A Panel Data Analysis Using Russian Subjective Data[END_REF], [START_REF] Frijters | Money does matter! Evidence from increasing real incomes and life satisfaction in East Germany following reunification[END_REF], [START_REF] Van Praag | Happiness Quantified[END_REF], and Zimmerman and [START_REF] Easterlin | Life Cycle Happiness and Its Sources[END_REF]. This interest in fixed individual characteristics has spilled over into the analysis of the relationship between well-being and age: in an econometric world plagued by accusations of endogeneity, age, sex and ethnicity typically stand out as exogenous variables, and have consequently received a great deal of attention. Early work emphasised that older individuals tended to be happier/more satisfied than younger individuals. More recent analyses have refined this approach by considering non-linear relationships between well-being and age. The results here differ somewhat between economics and psychology. [START_REF] Mroczek | The effect of age on positive and negative affect: A developmental perspective on happiness[END_REF] find that positive affect follows an upwardly curved profile with age, while [START_REF] Mroczek | Change in life satisfaction during adulthood: Findings from the Veterans Affairs Normative Aging study[END_REF] suggest that subjective well-being follows an inverted U-shape, peaking at around retirement age. At the same time, a vigorous literature in Economics has introduced terms in age and age-squared into well-being regressions, revealing a strong U-shaped relationship which tends to bottom out somewhere between the mid-thirties and the early forties. This curve has now been found many times in a wide variety of datasets across different countries. 1 Two competing interpretations of this U-shaped relationship have been proposed. One is that it reflects individuals passing through different life events; another is that it reflects a cohort effect, so that individuals born in the 1950s, say, have (and always will have) particularly low levels of subjective well-being. This paper uses two measures of well-being in fourteen waves of British panel data to test the hypothesis that the U-shape is a pure cohort phenomenon. Two types of test are presented, the first indirect, although intuitive, and the second direct. The tests are carried out on both unbalanced and balanced panel data.

The first intuitive test is based on the estimated minimum point of the U-shape. If this latter reflects a cohort phenomenon, then the point of lowest well-being should move to the right by one year from data wave t to wave t+1. The age of minimum well-being in Wave 14 should therefore be thirteen years greater than that in Wave 1. This turns out not to be the case. The conclusion is that there is an Aaging phenomenon@ in well-being: this is something that we will all (statistically) go through, no matter when we were born.

The second test is direct. Panel well-being regressions are estimated which control for unobserved individual fixed effects. These regressions, which control for all cohort effects, continue to show a U-shaped relationship between age and well-being. It is important to underline that the age effect here is obtained by examining the different levels of well-being of the same individual at different ages. Again, the conclusion is that the U-shaped relationship is at least partly driven by aging, rather than being a pure cohort effect.

The remainder of the paper is organised as follows. Section 2 describes the two tests of cohort effects, and the data to which they will be applied. Section 3 contains the main results regarding the persistence of the U-shaped relationship between well-being and age. Last, section 4 concludes.

2.

Cohort or Life-Cycle?

Empirical work linking age to measures of subjective well-being (such as life satisfaction or happiness) typically finds a U-shaped relationship, minimising somewhere in the mid-thirties to the early forties. As highlighted in [START_REF] Frey | Happiness and Economics[END_REF], there is less agreement on why this U-shape is so consistently found. One interpretation is that, loosely speaking, the U-shape reflects the different events that occur to individuals over the life cycle, and their reaction to these events. This was suggested by [START_REF] Argyle | The Psychology of Happiness[END_REF]; more recent discussions are found in [START_REF] Hayo | Subjective Economic Well-Being in Eastern Europe[END_REF] and [START_REF] Blanchflower | Well-Being over Time in Britain and the USA[END_REF]. Alternatively, we might argue that well-being is broadly flat over the life-cycle, and that the U-shape reflects unobserved individual heterogeneity or cohort effects. 2 In this case, with data from the 1990s, the hypothesis is that those in the late 1950s/early 1960s birth cohort report lower well-being scores than do those born earlier or later.

