Mehdi Senouci
email: mehdi.senouci@ecp.fr

The endogenous direction of technological change in a discrete-time Ramsey model

The endogenous direction of technological change in a

discrete-time Ramsey model

Empirical research on economic growth strongly supports the hypothesis that there exists a variety of regimes of growth. Historical evidence on the Industrial Revolution in the UK shows clear signs of a change of regime circa the 1860's, while the US economy witnessed a switch in the early 1980's called the Great Moderation. Neoclassical theory asserts that only increases in productivity can lead to sustained economic growth. In this perspective, the direction of technological change is the prime factor to account for the variety of growth regimes.

Since the seminal study by [START_REF] Greenwood | Long-Run Implications of Investment-Specific Technological Change[END_REF] it has become common among economists to talk about 'investment-specific technological change' (ISTC). ISTC refers to upgrading the quality of machinery and manifests itself in a downward trend of the relative price of capital goods over consumption goods. There is wide evidence that the relative price of equipment goods and the real investment rate are strongly, positively correlated with GDP per capita in cross-section 1 . We interpret this correlation as an indication of the importance of forms of technical progress linked to capital (and not only to labor) to sustain high growth rates over long periods. Although other explanations for this fact are possible (taxation, tariffs, etc.), some empirical analyses support the view that LDCs suffer from a disproportionate productivity gap in the capital sector relatively to the most advanced countries 2 . Figure 1 puts in regards the 2006 PPP GDP per capita and the 2006 relative deflator of investment over consumption in national accounts for 167 countries; the relationship is clearly negative.

The goal of this paper is to reconsider the message of the canonical neoclassical model of growth augmented with endogenous direction of technological change 3 , and to assess the validity of this 1 International comparison of relative deflators of each item of national accounts (consumption, investment, exports, etc.) has been made possible by the work of Robert Summers, Irving B. Kravis and Alan Heston which gave birth to the Penn World Table (PWT) [See in particular Summers, [START_REF] Summers | International Comparisons of Real Product and Its Composition: 1950-77[END_REF]; [START_REF] Kravis | New Insights into the Structure of the World Economy[END_REF] and [START_REF] Summers | A new set of international comparisons of real product and price levels estimates for 130 countries, 1950-1985[END_REF]. The PWT 8 can be freely downloaded at http://www.rug.nl/research/ggdc/data/penn-world-table .]. All the papers quoted here make use of this database.

The seminal work of [START_REF] Barro | Economic Growth in a Cross Section of Countries[END_REF] famously cited the real investment rate were a strong predictor of GDP per capita in a cross-section of countries. When taking into account that the relative deflator of investment over consumption differs between countries -so that the nominal investment/GDP ratio is not a good proxy for capital accumulation -investment has a positive and significant effect on growth. De Long and [START_REF] Delong | Equipment Investment and Economic Growth[END_REF][START_REF] Delong | Equipment Investment and Economic Growth: How Strong Is the Nexus?[END_REF] argue that this correlation indeed reflects causality, which they interpreted using an AK externality logic. [START_REF] Jones | Economic Growth and the Relative Price of Capital[END_REF] showed that, in a crosssection of countries, economic growth was negatively correlated with the relative price of equipment but not with the price of non-equipment investment goods, which leads him to advocate the tax rate on machinery as an important growth policy instrument. Taylor (1998) further corroborates this result using a panel of Latin American and Asian countries between 1970 and 1989. In the meta-analysis performed by Sala i Martin, Doppelhofer and Miller (2004), the relative price of investment is one of the most robust predictor of GDP per capita. [START_REF] Eaton | Trade in Capital Goods[END_REF] have shown that only a handful of countries are net supplier of capital goods -these countries also happen to be the most R&D intensive economies. See also Chari, Kehoe and McGrattan (1997), Restuccia and Urrutia (2001) and Armenter and Lahiri (2012).

2 See for instance Hsieh and Klenow (2007) and Herrendorf and Valentinyi (2012). 3 The framework we set up below pertains to the class of induced-innovation models. This class of models investigates the consequence of market approach .

Namely, we use a discrete-time Ramsey structure with a logarithmic utility function. We assume that there are two goods: a capital good and a consumption good. The capital good is simply produced from the consumption good, through a linear technology. We suppose that the agent is initially at the steady state without technical change. At date 0, he is given the opportunity to undertake either some marginal labor-augmenting technical change or investment-specific technical change (LATC vs. ISTC). LATC increases the effective supply of labor, while ISTC increases the efficiency of the process through which capital is produced 4 .

Each choice has specific dynamic consequences and, since the agent is impatient, he does not necessarily pick the high-growth option. In particular, we show that whenever the elasticity of substitution between capital and labor is greater than the product of the capital share and the rate of preference for the present σ > αβ , the agent might choose to undertake LATC even if ISTC increases relatively more steady-state consumption -in this case, we can say that ISTC is undervalued relatively to LATC. Since a great part of the gains from LATC is immediate while a great part of ISTC comes slowly through capital accumulation, this is no surprise that (for the usual values of the parameters) the agent might have a bias towards LATC. It is also no surprise that in such a framework, the more patient the agent is, the more likely he is to choose the ISTC option. This augmented Ramsey approach can explain why the relative price of machinery is higher in less-developed countries. But when we confront it with available macroeconomic data on the early British growth experience (~1770-1913), we find that the timing of the transition from an ISTC-led to an LATC-led growth regime circa 1860 is at odds with the theoretical predictions.

It comes out that the canonical neoclassical framework is more successful in accounting for each growth regime individually than in explaining the transition between them. We argue that, by assuming that the production function is inexorably fixed, the neoclassical model cannot depict capital-labor substitution as a technology choice, while the elasticity of substitution is the ultimate determinant of the direction of technological change.

The rest of the paper is organized as follows. Section 1 presents the augmented Ramsey model and its steady state; section 2 analyzes the endogenous direction of technical change; section 3 summarizes the conclusions and discusses the scope of validity of the approach; section 4 concludes.

Time is discrete and begins at date t = 0. The economy is populated by one eternal and impatient agent, whose preferences represented by:

V (C 0 , C 1 , . . .) = ∞ t =0 1 1 + θ t ln C t
with θ > 0. The agent supplies L units of raw labor at each period, and labor-augmenting productivity is constant at h , so the effective labor supply at each period is at h L . The agent is also initially endowed with a stock of capital K 0 > 0. Production at date t is:

Y t = F (K t , h L ) = h L f (k t )
where K t is the capital stock available at date t , F is a constant-returns-to-scale and diminishingmarginal-returns production function, f is defined by f (•) = F (•, 1), and k t = K t /h L stands for the effective capital-labor ratio. We denote by k = K 0 /h L the initial effective inputs ratio.

Capital is produced linearly from output: one unit of output can be turned into q units of capital q denotes investment-specific productivity. Capital then depreciates in one period, so:

∀t ≥ 0, K t +1 = q (F (K t , h L ) -C t )
or, in intensive notations:

∀t ≥ 0, k t +1 = q f (k t ) - c t h with c t = C t /L . The problem of the agent seen from date t = 0 is: max c 0 ,c 1 ,... ∞ t =0 1 1 + θ t ln c t (1) s.t. k t +1 = q f (k t ) - c t h ( 2
)
k 0 = k
The Lagrangian is:

= ∞ t =0 1 1 + θ t ln c t -λ t k t +1 -q f (k t ) - c t h
Optimality conditions are summarized by the following equations ( 3), ( 4) and ( 5):

∂ ∂ c t = 0 ⇐⇒ 1 c t = q h λ t (3) ∂ ∂ k t +1 = 0 ⇐⇒ λ t = q f k t +1 1 + θ λ t +1 (4) 1 (1 + θ ) t λ t c t t →∞ --→ 0 (5)
The system in the (k , c ) plane obeys to the law: This version of the Ramsey model, augmented with an investment-specific productivity term, admits the traditional phase diagram represented on figure 2. The only non-trivial steady state is saddle-path stable and characterized by:

c t +1 c t = q f ( k t +1) 1+θ k t +1 = q f (k t ) - c t h
q f (k * ) = 1 + θ (6) c * = h f (k * ) - k * q (7)
Notice that the steady-state effective capital-labor ratio k * depends only on q , not on h . Consequently:

h c * ∂ c * ∂ h = 1 (8)
Log-differentiating (6) with respect to q yields:

q k * ∂ k * ∂ q = -f (k * ) k * f (k * ) (9)
Let's denote by σ * the elasticity of substitution and α * the capital share around k * :