Cross-section data does not allow us to distinguish between the life-cycle and cohort components of well-being. Neither can twin data, as age and year of birth (the cohort effect) are identical across matched subjects. Progress can be made with panel data, however, in which we have repeated observations on individuals of the same birth cohort, over different ages, allowing the two effects to be identified separately. The increasing availability of long-run panel data has been a huge boon for social science. This is especially the case in terms of the research on aging or the life-cycle. This paper uses fourteen waves of British panel data to distinguish between lifecycle and cohort effects. Two separate tests that the U-shape represents a cohort effect are proposed.

The first, indirect, test relies on the prediction of the Acohort@ model that the whole Ushape should move one year to the right at every wave: the unhappy people who were born in 1955 will be unhappy at age 36 in Wave 1 (in 1991), but equally unhappy at age 37 in Wave 2, and so on. One measure of the position of the U-shaped relationship is its minimum. The first test thus consists in seeing whether the point of minimum well-being shifts to the right by one year per wave.

The second, direct, test involves controlling explicitly for fixed effects in panel wellbeing regressions. These fixed effects will include by definition the individual's year of birth: her cohort. Any effect of age variables in well-being regressions must then reflect life-cycle or aging effects: systematic changes in well-being that happen to the same individual as they age.

Data

The data come from the first fourteen waves of the British Household Panel Survey (BHPS), a general survey initially covering a random sample of approximately 10 000 individuals in 5 500 British households. The Wave 14 sample consists of around 15 000 individuals in 9 000 households. 3 The BHPS includes a wide range of information about individual and household demographics, employment, income and health. More information on this survey is available at http://iserwww.essex.ac.uk/ulsc/bhps/. There is both entry into and exit from the panel, leading to unbalanced data. The BHPS is a household panel: all adults in the same household are interviewed separately. The wave 1 data were collected in late 1991 -early 1992, the wave 2 data were collected in late 1992 -early 1993, and so on. The analysis in this paper refers to individuals aged between 16 and 64, and will be carried out on both unbalanced and balanced panel data.

The central question addressed here is whether individual well-being changes systematically over the life cycle. 4 Two measures of subjective well-being are considered: the General Health Questionnaire (GHQ), which appears in all waves of the BHPS, and overall life satisfaction, which appears in Waves 6-10, and then 12-14. There are 2 836 individuals who provided a GHQ score at every wave of the BHPS, so that the balanced panel analysis can be carried out on a maximum of 39 704 observations. Equally, 4015 individuals provided eight separate life satisfaction scores, for a maximum balanced sample of 32 120 observations. In practice, the balanced regression analysis will be based on slightly fewer observations due to missing values on some of the explanatory variables.

The GHQ-12 (see [START_REF] Goldberg | The Detection of Psychiatric Illness by Questionnaire[END_REF] reflects overall mental well-being. It is constructed from the responses to twelve questions (administered via a self-completion questionnaire) covering feelings of strain, depression, inability to cope, anxiety-based insomnia, and lack of confidence, amongst others (see Appendix A). Responses are made on a four-point scale of frequency of a feeling in relation to a person's usual state: "Not at all", "No more than usual", "Rather more than usual", and "Much more than usual". 5 The GHQ is widely used in medical, psychological and sociological research, and is considered to be a robust indicator of the individual=s psychological state. The between-item validity of the GHQ-12 is high in this sample of the BHPS, with a Cronbach=s alpha score of 0.90.

This paper uses the Caseness GHQ score, which counts the number of questions for which the response is in one of the two "low well-being" categories. This count is then reversed so that higher scores indicate higher levels of well-being, running from 0 (all twelve responses indicating poor psychological health) to 12 (no responses indicating poor psychological health). 6

The distribution of this well-being index in the BHPS sample is shown in Appendix B. The median and mode of this distribution is 12: no responses indicating poor psychological health.

However, there is a long tail: one-third of the sample have a score of 10 or less, and thirteen per cent have a score of 6 or less.

The second measure is satisfaction with life, which appears in Waves 6-10 and 12-14 of the BHPS. Respondents are asked "How dissatisfied or satisfied are you with your life overall", with responses measured on a scale of one (not satisfied at all) to seven (completely satisfied).

The distribution of replies is shown in the second panel of Appendix B. The median score is five, with a mode of six and a mean of 5.2.

The following section considers how both of these well-being measures are related to age, both with and without controls for cohort effects.

designed. On a practical level, the employed=s GHQ is more strongly correlated with job satisfaction levels in the BHPS data than with job satisfaction changes. Last, with fourteen years of balanced panel data, a relatively direct test of the usefulness of the GHQ score in this respect can be envisaged. If events become more Ausual@ as an individual ages then the standard deviation of GHQ scores (and of its individual components) will fall with age.

There is no evidence of this phenomenon in balanced BHPS panel data. 6 Alternatively, the responses to the GHQ-12 questions can be used to construct what is known as a Likert measure. This is the simple sum of the responses to the twelve questions, coded so that the response with the lowest well-being value scores 3 and that with the highest well-being value scores 0. This count is then reversed, so that higher scores indicate higher levels of well-being. The measure thus runs from 0 (all twelve responses indicating the worst psychological health) to 36 (no responses indicating poor psychological health). Practically, the results are very similar between the Caseness and Likert measures.

Well-Being and Age: Pooled and Panel Results

Well-Being and Age in Pooled Data

Table 1 sets the scene by presenting the results from "standard" well-being equations estimated on pooled data. All of the regressions in this paper are estimated using linear techniques. The pooled analysis in Table 1 comes from OLS estimation; the panel results below come from "within" regressions. It can, of course, be objected that the assumption of cardinality required for OLS is unlikely for well-being measures (is someone with a life satisfaction score of six exactly twice as happy as someone with a life satisfaction score of three?). However, Ferreri-Carbonell and Frijters (2004) have shown that, practically, the difference between the ordinal and cardinal estimation of subjective well-being is small compared to the difference between pooled and panel results. More pragmatically, all of the results presented in this paper can be reproduced using appropriate ordinal estimation methods (ordered probit for the pooled analysis, and conditional fixed effect logits for the panel regressions).

Column 1 of Table 1 shows the results from pooled cross-section regressions of GHQ scores, while column 2 refers to life satisfaction. The regressions include age and age-squared as explanatory variables, as well as a number of other controls. The very significant coefficients on age (negative) and age-squared (positive) reveal that, ceteris paribus, well-being is U-shaped in age. Some simple algebra shows that the age of minimum well-being is 39 for GHQ and 42 for life satisfaction.

The estimated coefficients on the other right-hand side variables are all fairly standard in the empirical literature on well-being. Unemployment, marital status and health have large impacts on both measures of well-being in the expected direction. Three variables have opposing effects on GHQ and life satisfaction. Both income 7 and self-employment are associated with higher life satisfaction scores, but greater mental stress. On the contrary, men report lower life satisfaction scores, but less mental stress. 8 It is, in particular, contentious to include health as a right-hand side variable 9 , although this is widespread in the literature. The implication here is that we are comparing individuals of working age, but with the same level of health. All of the main results below can be reproduced in analyses which do not include health as an explanatory variable.

Test 1: Does the U-shape move to the right by one year per wave?

If the U-shape in age is a cohort phenomenon, then the whole distribution of well-being should shift to the right by one year per wave. This hypothesis is tested by re-running Table 1's regressions separately for each of the fourteen waves of the BHPS, and calculating the estimated age of minimum well-being in each wave. 10 The results are summarised in the top panel of Figure 1. If the U-shape reflects only cohort effects, then the estimated age of minimum well-being should increase by one year per wave, tracing out a 45-degree line. Figure 1 shows little evidence of this. Although the estimated minimum of GHQ does rise a little at the very beginning of the sample period, there is no strong trend thereafter.