σ * = - f (k * ) f (k * ) -k * f (k * ) k * f (k * ) f (k * ) α * = k * f (k * ) f (k * )
Through these definitions, equation ( 9) can be rewritten like:

∂ k * ∂ q = σ * 1 -α * (10)
The steady-state increase in consumption, through equation ( 6) is then:

q c * ∂ c * ∂ q = q f (k * )-k * q ∂ f (k * )-k * q ∂ q = 1 f (k * )-k * q k * f (k * ) σ * 1-α * -k * q σ * 1-α * -1
With the definitions of α * and with the quasi-golden rule identity (6), we can finally write that:

q c * ∂ c * ∂ q = α * + θ α * 1-α * σ * 1 -α * + θ (11)
Let U k , h , q be the value function associated with the problem [START_REF] Acemoglu | Directed Technical Change[END_REF]. Then, by the definition of U :

U k * (q ), h , q = 1 + θ θ ln c * = 1 + θ θ ln h + ln f (k * ) - k * q

Properties of the value function

Let's now see problem (1) from a recursive angle. Bellman's principle of optimality tells us that, at optimum, for each date t ≥ 0:

U k t , h , q = ln c t + 1 1 + θ U k t +1 , h , q (12)
Through equation ( 2), we can write the problem in terms of k t +1 only:

U k t , h , q = max k t +1 ln h f (k t ) - k t +1 q + 1 1 + θ U k t +1 , h , q (13)
The envelope theorem applied to [START_REF] Fellner | Two Propositions in Theory of Induced Innovation[END_REF], taking k t as a parameter, leads to the following identity valid for all t ≥ 0:

∂ U ∂ k (k t ,h ,q ) = f (k t ) f (k t ) - k t +1 q = h f (k t ) c t ( 14
)
By taking k t = k * and c t = c * in equation ( 14) we can derive the value of ∂ U ∂ k at steady state:

∂ U ∂ k (k * (q ),h ,q ) = h f k * (q ) c * (h , q ) or: ∂ U ∂ k (k * (q ),h ,q ) = 1 + θ q f (k * ) -k * q (15)
Let's simply write k = k . Via equation ( 14) and the capital accumulation relationship (2), equation (12) can be written in terms of k only:

U k , h , q = ln h f (k ) ∂ U ∂ k + 1 1 + θ U q f (k ) - f (k ) ∂ U ∂ k , h , q (16)
This equation characterizes the value function, but -to our knowledge -it cannot be solved analytically. In any case, this characterization will bring the results we are looking for.

Impatience and the direction of technological change

Suppose now that the agent is initially at the steady state corresponding to (h , q ). At date t = 0, the agent is offered the following choice : he can choose either to increase q marginally by a factor dz q (dq = q dz q ), or increase h marginally by a factor dz h (dh = h dz h ). In the environment of perfect certainty that we consider here, the agent shall choose the option that increases the most his indirect utility function. Let's analyze successively the two options.

An increase in q

If the agent chooses to increase q at date t = 0, the resulting increase in his indirect utility is 5 :

dU | q = ∂ U ∂ q (k * (q ),h ,q ) dq So we need to derive ∂ U ∂ q (k * (q ),h ,q )
. Let's differentiate equation ( 16) with respect to q , for any k > 0:

∂ U ∂ q ( k ,h ,q ) = - ∂ 2 U ∂ q ∂ k | (k ,h ,q ) ∂ U ∂ k | (k ,h ,q ) + 1 1 + θ     f (k ) - f (k ) ∂ U ∂ k | (k ,h ,q ) + q f (k ) ∂ 2 U ∂ q ∂ k | (k ,h ,q ) ∂ U ∂ k | (k ,h ,q ) 2   ∂ U ∂ k (k + ,h ,q ) + ∂ U ∂ q (k + ,h ,q )   (17)
with

k + = q f (k ) - f (k ) ∂ U ∂ k .
Let's evaluate equation [START_REF] Herrendorf | Which Sectors Make Poor Countries so Unproductive?[END_REF] around steady state, i.e. by taking

k = k * (q ) = k + . Since q f k * (q ) = 1 + θ , the term in cross derivative ∂ 2 U ∂ q ∂ k | (k ,h ,q
) cancels out at steady state, so we simply get:

∂ U ∂ q ( k * (q ),h ,q ) = 1 1 + θ   f k * (q ) ∂ U ∂ k (k * (q ),h ,q ) -f k * (q ) + ∂ U ∂ q ( k * (q ),h ,q )  
Equations ( 6) and ( 15) then yield:

∂ U ∂ q ( k * (q ),h ,q ) = 1 + θ θ 1 q   f (k * ) f (k * ) -k * q -1   = 1 + θ θ 1 q k * q f (k * ) -k * q
Again with (6) and the definition of capital share, we can write down the final result like:

∂ U ∂ q ( k * (q ),h ,q ) = 1 + θ θ 1 q α * 1 -α * + θ
So a marginal increase in q , when the agent is initially at steady state, brings the following increase in indirect utility:

dU | q = 1 + θ θ α * 1 -α * + θ dz q (18)
Remark that, surprisingly enough, this formula is independent of the elasticity of substitution at steady state σ *6 . Thus, even when σ = 0, utility gains from ISTC are strictly positive, while in this case the output never increases when q gets higher. In this case, the agent takes advantage of ISTC by saving on the resources devoted to the production of capital goods and increasing the resources devoted to the production of consumption goods7 .

An increase in h

Suppose now that the agent instead chooses to increase h by a factor dz h . This choice has two immediate consequences: (i) to expand the production possibility frontier, and (ii) to make the effective capital-labor ratio decrease by dk = -k * dz h .

To derive dU | h , it is useful to separate these two effects:

dU | h = ∂ U ∂ k (k * ,h ,q ) dk + ∂ U ∂ h (k * ,h ,q ) dh ( 19
)
Via equations [START_REF] Greenwood | Long-Run Implications of Investment-Specific Technological Change[END_REF], the first part of this expression is simply:

∂ U ∂ k (k * ,h ,q ) dk = - 1 + θ q f (k * ) -k * q k * dz h = - k * f (k * ) f (k * ) - k * f (k * ) 1+θ dz h
and so:

∂ U ∂ k (k * ,h ,q ) dk = -(1 + θ ) α * 1 -α * + θ dz h
The second term in the right-hand side of ( 19) is the increase in the value function when h increases but keeping the effective capital-labor ratio constant. The problem like formulated in (1) makes it clear that, when the initial effective capital-labor ratio is held constant, optimal consumption at any date increases proportionally with h ; so if c t denotes optimal consumption at date t , then:

∂ c t ∂ h = c t h Thus, for any effective capital-labor ratio k , the value function verifies:

∂ U ∂ h (k ,h ,q ) = ∞ t =0 1 1+θ t ∂ ln c t ∂ h = 1+θ θ 1 h
In particular, this is true when the economy is initially at steady state, so:

∂ U ∂ h (k * ,h ,q ) dh = 1 + θ θ dz h ( 19
) then gives the final result:

dU | h = 1 + θ θ 1 - θ α * 1 -α * + θ dz h (20)

Synthesis

The relative (subjective) benefits of ISTC over LATC are, then:

dU | q dU | h = α * 1-α * +θ 1 -θ α * 1-α * +θ dz q dz h = 1 1 + θ α * 1 -α * dz q dz h (21)
The agent:

-chooses to undertake ISTC if

1 1+θ α * 1-α * dz q > dz h ; -chooses to undertake LATC if 1 1+θ α * 1-α * dz q < dz h .
So -everything else equal -the more impatient the agent is (i.e. the higher θ ), the less likely he is to undertake ISTC.

Let's compare criterion [START_REF] Jones | Economic Growth and the Relative Price of Capital[END_REF] with the relative increase in steady-state consumption 8 . From ( 8) and ( 11), we have:

q ∂ c * ∂ q h ∂ c * ∂ h = α * + θ α * * σ * 1 -α * + θ ( 23
)
so the increase in q brings a higher increase in steady-state consumption than the increase in h if and only if:

α * + θ α * 1-α * σ * 1 -α * + θ dz q > dz h
When θ > 0, it is straightforward to see that:

dU | q d U | h < q ∂ c * ∂ q h ∂ c * ∂ h ⇐⇒ 1 1 + θ α * 1 -α * < α * + θ α * 1-α * σ * 1 -α * + θ ⇐⇒ σ * > α * 1 1 + θ ( 24
)
This result can be stated the following way: when the elasticity of substitution is greater than the product of the capital share and the rate of preference for the present, the agent grants relatively less value to ISTC than the relative effect of ISTC on steady-state consumption. If we denote by β = 1/1 + θ the rate of preference for the present, this criterion can be simply written "σ > αβ ".