The BHPS is an unbalanced panel. It is therefore theoretically possible that the pattern of well-being amongst those who enter and exit the data has disguised the true relationship between 7 Income is measured in real terms, having been deflated by the CPI. 8 Nolen-Hoeksema and Rusting (1999) conclude in their survey article that women show higher rates for almost all of the mood and anxiety disorders, but in general report higher levels of happiness. 9 [START_REF] Blanchflower | Well-Being over Time in Britain and the USA[END_REF] explicitly do not control for health in their statistical analysis. 10 Alternatively, interaction terms between age and wave can be introduced into Table 1's regressions; these give qualitatively very similar results.

well-being and age. To investigate, the bottom panel of Figure 1 shows the same age of minimum well-being estimated on a balanced panel (over all fourteen waves for GHQ, and over eight waves for life satisfaction). The balanced results again provide little support for the hypothesis that the age distribution shifts to the right by one year per wave.11

This evidence points to at least some role for a pure life-cycle effect, whereby for the The main result from the panel analysis is that, even controlling for individual fixed effects, well-being continues to show a U-shaped relationship with age. This is true for both GHQ and life satisfaction, although the U-shape is more pronounced for the former than for the latter. The age of minimum well-being is in the forties for both measures of well-being. The estimated relationships between well-being and age are illustrated in Figure 2. It is notable that the left-hand side of the "U" in the life satisfaction regressions depends entirely on the drop in well-being between ages 16-19 and 20-24. Thereafter, life satisfaction stays fairly flat up to the end of the forties.

As with test 1, on the estimated age of minimum well-being by wave, it is important to take panel exit and entry into account. Table 3 therefore repeats the analysis described in Table 2, but now estimated only on the balanced sample (4000 individuals for the GHQ score, and just under 3000 for life satisfaction). Even in this much smaller balanced sample, the top panel of Table 3 shows a persistent U-shaped relationship between age and GHQ. In the bottom panel, the relationship between age and life satisfaction is now less evident. Even so, the U-shape persists in the estimated coefficients, which are jointly significant. In addition, the real test of the U-shape is whether the ends are greater than the middle. This test passes at the ten per cent level comparing the youngest age-group to those in their mid-forties, and at almost the one percent level when we compare the oldest age-group to those in their mid-forties.

Interpreting the results.

Well-being continues to be U-shaped in age even controlling for cohort effects (via individual fixed effects). As such, the well-being of any individual, no matter when they were born, will trace out the profile given by the "panel" lines in Figure 2. While it is easy to think of some aspects of life which might systematically be difficult between the ages of 35 and 45, it is worth noting that the multivariate analyses controls for a number of these (labour force and marital status, home ownership, and number of children). One open question is therefore the identification of the life cycle events that hit hard between the ages of 35 and 45. One possibility is stress at work (perhaps combined with young or adolescent children), although it is difficult to measure such phenomena accurately in large-scale datasets.

As a by-product of column 3 of Table 2, we can examine the distribution of the estimated fixed effects by birth cohort. Graphs showing the pattern of these fixed effects by year of birth are presented in Appendix C. A small number of birth years for which there were fewer than 20 individuals in the cell have been dropped: this applies particularly to the graphs by level of education. The shape of the cohort effects can be inferred from Figure 2. For GHQ, both pooled and panel results are U-shaped, but the distance between the two lines is greater for the older age-groups than for the younger age-groups: in other words, those born earlier report higher levels of well-being on the GHQ scale, independent of their current age. 12 However, the size of this estimated cohort effect is not overwhelmingly large: the difference between the average fixed effect of those born in the 1930s and those born in the 1970s is about one-eighth of a GHQ point, on the 0-12 scale. By way of comparison, this is the size of the estimated effect of divorce on well-being in the pooled cross-section estimates in Table 1, but is only about one-third of the size of the effect of separation or unemployment.