Since the agent is impatient, he does not only take into account the steady-state consequences of each choice, but rather perfectly anticipates the dynamic consequences of each choice.

When the agent chooses to increase h , he gets some immediate benefits since production increases at date t = 0; the rest of the benefits come from capital accumulation.

If he chooses to increase q , he gets some positive benefits at date t = 0 because capital is cheaper to produce, so everything else equal the agent can consume more. But most of the benefits come through capital accumulation, from t = 1 on. 8 The steady-state increases in consumption-value of product per capita y * = h f (k * ) after the h -and q -shock are:

h y * ∂ y * ∂ h = 1 q y * ∂ y * ∂ q = α * 1-α * σ * (22)
We have proved that when the hypothesis σ > α 1 1+θ holds, the agent prefers the consumption path resulting from an increase in h rather than the one resulting from an increase in q . With the usual values of the parameters (σ ≈ 1, α ≈ 1/3, θ > 0) this inequality is fulfilled.

In the contrary, when θ = 0 the agent attempts to increase the steady-state value of consumption, without valuing transitional dynamics. In this case, ( 21) and ( 23) are both equal to α * /1 -α * .

The Cobb-Douglas case

The case where the production function is Cobb-Douglas of parameter α takes a very simple form. With σ * = 1, equation ( 23) becomes:

q ∂ c * ∂ q h ∂ c * ∂ h = α + θ α 1-α 1 -α + θ = α 1 + θ 1-α (1 -α) 1 + θ 1-α = α 1 -α
Thus, while increasing q rather than h brings an increase in steady-state consumption equal to α 1-α dz q rather than dz h , the agent only chooses to increase q rather than h if

1 1+θ α 1-α dz q dz h > 1.
3 How useful is the Ramsey approach to the endogenous direction of technological change?

The 'story' told by the framework set above might be summarized as follows: different forms of technological change lead to qualitatively different paths of consumption, so an impatient Ramsey agent will not necessarily pick the highest-growth option. When the parameters of the model satisfy a certain criterion -'σ > αβ ' -the consumption growth path resulting from the disembodied form of technological change (LATC) is preferred to the consumption growth path resulting from the embodied form of technological change (ISTC); in this case ISTC is undervalued relatively to LATC.

So the Ramsey approach to the direction technological change is able to account for the negative relationship between GDP per capita and the relative price of investment in cross-section. The model depicts rich countries as rich because they have undertaken superior -albeit longerforms of technological change in the past, resulting both in a lower relative price of investment goods over consumption goods and in a higher GDP per capita today.

If the approach we have presented above is correct, then the right policy to stimulate growth is to tax labor and/or subsidize capital, with the objective of increasing the relative profitability of ISTC over LATC. However, the predictions of the model lie in contrast with both cross-section evidence as well as with the early growth episode in Britain (~1770-1913). However appealing, it comes out that the Ramsey approach is a bad guide to the transition from a low-growth to a highgrowth path.

Growth and capital shares in cross-sections of countries

The model predicts that countries with a labor share above some threshold [1 -α > 1 -ᾱ] will choose to undertake LATC and grow at rate dz h , while countries with a labor share below this threshold [1 -α < 1 -ᾱ] will choose to undertake ISTC and grow at the rate α 1-α dz q . We have seen that if the parameters of the model satisfy the σ > αβ assumption, then the rate of economic growth is always greater under ISTC than under LATC. The situation is represented on figure 3 9 .

Figures 4 represents the cross-section relationship between GDP per capita and the labor share in all OECD member economies except Chile 10 . Data on the labor share is taken from the OECD economic database, which is accessible at http://stats.oecd.org/. We use the 'Labor Income Share' variable, which is available in 2006 for all thirty-three countries. As we see, there is no clear 9 By restricting to the problem around steady state, we did not explore the dynamic feedback between the direction of technological change and the capital share -in this case the expected elasticity of substitution might influence the dynamic path of the economy, contrary to the criterion resting only on the capital share that we derived in equation [START_REF] Jones | Economic Growth and the Relative Price of Capital[END_REF].

Unfortunately, we are unable at this stage to solve the dynamic problem of the endogenous direction of technological change that would lead to asymptotic economic growth :

                 max (ct ,ψ t ) t ≥0 ∞ t =0 1 1+θ t ln c t s.t. k t +1 = q t f (k t ) - c t h , k 0 = k q t +1 = (1 + zq ) ψ t q t , q 0 = q h t +1 = (1 + zh ) 1-ψ t h t , h 0 = h 0 ≤ ψ t ≤ 1 (25)
Still, there is strong presumption that, in the dynamic framework in [START_REF] Piketty | Capital is Back: Wealth-Income Ratios in Rich Countries 1700-2012[END_REF], if the elasticity of substitution is everywhere less than unity, then the relative benefits of ISTC decrease when q increases (i.e. as ISTC is undertaken), and that if the elasticity of substitution is everywhere greater than unity then the relative benefits of ISTC increase when q increases. See chapter 2. 10 Countries represented on figure 4 are: Australia, Austria, Belgium, Canada, the Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Israel, Italy, Japan, Korea, Luxembourg, Mexico, the Netherlands, New Zealand, Norway, Poland, Portugal, the Slovak Republic, Slovenia, Spain, Sweden, Switzerland, Turkey, the United Kingdom, and the United States. relationship between income per capita and the labor share for the whole sample. Contrary to what was predicted by theory, this relationship is rather positive than negative and the least advanced countries of our sample display on average higher capital shares.

Nevertheless, the right part of figure 3 is clearly observable. If we remove the seven countries with a GDP per capita less than US$20,000 11 , the relationship becomes clearly negative and significant, like shown on figure 5. Thus, when we restrict our attention to the most advanced countries only, there is a positive link between the capital share and GDP per capita, as predicted by the neoclassical growth framework under ISTC 12 .

The Industrial Revolution in Great Britain: ~1770-1913

We have seen in chapter 3 that the first phase of the Industrial Revolution, from the late eighteenth to the mid-nineteenth century, was best described as fueled by ISTC, coupled with capital/labor gross substitutability. From the point of view of the distribution of income, the capital share increased at the expense of the labor and of the land share 13 . During this period, the investment rate increased 14 , while according to recent estimates by Piketty and Zucman (2014) the ratio of non-housing and non-land domestic capital to GDP increased from around 200 percent in 1750 to around 300 percent in 1855 (figure 6), as one would expect in a growth regime led by ISTC coupled 11 These countries are the Czech Republic, Estonia, Hungary, Mexico, Poland, the Slovak Republic and Turkey. 12 The same pattern emerges when we plot PPP GDP per capita instead of GDP per capita. 13 See Allen (2009). Indeed, real wages did not increase at all before the mid-nineteenth century. 14 [START_REF] Crafts | British Economic Growth during the Industrial Revolution[END_REF] estimates that the investment rate grew from 6.0 percent in 1760 to 11.7 percent in 1831.

with capital-labor gross substitutability (σ > 1).

From the mid-nineteenth century to the First World War, human capital clearly become the prime engine of growth. Literacy rate soars and the investment rate stabilizes 15 , while according to Piketty and Zucman (2014) the ratio of non-housing/non-land domestic capital stabilizes as well (figure 6). This is beyond dispute that the early economic growth episode in Britain has witnessed two distinct phases, the first one being dominated by physical capital and the second one by human capital. The neoclassical framework does a good job at explaining each economic growth regime. But how well does our augmented Ramsey framework perform in accounting for the transition between the first to the second one?

Poorly, indeed. First, if we accept that the Industrial Revolution began somewhere in the eighteenth century, then according to Allen's (2009) data we must recognize that it happened at a moment when the labor share was relatively high. In the contrary, the transition to a human capitalled path by the mid-nineteenth century happened when the labor share was relatively low -exactly the opposite of what is expected from theory. Secondly, we do not see any growth collapse with the transition to the human capital regime; instead, as is well-known the rate of economic growth seems to have been higher in the latter part of the nineteenth century than during the period from around 1770 to around 1850 16 .