The same approach for life satisfaction suggests that the fixed effects must be U-shaped.

The pooled results are markedly U-shaped, but the panel results are less so. The difference between them is the fixed effect, which must therefore itself have a U-shaped distribution. In this case, it does seem to be true that those born around the late 1950s to early 1960s have (fixed) lower levels of life satisfaction.

Why do the fixed effects have this pattern? Any attempt at explanation will be speculative, as by definition fixed effects reflect unobserved differences between individuals.

With respect to the shape of GHQ fixed effects, one such piece of speculation as to why those born earlier have, ceteris paribus, higher levels of mental well-being appeals to social comparisons. Researchers in a number of social science disciplines have emphasised the importance of comparisons to reference groups [START_REF] Adams | Inequity in Social Exchange[END_REF][START_REF] Frank | Frames of Reference and the Quality of Life[END_REF][START_REF] Kapteyn | Individual Welfare Functions and Social Reference Spaces[END_REF][START_REF] Pollis | Reference Group Re-examined[END_REF]. It seems likely that one type of comparison may occur with respect to the past, and perhaps even to a certain defined period (parents= situation during the individuals= childhood, or the individual=s first job, for example). Secularly rising living standards will then imply that older cohorts compare current outcomes to lower reference levels, and will therefore report higher well-being scores. 13 13 Such comparisons to the past imply that, in the long run, the correlation between GDP per capita and individual well-being may well be small. For some recent empirical contributions to this debate, see [START_REF] Diener | Money and happiness: Income and subjective well-being across nations[END_REF], [START_REF] Easterlin | Will Raising the Incomes of All Increase the Happiness of All?[END_REF] and [START_REF] Oswald | Happiness and economic performance[END_REF]. This literature is surveyed in [START_REF] Clark | Income and Happiness: Evidence, Explanations and Economic Implications[END_REF].

Is Everyone the Same?

It is of interest to carry out the above analysis separately for different demographic groups. Specifically, Table 2's regressions were re-run for men and for women, and for three different educational groups (where high education corresponds to qualifications obtained in higher education, and medium education to A-Level, O-Level or Nursing qualifications).

Both the estimated U-shape and the fixed effects profile differ by demographic group.

The U-shaped relationship between well-being and age is much more prominent for men than for women. In addition, the negative trend in the GHQ fixed effect (so that older cohorts are happier than younger cohorts) is found only for women. Regarding education, the U-shape is stronger for the higher-educated than for the other groups, and the negative trend in the GHQ fixed effect is found only for the higher-educated. However, the U-shaped fixed effect in life satisfaction is found for all demographic groups.

Conclusion

This paper has used fourteen waves of British panel data to confirm that subjective wellbeing is U-shaped in age in pooled data. The application of panel analysis techniques allows us to distinguish the life-cycle or ageing component of this relationship from the fixed effect or cohort part. The results show that, even controlling for individual fixed effects, both life satisfaction and GHQ scores remain U-shaped in age. The analysis of the fixed effect in GHQ scores shows that individuals from earlier cohorts (i.e. those who were born earlier) will have, ceteris paribus, distinctly higher levels of subjective well-being, as measured by the GHQ-12 score, than those from later cohorts. This pattern is markedly different by sex, and by level of education. The fixed effects in life satisfaction exhibit a U-shaped relationship.

The main result of this analysis may be considered as essentially negative: whereas we previously thought that there was only one phenomenon to explain (the U-shape), there are now two: a U-shaped life-cycle or aging effect, and the cohort profiles. This paper has not explicitly tested any theories of why these data shapes pertain, although the GHQ fixed effects results are consistent with reference group theory, in that those born earlier may have lower standards of comparison.

This paper=s conclusions are based on British data, although the robust U-shaped relationship is found across two rather different measures of well-being (while weaker for life satisfaction than for the GHQ). It may be that other datasets will produce different results. The simple method used in this paper can be easily applied to any panel data set of sufficiently long duration. The search for consistent patterns in well-being data should perhaps now pay more attention to the structure of the fixed effect, and in particular to its relationship with year of birth.

1.

Here are some questions regarding the way you have been feeling over the last few weeks. For each question please ring the number next to the answer that best suits the way you have felt.

  same individual well-being changes systematically with age. It should be borne in mind, however, that the age of minimum well-being, which is the ratio of two estimated coefficients, is likely measured with a certain degree of error; as such we cannot consider Test 1's results to be definitive, but rather suggestive. What follows is a direct test of the importance of cohort effects which escapes this criticism.Table2shows three sets of regression results for both GHQ and life satisfaction. The first two regressions are estimated on pooled data. The first includes only the age dummies on the right-hand side, and thus provides a non-parametric estimate of the relationship between wellbeing and age. Well-being is U-shaped, with minimum well-being occurring at age 40-44 for both measures. The second column introduces the other demographic controls used in Table1.These controls make the U-shape more pronounced, if anything, and do not change the age of minimum well-being.The last column introduces individual fixed effects. The estimated coefficients on the age dummies in column 3 therefore represent the different levels of well-being reported by the same individual as they go through the life cycle. Of course, even with fairly long-run panel data, we do not observe the complete sequence of ages for any respondent. In the BHPS, any one individual can appear in a maximum of four different age categories in the GHQ regressions (over 14 waves) and three different age categories in the life satisfaction regressions (between waves 6 and 14).

Test 2: Introducing individual fixed effects.

A perhaps simpler approach to the question is to introduce controls for unobserved heterogeneity. To allow a flexible relationship between well-being and age, ten age dummies are created. The first refers to age 16-19, then 20-24, 25-29 and so on up to 60-64. The youngest age group is the omitted category, so all of the estimated coefficients in Table

2

are to be read as relative to the well-being of the youngest.

Age of Minimum Well-Being by Wave

  Not at all ................................. 1 No more than usual.................. 2 Rather more than usual............ 3 Much more than usual ............. 4 More so than usual ................. 1 About same as usual ................ 2 Less so than usual ................... 3 Much less than usual................ 4 TABLE 1. WELL-BEING REGRESSIONS. BHPS WAVES 1 TO 14 POOLED FIGURE 1. THE ESTIMATED MINIMUM OF THE U-SHAPE, BY WAVE

				Caseness GHQ	Life Satisfaction
			Age	-0.089**	-0.075**
				(0.005)	(0.003)
			Age-squared/100	0.115** GHQ Life Satisfaction	0.091**
				(0.006)	(0.003)
	then	Have you recently.... a) been able to concentrate on whatever you're doing ? (0.025) Wave Married 0.058* 0.256** (0.013) 1 2 3 4 5 6 7 8 9 10 11 12 . 4 Total Income -0.027** 43 0.018** (0.008) (0.004) Self-Employed -0.088** 0.054** (0.029) (0.016) Unemployed -0.985** -0.401** (0.038) (0.021) Retired -0.062 0.107** (0.043) (0.022) Other LFS -0.684** -0.202** (0.022) (0.012) Male 0.433** (0.017) (0.009) 35 -0.092** 36 37 38 39 40 41 42 Age of Minimum Well-Being	13	14
			Separated	-0.917**	-0.273**
		b)	lost much sleep over worry ? (0.054)	(0.029)
		e)	felt constantly under strain ? Divorced -0.348**	-0.098**
		f) i)	felt you couldn't overcome your difficulties ? been feeling unhappy or depressed ? (0.035) Widowed -0.627** Age	(0.018) -0.134**
		j)	been losing confidence in yourself ? (0.062)	(0.033)
		k)	One Child been thinking of yourself as a worthless person ? -0.023 (0.024)	-0.051** (0.013)
	Two Children with the responses:	0.095** (0.026)	-0.043** (0.014)
			Three+ Children	0.056	-0.049*
				(0.038)	(0.021)
			Renter	-0.166**	-0.163**
				(0.019)	(0.010)
			Education: High	-0.264**	-0.119**
	then			(0.022)	(0.012)
		c)	felt that you were playing a useful part in things ? Education: Medium -0.100**	-0.082**
		d) g) h) l)	(0.021) felt capable of making decisions about things ? Health: Excellent 2.121** been able to enjoy your normal day-to-day activities ? (0.011) 1.039** (0.022) (0.012) been able to face up to problems ? Health: Good 1.688** 0.690** been feeling reasonably happy, all things considered ? (0.019) (0.010)
	Wave Dummies with the responses: Region Dummies	Yes Yes	Yes Yes
			Observations	132665	82096
			Standard errors in parentheses. * significant at 5%; ** significant at 1%
			Estimated Age of	38.8	41.5
			Minimum Well-Being