So our approach does a poor job at accounting for the switch of economic growth regime by the mid-nineteenth century in the UK.

Conclusion

The analysis of economic growth through the variety of directions of technological change is appealing because it might bring insights into the diversity of growth regimes without resorting to any externality or non-linearity.

In this paper, we have analyzed the problem of an impatient Ramsey agent (with logarithmic utility) facing the choice between improving marginally either on labor-augmenting productivity or on investment-specific productivity while initially at steady state. We have found that this choice was governed by a very simple criterion involving only the ratio of the capital share to the labor share and the impatient rate. The prime message is that the benefits from LATC depend on the labor share -i.e. on the elasticity of output with respect to labor -while the benefits from ISTC depend on the capital share -i.e. on the elasticity of output with respect to capital.

The main benefit of the Ramsey approach to the direction of technological change is that it can be brought to data. As we have seen in section 3, it seems that this approach performs well in explaining the behavior of economies inside each regime, but does a bad job in accounting for the transition between different growth regimes led by different forms of technical change.

Indeed, the evidence we have exhibited here is completely opposed to the conclusions of the model: poor countries seem to display higher capital shares than rich countries, while the early British growth experience shows a change of regime towards ISTC by the end of the eighteenth 15 See the review of literature in [START_REF] Galor | From Stagnation to Growth: Unified Growth Theory[END_REF]. 16 Crafts and Harley (1992). century -when the labor share seem to have been relatively high -and eventually transited to an LATC-led path by the mid-nineteenth century, when the labor share was relatively low.

To our knowledge, the only piece of theory able to account for this fact is found in Habakkuk (1962) -who argue that the dearness of labor in the USA was itself an incentive for designing laboreconomizing machines. The 'Habakkuk thesis' is not neoclassical in nature, since it assumes that wages are determined independently from the marginal productivity of labor -namely, by the option value of working in agriculture.

Figure 1 :

 1 Figure 1: The negative relationship between the relative deflator of investment over consumption and PPP GDP per capita; source: Penn World Table 8.0 (p I /p C ) and IMF WEO (PPP GDP per capita).

Figure 2 :

 2 Figure 2: The phase diagram in the absence of technical change.

Figure 3 :

 3 Figure 3: The model's predicted relationship between the labor share and the rate of economic growth under the assumption that σ > αβ .

Figure 4 :

 4 Figure 4: The cross-section relationship between GDP per capita and the labor share in 33 OECD member countries; source: OECD StatExtracts (labor share) and IMF WEO October 2013 (GDP per capita).

Figure 5 :

 5 Figure 5: Idem figure 4 for countries with GDP per capita > US$20,000 (26 countries).

Figure 6 :

 6 Figure 6: The estimated ratio of non-land, non-housing domestic capital to GDP, UK, 1700-1913; source: Piketty and Zucman (2014).

The augmented Ramsey steady stateIn this section, we present the model with no technological change.equilibrium on the direction of technical change. The prime intuition is due toHicks (1932), who claimed that 'a change in the relative share of factors of production is itself a spur to innovation and inventions of a particular type -directed at economizing the use of a factor which has become relatively expensive.[START_REF] Fellner | Two Propositions in Theory of Induced Innovation[END_REF],[START_REF] Kennedy | Induced Bias in Innovation and the Theory of Distribution[END_REF],[START_REF] Samuelson | A Theory of Induced Innovation Along Kennedy-WeisÃd'cker Lines[END_REF],[START_REF] Drandakis | A Model of Induced Innovation, Growth and Distribution[END_REF],[START_REF] Nordhaus | The Optimal Rate and Direction of Technical Change[END_REF][START_REF] Nordhaus | Some Skeptical Thoughts on the Theory of Induced Innovation[END_REF] and[START_REF] Acemoglu | Directed Technical Change[END_REF]

2003) have tackled this issue. To our knowledge, what follows constitutes the first analysis of induced innovation considering investment-specific (rather than capital-augmenting) technical change.[START_REF] Armenter | Accounting for Development through Investment Prices[END_REF] This distinction is consistent with the idea that one can improve on his skills or on his tools to become more productive.

Remark that initial effective capital-labor ratio is unaffected, since the capital available at date t = 0 has been accumulated at date t = -1, and that neither h nor L is altered.

This result is even more intriguing when we refer to equation[START_REF] Drandakis | A Model of Induced Innovation, Growth and Distribution[END_REF], which shows that the steady-state consequences of an investment-specific technological shock do depend on the elasticity of substitution. Indeed, σ * does not even appear in the derivation of dU | q ; by the method used above the second derivative of f does not come up.

See chapter 1.

We believe that endogenizing the elasticity of substitution σ and thereby, the production function, would constitute a bridge between the theory that explains well the shape of the growth regimes independently (the neoclassical growth theory) and the theory that explains the switches between regimes (the Habakkuk thesis).


Mehdi Senouci
email: mehdi.senouci@ecp.fr

The Habakkuk hypothesis in a

Keywords: Economic growth, labor share, capital-labor substitution, endogenous growth theory

neoclassical framework (To make more with less or to make more with more: that is the question)

Introduction

In his famous 1962 essay American and British Technology in the Nineteenth Century: The Search for Labour-Saving Inventions, Habakkuk argued that high wages in the USA relatively to Great Britain encouraged American entrepreneurs to take on relatively more labor-saving methods than their British counterparts, which ultimately resulted in the American production system using relatively more capital and less labor.

This point -which was raised by [START_REF] Rothbarth | Causes of the Superior Efficiency of U.S.A. Industry as Compared with British Industry[END_REF] before -has since then gained considerable influence on the history of technology 1 . The Habakkuk hypothesis is also deeply seated in the field of economic growth, though theorists do not agree on the exact meaning to give to a labor-saving invention, nor on the exact framework to have in mind 2 .

Models in neoclassical spirit usually assume that technological change is representable by the alteration of some productivity term (Hicks-neutral, Harrod-neutral, investment-specific, etc.) 3 . The purpose of this paper is to show that by not restricting ourselves to a fixed production function, we are able to unveil some new and interesting growth regimes, and shed new light on the issue of how economies switch from one growth regime to another 4 .

The structure we use is the simple discrete-time Ramsey model with 100 percent depreciation rate and a strictly positive impatience rate. We suppose that while initially at steady state, the Ramsey agent might rotate the production function clockwise or anticlockwise. In the first case, the agent switches to a production technique that is more efficient than the old technique only for lower levels of capital; in the second case the agent switches to a production technique that is more efficient than the old technique only for higher levels of capital 5 .

Thus, the central assumption we take is that technological change do not affect the product immediately but only with the convergence to the new steady state. We respectively dub 1 Among many: [START_REF] Hayami | Factor Prices and Technical Change in Agricultural Development: The United States and Japan, 1880-1960[END_REF] compare the patterns of technical change in Japan and the US between 1880 and 1960 and conclude that factor prices (land and labor) were the main drivers of the direction of innovation; [START_REF] Allen | The British Industrial Revolution in Global Perspective[END_REF] argues that relatively high wages and low energy prices in Great Britain vis-à-vis the continent explains why the Industrial Revolution (which the author sees as an episode of substitution of capital and power for labor) originated in this country. For a conflicting view, see [START_REF] Temin | Labor Scarcity and the Problem of American Industrial Efficiency in the 1850's[END_REF][START_REF] Temin | Labor Scarcity in America[END_REF] and [START_REF] Field | Land Abundance, Interest/Profit Rates, and Nineteenth-Century American and British Technology[END_REF]. 2 Acemoglu (2010) analyzes the link between labor scarcity and the rate of innovation in a range of neoclassical models, and shows that labor scarcity makes innovation accelerate when technology advances are 'strongly labor complementary', i.e. when they make the marginal product of labor decrease. 3 One reason for the success of the approach based on productivity terms lies in the Uzawa-Robinson, which states that if the growth rate of a neoclassical growth model is constant, then technical change (i ) is representable by a steady increase in some productivity term, and (i i ) that this productivity term is the Harrod-neutral productivity term. What the traditional interpretation of this result misses is the fact that the growth rate of a growth model does not have to be strictly constant to reproduce the Kaldor facts. We know from [START_REF] Jones | The Shape of the Production Function and the Direction of Technical Change[END_REF] and León-Ledesma and Satchi (2013) that there exist neoclassical growth models where technical change reflects itself in alterations of the production function -rather than in increases of a productivity index of some unaltered production function -that generate the Kaldor facts -see also the parts of [START_REF] Jones | Growth, Capital Shares, and a New Perspective on Production Functions[END_REF] that was not published in [START_REF] Jones | The Shape of the Production Function and the Direction of Technical Change[END_REF], Growiec (2008) and Allen (2012). 4 We have seen in [START_REF] Senouci | The Endogenous Direction of Technological Change in a Discrete-Time Ramsey Model[END_REF] that the switches of regime in 18 th and 19 th century Great Britain were not consistent with a laboraugmenting vs. investment-specific technical change approach, and that among OECD countries, the relatively less developed countries display higher capital shares than advanced countries. Our approach is consistent with these empirical facts. On the relationships between capital share and GDP per capita, see [START_REF] Gollin | Getting Income Shares Right[END_REF], [START_REF] Rodriguez | Are Capital Shares Higher in Poor Countries? Evidence from Industrial Surveys[END_REF] and [START_REF] Maarek | Labor Share, Informal Sector and Development[END_REF]. 5 Since different production techniques most often make use of qualitatively different capital goods, we suppose that each type of capital is produced from the consumption good through a linear technology, then we scale all production functions to define them over the argument 'consumption-goods-expenses on capital goods' rather than 'physical units of capital', in order to have a common unit for inputs.

In what follows we simply refer to this variable as 'capital'.

as 'complementation' and 'substitution' the first and second type of technological change 6 .

The distinction between capital-saving and labor-saving forms of technical change is common in economic thought 7 . However, neoclassical theory so far has not seen a satisfying, simple way to incorporate these forms of technical change into standard models 8 .

Complementation increases the marginal product of labor (without affecting product) around the initial steady state, and triggers convergence towards a new steady state with lower capital expenses and lower product; while substitution increases the marginal product of capital (without affecting product) around the initial steady state and triggers convergence towards a new steady state with higher capital expenses and higher product. Since we assume that the quantity of labor is fixed in the economy, these changes ultimately reflect themselves respectively into lower and higher capital-intensity. In this sense, we can qualify complementation as the capital-saving direction, and substitution as the labor-saving direction of technical change.

Thus, both types of technological change lead to totally different dynamic (and steadystate) outcomes; however, this is very simple to prove that both types of technological change are desirable from the point of view of the agent -both strictly increase his intertemporal utility.

We investigate two types of paths, both from a successive-steady states approach: (a) paths driven by exogenous technological change, and (b) paths with endogenous choice of the direction of technological change. The first issue needs only the specification of the family of production functions that we consider. The second issue requires to have knowledge of the value function of the Ramsey problem, which (as is well-known) we do only under very narrow assumptions. Thus, at this stage we are able to analyze exogenous-technological change paths under the assumptions that the production functions are either Cobb-Douglas or CES; and we are able to analyze endogenous-direction of technological change paths under the assumptions that the production functions are Cobb-Douglas and that the instantaneous utility function is logarithmic.

We prove the following results. When considering exogenous technological change, we find that under 'steady comple- 6 For instance, the early cars constituted a substitution change (away from horses), while later the invention of less expensive cars constituted a complementation change. 7 See the excellent discussion by [START_REF] Rosenberg | Capital Goods, Technology, and Economic Growth[END_REF]. 8 Atkinson and Stiglitz (1969) first introduced the concept of "localized" technical progress, and qualitatively analyzed its influence on the production function. [START_REF] Barrère | Capital Intensity and the Combination of Factors of Production[END_REF], [START_REF] Champernowne | A Dynamic Growth Model Involving a Production Function[END_REF] and [START_REF] Zeira | Workers, Machines, and Economic Growth[END_REF][START_REF] Zeira | Machines as Engines of Growth[END_REF] investigate models where growth is driven by substitution of capital for labor; [START_REF] Seater | Share-Altering Technical Progress[END_REF], Zuleta (2008) and Peretto and Seater (2013) present frameworks where technical change closely resemble the one we consider here. The "putty-clay" literature has also tackled the issue of the determination of capital-intensity as a technology phenomenon -see, for instance, Caballero and Hammour (1998) and [START_REF] Gilchrist | Putty Clay and Investment: A Business Cycle Analysis[END_REF]. [START_REF] Allen | Technology and the Great Divergence: Global Economic Development Since 1820[END_REF] investigates the empirical evolution of the World production function since 1820 and concludes that:

"It is remarkable that countries in 1990 with low capital labor ratios achieved an output per worker that was no higher than countries with the same capital labor ratio in 1820. In the course of the last two hundred years, the rich countries created the production function of the world that defines the growth possibilities of poor countries today." (Allen (2012, abstract))

mentation' consumption and output converge (from above) to constants, while capital expenses and the capital share both tend to zero, in both the Cobb-Douglas and CES cases. In the 'steady substitution' regime, the growth rates of output, capital expenses and consumption converge to some common strictly positive limit, while the capital share tends to one in the limit -in both the Cobb-Douglas and CES cases 9 .

When the direction of technological change is endogenous, we find a bias towards complementation whenever the agent is impatient: in the most general case with whatever family of production functions, the agent might increase more steady-state consumption through substitution than through complementation but still choose complementation for the shortterm rewards it brings. With the assumptions that the production functions are Cobb-Douglas and that the utility function is logarithmic, we find support for the Habakkuk hypothesis in the sense that the agent never undertakes pure complementation asymptotically: whenever the labor share decreases below some threshold, the agent switches to substitution -which increases capital share. Furthermore, when the potential rate of complementation is greater than the potential rate of substitution, the agent periodically changes the direction of technological change; this regime yields dynamics that fulfill the Kaldor facts while depicting the stationarity of the labor share as an endogenous, technological phenomenon.

The rest of the paper is organized as follows. Section 1 sets the basic Ramsey framework with no technological change. Section 2 introduces the two forms of technological change. Section 3 analyzes the patterns of exogenous growth with Cobb-Douglas and CES production functions. Section 4 treats the endogenous direction of technological change with Cobb-Douglas production functions and logarithmic utility and the resulting growth paths. Section 5 concludes.

The Ramsey framework and the value function

We use the discrete-time version of the Ramsey model with total depreciation. In the absence of technological change, the problem of the agent is:

max (c 0 ,c 1 ,...) ∞ t =0 1 1+θ t u (c t ) s.t. x t +1 = f ϕ (x t ) -c t x 0 = x (1)
where f ϕ is a neoclassical production function. This is well-known that problem (1) converges through a saddle path to the interior steady state10 :

f ϕ x * ϕ = 1 + θ
We call A * ϕ the value of production at steady state:

A * ϕ ≡ f ϕ x * ϕ
Consumption at steady state is then:

c * ϕ = A * ϕ -x * ϕ
The competitive capital share at any date t is:

α t = x t f ϕ (x t )
f ϕ (x t ) , so at steady state:

α * ϕ = x * ϕ f ϕ x * ϕ f ϕ x * ϕ = (1 + θ ) x * ϕ A * ϕ > 0 (2)
Let's denote by U x 0 ; ϕ the value function corresponding to problem (1) when the production function is f ϕ . We already know that:

U x * ϕ ; ϕ = 1 + θ θ u c * ϕ (3)
Besides, the recursive formulation of problem (1) is:

U x 0 ; ϕ = max x 1 u f ϕ (x 0 ) -x 1 + 1 1 + θ U x 1 ;
ϕ so the envelope theorem applied to any x 0 > 0 yields:

∂ U x 0 ; ϕ ∂ x 0 = f ϕ (x 0 ) u f ϕ (x 0 ) -x * 1 (x 0 ) = f ϕ (x 0 ) u c * 0 (x 0 ) , (4)
where c * 0 (x 0 ) stands for the optimal choice of c 0 given x 0 in problem (1) and where

x * 1 (x 0 ) = f ϕ (x 0 ) -c * 0 (x 0 ).
In particular, equation (4) applied at steady state x 0 = x * ϕ yields:

∂ U ∂ x x * ϕ ;ϕ = (1 + θ ) u c * ϕ (5)

The technological opportunities: complementation vs. substitution

Suppose now that the agent initially stands at the steady state corresponding to technique ϕ. At date t = 0, function f ϕ is altered and turned into function f ϕ . The specific assumption we make on f ϕ is: Assumption. The new production function yields exactly the same output than the old production function at the old steady state:

f ϕ x * ϕ = f ϕ x * ϕ = A * ϕ
Thus the altering of the production function does not influence production at date t = 0 and only does so through capital decumulation or accumulation. The agent will converge to the new steady state characterized by:

f ϕ x * ϕ = 1 + θ
As shown on figures 1 and 2, the location of the new steady state depends on the value of the derivative of the new production function f ϕ at the original capital level x * ϕ . Precisely:

-if f ϕ (x * ϕ ) < 1 + θ , then x * ϕ < x * ϕ and A * ϕ < A * ϕ ; -if f ϕ (x * ϕ ) > 1 + θ , then x * ϕ > x * ϕ and A * ϕ > A * ϕ .
In the first case (figure 1), we say that technological change takes the 'complementation' form; while in the second case (figure 2) we talk about 'substitution'. The net welfare gains due to the passage from technique ϕ to ϕ while initially at the ϕsteady state are:

∆U ϕ→ϕ = U x * ϕ , ϕ -U x * ϕ , ϕ = U x * ϕ , ϕ - 1 + θ θ u c * ϕ Notice that if f ϕ (x * ϕ ) = 1 + θ
, the new steady state coincides with the original one so the altering of the production function has strictly no effect on the consumption path, and in particular ∆U ϕ→ϕ = 0.