Better than usual .......................1 Same as usual .......................... 2 Less than usual ........................ 3 Much less than usual...............

of Minimum Well-Being by Wave: Balanced Data

		FIGURE 2. AGE AND WELL-BEING: POOLED AND PANEL RESULTS
							GHQ		Life Satisfaction
		55
	Age of Minimum Well-Being	35 40 45 50
		30
		1	2	3	4	5	6	7	8	9	10	11	12	13	14
								Wave

This is the conclusion reached by[START_REF] Easterlin | Income and Subjective Well-Being Over the Life-Cycle[END_REF], using twenty years of cohort data from the General Social Survey.[START_REF] Cribier | Changes in the experience of life between two cohorts of Parisian pensioners, born in circa 1907 and 1921[END_REF] is an evocative account of the differences in life experience between two cohorts of French workers born only 14 years apart.

The wave 1 panel was drawn from 250 areas of Great Britain. Additional samples of 1,500 households in each of Scotland and Wales were added to the main sample in 1999, and in 2001 a sample of 2,000 households was added in Northern Ireland.

More precisely: whether subjective well-being changes systematically in a way that cannot be explained by the standard set of explanatory variables (covering income, employment, health, demographics etc.).

A first reaction might be that the GHQ is singularly unsuitable for this kind of analysis, as its constituent parts are explicitly phrased in terms of comparisons to usual. It is worth noting that the empirical literature on GHQ scores treat them unambiguously as indicators of the level of well-being, and it was for this purpose that the instrument was

[START_REF] Blanchflower | Well-Being over Time in Britain and the USA[END_REF] carry out a similar test on American General Social Survey data, and conclude that there is only slight evidence that the minimum moves to the right over time. The GSS is not, however, a panel.

[START_REF] Easterlin | Life Cycle Happiness and Its Sources[END_REF] reaches a similar conclusion for happiness in US cohorts.

The BHPS data were made available through the ESRC by the ESRC Research Centre on Micro-social Change at the University of Essex. Neither the original collectors of the data nor the Archive bear any responsibility for the analyses or interpretations presented here. Paris-

Data Archive. The data were originally collected

Appendix A

The twelve questions used to create the GHQ-12 measure appear in the BHPS questionnaire as follows:

Appendix B

The Distribution of Well-Being in the BHPS (Inverted Caseness index of the GHQ-12)

Well 1930 1940 1950 1960 1970 1980 1990 Year of Birth All -1.5-1-.5 0 .5 1 Fixed Effects 1930Effects 1940Effects 1950Effects 1960Effects 1970Effects 1980Effects 1990 Year of Birth Women -1.5-1-.5 0 .5 1 Fixed Effects 1930Effects 1940Effects 1950Effects 1960Effects 1970Effects 1980Effects 1990 Year of Birth 1930 1940 1950 1960 1970 1980 1990 Year of Birth Medium-Educated -1.5-1-.5 0 .5 1 Fixed Effects 1930 1940 1950 1960 1970 1980 1990 Year of Birth Low -Educated

GHQ Life Satisfaction

No controls Demographic controls

Demographic controls plus Individual Fixed Effects

Age 20-24 -0.