In all other cases, we can state a strikingly simple result:

Theorem 1. If Assumption 2 holds and if f ϕ x * ϕ = 1 + θ , then: ∆U ϕ→ϕ > 0.
Proof.

At date t = 0, the agent owns x * ϕ capital with which (Assumption 2) he manages to produce

A * ϕ = f ϕ (x * ϕ ).
If he saves x * ϕ at each period, he will be able to consume c * ϕ at each period and so ensure inter-temporal utility 1+θ θ u c * ϕ . So under Assumption 2, the consumption path ∀t ≥ 0, c t = c * ϕ is feasible through technique ϕ (see figures 1 and 2). Then: ∆U ϕ→ϕ ≥ 0.

If further f ϕ (x * ϕ ) = 1+θ , then the agent can strictly increase his inter-temporal utility by marginally adjusting the capital level towards x * ϕ = x * ϕ . More precisely:

-If f ϕ (x * ϕ ) < 1 + θ , the agent strictly increases inter-temporal utility by consuming marginally more than c * ϕ and decreasing capital level marginally below x * ϕ at date t = 0; -If f ϕ (x * ϕ ) > 1 + θ , the agent strictly increases inter-temporal utility by consuming marginally less than c * ϕ and increasing capital level marginally above x * ϕ at date t = 0.

Thus, whenever

f ϕ (x * ϕ ) = 1 + θ , ∆U ϕ→ϕ > 0.
So the two directions of technological change increase the subjective welfare of the agent. The generality of this result is noteworthy, and indicates that the switch from technique ϕ to ϕ while initially at steady state corresponding to ϕ must represent an economy on scarce factors of production.

Both technological shocks leave product unchanged at date t = 0 but influence consumption at date t = 0. The agent initiates convergence to the new steady state x * ϕ , A * ϕ by choosing some consumption level c * 0ϕ which is strictly less than c * ϕ right after a complementary shock, and strictly greater than c * ϕ right after a substitution shock. It is also true in some sense that the most 'aggressive' the alteration of the first derivative of the production function around the original steady state, the higher the increase in indirect utility will be. We make it clear in the following proposition: Proposition 1. Let ϕ be a family of neoclassical production functions satisfying Assumption 2, and such that for all functions f ϕ 1 , f ϕ 2 ∈ ϕ , f ϕ 1 and f ϕ 2 only cross at x * ϕ . Then:

-If f ϕ 2 (x * ϕ ) < f ϕ 1 (x * ϕ ) < 1 + θ , then: ∆U ϕ→ϕ 2 > ∆U ϕ→ϕ 1 (> 0); -If f ϕ 2 (x * ϕ ) > f ϕ 1 (x * ϕ ) > 1 + θ , then: ∆U ϕ→ϕ 2 > ∆U ϕ→ϕ 1 (> 0) .
Proof. See appendix.

The situation is depicted on figures 3 and 4. The formal proof reflects the property that, with the notations we have taken, f ϕ 2 always lies above f ϕ 1 in the interval of capital that is reached by the agent after each type of shock: in the complementation case (figure 3), the agent tends to decrease x , but f ϕ 2 lies above f ϕ 1 over the (0, x * ϕ ) interval so ϕ 2 is preferred; in the substitution case (figure 4), the agent tends to increase x , but f ϕ 2 lies above f ϕ 1 over the (x * ϕ , +∞) interval so ϕ 2 is preferred. At this level of generality regarding the production functions and the utility function, we have proven through theorem 1 that the agent always benefits from any 'rotation' of the production function around the steady state (x * ϕ , A * ϕ ) corresponding to some technique ϕ, and through proposition 1 that for any direction, he prefers more rotation than less.

The steady-state consequences of complementation and substitution

Again at this level of generality, we can derive some results on the steady-state consequences of technological change:

Proposition 2. If f ϕ > 1 + θ (substitution case), then: c * ϕ > c * ϕ .
Proof. In all cases, variation in inter-temporal utility can be written:

∆U ϕ→ϕ = U x * ϕ ; ϕ -U x * ϕ ; ϕ - x * ϕ x * ϕ ∂ U (x ; ϕ ) ∂ x dx = 1 + θ θ u c * ϕ -u c * ϕ - x * ϕ x * ϕ f ϕ (x ) u c * 0ϕ (x ) dx
where c * ϕ is the new steady-state value of consumption and where c * 0ϕ (x ) denotes the optimal choice of consumption at date t = 0 in problem (1) with technique ϕ and initial capital x > 0.

When

f ϕ > 1 + θ , x * ϕ < x * ϕ . So x * ϕ x * ϕ f ϕ (x ) u (c * 0ϕ (x )) (>0)
dx > 0 in this case (substitution).

From theorem 1,

∆U ϕ→ϕ > 0. If c * ϕ ≤ c * ϕ , then u (c * ϕ ) ≤ u (c * ϕ )
then ∆U ϕ→ϕ < 0 from the formula above, which constitutes a contradiction to theorem 1.

If f ϕ < 1 + θ (complementation case), c *
ϕ might be greater or less c * ϕ . Indeed, when θ > 0, the agent may increase steady-state consumption by economizing on capital, but his strictly positive impatience might also lead him to take advantage of the short-term opportunity of capital decumulation at the expense of steady-state consumption.

Exogenous growth with Cobb-Douglas/CES production functions

We now proceed to the analysis of the exogenous growth paths under the assumption that the production functions are all either Cobb-Douglas, or CES.

The Cobb-Douglas case

In this part we assume that all the production functions f ϕ we consider are of Cobb-Douglas type.

For any x * > 0, A * > 0 and α ∈ (0, 1), the only Cobb-Douglas production function of capital share parameter α and for which f (x * ) = A * and for is given by formula:

∀x > 0, f (x ) = A * x x * α
We follow this method and write down all functions f ϕ with respect to the steady-state corresponding to problem (1):

∀x > 0, f ϕ (x ) = A * ϕ x x * ϕ α ϕ with A * ϕ > 0,
x * ϕ > 0 and α ϕ ∈ (0, 1). In both the complementation and the substitution cases, the opportunity that is given to the agent is to alter parameter α ϕ , that is, to move to technique ϕ such that:

∀x > 0, f ϕ (x ) = A * ϕ x x * ϕ α + (6) so if α + < α ϕ , then f ϕ (x * ϕ ) < 1 + θ so technological change takes the form of complemen- tation, while if α + > α ϕ then f ϕ (x * ϕ ) > 1 + θ ,
which means the agent undergoes substitution.

Production function f ϕ can also be written with respect to the steady state corresponding to technique ϕ :

∀x > 0, f ϕ (x ) = A * ϕ x x * ϕ α ϕ (7)
The identification of parameters in ( 6) and ( 7) yields:

α ϕ = α + (8) A ϕ A ϕ = x ϕ x ϕ α ϕ (9)
On the other hand, equation ( 2) applied to techniques ϕ and ϕ yields:

α ϕ = (1 + θ )x * ϕ A * ϕ ( 10
)
α ϕ = (1 + θ )x * ϕ A * ϕ ( 11
)
We get the final result from equations ( 2), ( 8) and ( 9); when the agent alters technique ϕ by turning α ϕ into α + = α ϕ , the parameters of the new production function are:

x * ϕ = x * ϕ α + α ϕ 1 1-α + = x * ϕ α ϕ α ϕ 1 1-α ϕ ( 12
)
A * ϕ = A * ϕ α + α ϕ α + 1-α + = A * ϕ α ϕ α ϕ α ϕ 1-α ϕ ( 13
)
Figure 5: Post-complementation shock dynamics (Assumptions:

α 0 = 0.3; A * ϕ = 1; θ = 0.1; α ϕ = (α ϕ ) 1.5 ).
We can check that if α + < α ϕ , then

x * ϕ < x * ϕ and A * ϕ < A * ϕ ; while if α + > α ϕ , then x * ϕ > x * ϕ and A * ϕ > A * ϕ .
Equations ( 10), ( 11), ( 12) and ( 13) permit to derive the variation of steady-state consumption:

c * ϕ = A * ϕ -x * ϕ = A * ϕ α ϕ α ϕ α ϕ 1-α ϕ -x * ϕ α ϕ α ϕ 1 1-α ϕ =⇒ c * ϕ = c * ϕ 1 + θ -α ϕ 1 + θ -α ϕ α ϕ α ϕ α ϕ 1-α ϕ ( 14
)
By proposition 2 we know that when α ϕ > α ϕ it always holds that c * ϕ > c * ϕ . We can also see why we cannot conclude if α ϕ < α ϕ .

Post-shock dynamics with Cobb-Douglas production functions and log utility -Solow

When the production function is Cobb-Douglas and when utility is logarithmic, this is wellknown that substitution and wealth effects cancel out so the agent simply chooses the constant saving rate s = α/1 + θ at each period. The model is then isomorphic to a Solow model and we can easily simulate convergence to the new steady state 11 . Figure 5 shows one convergence path for a complementation case by assuming α ϕ = (α ϕ ) 1.5 , figure 6 does the same in the substitution case assuming α ϕ = (α ϕ ) 0.5 . In both cases we assume α ϕ = 0.3, A * ϕ = 1 and θ = 0.1. We represent the system of the system from date t = -1 on, where t = -1 that corresponds to the ϕ-steady state.

In all cases production (y t ) and capital (x t ) remain unaffected at date t = 0. However, consumption changes from date t = 0.

The complementation shock (figure 5) induces the agent to increase consumption (and so make capital decrease) at t = 0; the new steady state features lower GDP, lower capital and, in this case, higher consumption. Figure 7 changes the value of the impatience rate to θ = 2; in this case, the new steady-state consumption is strictly less than the original steady-state consumption.

The substitution shock (figure 6) induces the agent to decrease consumption (and so make capital increase) at t = 0 and the new steady state has higher GDP, higher capital and higher consumption (proposition 1). 11 The software used is Microsoft Excel 2010. a) The complementation case -Suppose now that the agent undergoes successive technological shocks of the form: α +,n = α 1+µ n with µ > 0. If we denote respectively by A n , x n and c n the values of production, capital and consumption at each steady state, then from equations ( 8), ( 12), ( 13) and ( 14):

The sequence of steady states with Cobb-Douglas production functions

α n+1 = α 1+µ n ( 15
)
A n +1 A n = α µ α 1+µ n 1-α 1+µ n n ( 16
)
x n +1 x n = α µ 1 1-α 1+µ n n ( 17
)
c n +1 c n = 1 + θ -α 1+µ n 1 + θ -α n α µ α 1+µ n 1-α 1+µ n n ( 18
)
with the initial conditions α 0 , A 0 and x 0 =

α 0 A 0 1+θ .
From ( 15), (log α n ) is a negative geometric sequence with common ratio 1

+ µ > 1, so α n = α (1+µ) n 0 n→∞ --→ 0. Also since µ α 1+µ n 1-α 1+µ n
> 0 and α n ∈ (0, 1), then from equation ( 16):

(0 <) A n+1 < A n . So (A n ) has a positive limit. Since α n = α (1+µ) n 0
we can transform equation ( 16) into:

log A n+1 A n = - α (1+µ) n +1 0 1 -α (1+µ) n +1 0 µ(1 + µ) n log 1 α 0
The series (1+µ

) n α (1+µ) n +1 0 1-α (1+µ) n +1 0 is convergent, because (1+µ) n α (1+µ) n +1 0 1-α (1+µ) n +1 0 > 0 and n 2 (1+µ) n α (1+µ) n +1 0 1-α (1+µ) n +1 0 n→∞ --→ 0
the term in n 2 (1+µ) n being dominated by the term α

(1+µ) n +1 0
. Consequently, output converges to A ∞ such that:

ln A ∞ = ln A 0 -µ log 1 α 0 ∞ n =0 (1 + µ) n α (1+µ) n +1 0 1 -α (1+µ) n +1 0 (19)
Since ∀n,

x n = α n A n 1+θ , then x n n→∞ --→ 0. Consequently, c n = A n -x n n →∞ --→ A ∞ (> 0).
The value of A n , x n and c n at the successive steady states are represented on figure 8, by taking α 0 = 0.3, A 0 = 1, θ = 0.1 and µ = 0.1. A n and x n are on the left-hand scale and c n is on the right-hand scale. For these values of the parameters, c n first decreases and then increases, in other cases (notably for high θ ), c n+1 < c n for all n > 0.

b) The substitution case -Suppose now that the agent undertakes successive steps of substitution, so:

α +,n = α 1-λ n
with λ ∈ (0, 1), so now α +,n > α n . Then:

α n +1 = α 1-λ n (20) A n+1 A n = α -λ α 1-λ n 1-α 1-λ n n (21) x n+1 x n = α -λ 1 1-α 1-λ n n ( 22
)
c n+1 c n = 1 + θ -α 1-λ n 1 + θ -α n α -λ α 1-λ n 1-α 1-λ n n ( 23
)
Now (log α n ) is a negative geometric sequence with common ratio 1 -λ ∈ (0, 1), so:

α n = α (1-λ) n 0 n →∞ --→ 1.
To derive the limit growth rate of A, we start by applying the logarithm to (21):

ln A n+1 A n = λ α 1-λ n 1 -α 1-λ n ln 1 α n
Then, we use the identity α n = α

(1-λ) n 0 to write down:

ln A n +1 A n = λ α (1-λ) n +1 0 1 -α (1-λ) n +1 0 (1 -λ) n ln 1 α 0
In this last identity, notice that α

(1-λ) n +1 0 n→∞ --→ 1, so: ln A n+1 A n ∼ n ∞ λ (1 -λ) n 1 -α (1-λ) n +1 0 ln 1 α 0 ( 24
)
The term in (1-λ) n

1-α

(1-λ) n +1 0 is an indeterminate form. We use de l'Hôpital's rule: if any, (1-λ) n

1-α

(1-λ) n +1 0 has the same limit than:

ln(1 -λ) (1 -λ) n -ln(1 -λ) ln α 0 (1 -λ) n +1 α (1-λ) n+1 0 which has limit 1 (1-λ) ln 1 α 0
as n tends to infinity. We inject this result in equation [START_REF] Jones | Growth, Capital Shares, and a New Perspective on Production Functions[END_REF] to get the limit growth rate of output:

ln A n +1 A n n →∞ --→ λ 1 -λ ( 25
)
Since ∀n, x n = α n A n 1+θ , equation (25) yields: Finally, the term 1+θ -α 1-λ n 1+θ -α n tends to one as n tends to infinity, so from (23) we also have:

ln x n+1 x n n→∞ --→ λ 1 -λ (26)
ln c n +1 c n n →∞ --→ λ 1 -λ (27)
Through continuous substitution, output, capital and consumption all grow asymptotically at the same gross rate of e λ 1-λ > 1. The capital share tends to one and, in this framework, substitution possibilities are never exhausted. This pattern is represented on figure 9.

We could have expected the substitution path to meet some bottleneck too 12 . Indeed, the sustainability of economic growth through pure substitution is the result of the assumption that we took on the transition between α n and α n +1 . When assuming that this transition takes the form in [START_REF] Gilchrist | Putty Clay and Investment: A Business Cycle Analysis[END_REF], α n converges quickly enough to 1 so growth can be sustained -for some other specifications of α +,n as a function of α n this result would of course not hold.

Still, we believe that the fact that our specification brings not only sustainability but also asymptotic balance makes it valuable as a reference path.

The CES case

Suppose now that the production functions f ϕ , instead of being Cobb-Douglas, are all CES with elasticity of substitution (ES) σ ∈ (0, 1).

Moysan and Senouci (2013) have proven that, for any x * > 0, A * > 0 and α * ∈ (0, 1), the only CES production function with ES σ and for which f (x * ) = A * and x * f (x * )/ f (x * ) = α * is given by formula:

∀x > 0, f (x ) = A * α * x x * σ-1 σ + 1 -α * σ σ-1
so we write function f ϕ like:

f ϕ (x ) = A * ϕ   α * ϕ x x * ϕ σ-1 σ + 1 -α * ϕ   σ σ-1
The relationship

f ϕ (x * ϕ ) = 1 + θ is equivalent to: α * ϕ = (1+θ )x * ϕ A * ϕ .
Like in the Cobb-Douglas case, the agent can only alter parameter α * ϕ . The new production function f ϕ can first of all be expressed through the old steady state: [START_REF] Moysan | A general characterization of neoclassical production functions and an application to physical capital-based growth models[END_REF] It can also be expressed through the new steady state:

f ϕ (x ) = A * ϕ   α + x x * ϕ σ-1 σ + 1 -α +   σ σ-1
f ϕ (x ) = A * ϕ   α * ϕ x x * ϕ σ-1 σ + 1 -α * ϕ   σ σ-1 (29)
The identification of the terms of ( 28) and ( 29) yields two indepedent equations in terms of unknown x * ϕ , A * ϕ , α * ϕ :

A * ϕ σ-1 σ (1 -α * ϕ ) = A * ϕ σ-1 σ (1 -α + ) (30)
A * ϕ x * ϕ σ-1 σ α * ϕ = A * ϕ x * ϕ σ-1 σ α + (31)
Another independent relationship between x * ϕ , A * ϕ and α * ϕ is given by equation ( 2) applied to the steady state corresponding to the production function f ϕ :

α * ϕ = (1 + θ )x * ϕ A * ϕ ( 32
)
Equations ( 30), ( 31) and (32) then constitute a set of three independent equations with three unknowns x * ϕ , A * ϕ , α * ϕ . This system solves in:

α * ϕ = (α + ) σ α * ϕ 1-σ (33) A * ϕ = A * ϕ 1 -α + 1 -α * ϕ σ σ-1 = A * ϕ 1 -α + 1 -(α + ) σ α * ϕ 1-σ σ σ-1 (34) x * ϕ = α * ϕ 1 + θ A * ϕ ( 35
)
Now, α + = α * ϕ because as the agent transits on the new production function towards the new steady state, factor shares are altered. Equation [START_REF] Seater | Share-Altering Technical Progress[END_REF] shows that there is however a simple relationship between α * ϕ on the one hand and α * ϕ and α + on the other hand: α * ϕ is the geometric mean of α + with weight σ and α * ϕ with weight 1 -σ.

By using equation ( 2) applied to function f ϕ we can express x * ϕ in equation ( 35) through x * ϕ :

x * ϕ = x * ϕ 1 -α + 1 -α * ϕ σ σ-1 α + α * ϕ σ   = x * ϕ 1 -α + 1 -(α + ) σ α * ϕ 1-σ σ σ-1 α + α * ϕ σ   (36)
Since σ ∈ (0, 1), we can verify that:

α + < α * ϕ =⇒ α * ϕ < α * ϕ and A * ϕ < A * ϕ and x * ϕ < x * ϕ α + > α * ϕ =⇒ α * ϕ > α * ϕ and A * ϕ > A * ϕ and x * ϕ > x * ϕ 3.2.

The sequence of steady states with CES production functions

Like in subsection 3.1.2 in the Cobb-Douglas case, we know turn to the analysis of the sequence of steady states with continual complementation or substitution. From equations ( 33), ( 34) and ( 35), steady-state variables α n , A n and x n evolve according to:

α n +1 = (α + (n)) σ α 1-σ n ( 37
)
A n +1 = A n 1 -α + (n ) 1 -(α + (n )) σ α 1-σ n σ σ-1 (38) x n +1 = α n+1 1 + θ A n +1 (39)
with the initial conditions α 0 , A 0 and x 0 =

α 0 A 0 1+θ .
a) The complementation case -With continual complementation:

α + (n ) = α 1+µ n
So equation [START_REF]The Least Developed Countries Report: Growth with employment for inclusive and sustainable development, United Nations[END_REF] becomes:

α n +1 = α 1+σµ
n so for all n :

α n = α (1+σµ) n 0 n→∞ --→ 0
Equation (38) yields:

A n+1 = A n 1 -α 1+µ n 1 -α 1+σµ n σ σ-1 (40)
The series ln

1-α (1+σµ) n (1+µ) 0 1-α (1+σµ) n +1 0 is convergent, since: ln 1 -α (1+σµ) n (1+µ) 0 1 -α (1+σµ) n +1 0 ∼ n∞ 1 -α (1+σµ) n (1+µ) 0 1 -α (1+σµ) n +1 0 -1 ∼ n∞ α (1+σµ) n 0 α 1+σµ 0 -α 1+µ 0
Thus, like in the Cobb-Douglas case, through continual complementation output converges to A ∞ such that:

ln A ∞ = ln A 0 + σ σ -1 ∞ n =0 1 -α (1+σµ) n (1+µ) 0 1 -α (1+σµ) n +1 0 (< ln A 0 )
The capital share tends to zero, capital x tends to zero, and steady-state consumption tends to A ∞ .

b) The substitution case -We take:

α + (n) = α 1-λ n
So equation [START_REF]The Least Developed Countries Report: Growth with employment for inclusive and sustainable development, United Nations[END_REF] becomes:

α n+1 = α 1-σλ n
Then, for all n :

α n = α (1-σλ) n 0 n →∞ --→ 1
Equation [START_REF] Zeira | Workers, Machines, and Economic Growth[END_REF] in the case of substitution becomes:

A n +1 = A n 1 -α 1-λ n 1 -α 1-σλ n σ σ-1 = A n 1 -α (1-σλ) n (1-λ) 0 1 -α (1-σλ) n +1 0 σ σ-1 (41)
The term

1-α 1-λ n 1-α 1-σλ
n is an indeterminate form. We apply de l'Hôpital's rule, so this term has the same limit than:

(1 -λ) ln α 0 ln(1 -σλ) (1 -σλ) n α (1-σλ) n (1-λ) 0 ln(1 -σλ) ln α 0 (1 -σλ) n+1 α (1-σλ) n +1 0
which tends to 1-λ 1-σλ when n tends to infinity.

Consequently:

A n +1 A n n →∞ --→ 1 -λ 1 -σλ σ σ-1 (> 1)
From equation ( 39) and since

α n +1 α n n →∞
--→ 1 we also have:

x n +1 x n n →∞ --→ 1 -λ 1 -σλ σ σ-1
while from the relationship c n = A nx n and from [START_REF] Zeira | Machines as Engines of Growth[END_REF] we get:

c n +1 c n = A n +1 A n 1 - α n +1 1+θ 1 - α n 1+θ n →∞ --→ 1 -λ 1 -σλ σ σ-1
So in the CES case (like in the Cobb-Douglas case), continuous substitution leads to asymptotically balanced growth. The limit growth rate of output, capital and consumption is the same and depends on parameters λ and σ only.

The endogenous direction of technological change with Cobb-Douglas production functions and logarithmic utility

In the preceding section we have seen that if the production functions were either all Cobb-Douglas or all CES with the same ES, the agent could enjoy steady growth by undertaking substitution.

However there are reasons to think that the agent will always do so, if he has the choice of undertaking either complementation or substitution. We have seen that the increase in subjective welfare due to the passage from technique ϕ to technique ϕ while initially at the ϕ-steady state is:

∆U ϕ→ϕ = 1 + θ θ u c * ϕ -u c * ϕ - x * ϕ x * ϕ f ϕ (x ) u c * 0ϕ (x ) dx
In a nutshell, the agent does not simply prefer the option that brings the highest increase in steady-state consumption. The term

x * ϕ x * ϕ f ϕ (x ) u (c * 0ϕ (x ))
dx might be dubbed the cost of convergence to the new steady state, and is > 0 in the case of substitution (x * ϕ > x * ϕ ) and < 0 in the case of complementation (x * ϕ < x * ϕ ). Whenever θ > 0, the agent gives attention to the value of his consumption in these transitional dynamics, which tend to be higher under complementation than under substitution.

In principle, we could solve the problem of the endogenous direction of technological change for any production functions f ϕ , however the computation of ∆U ϕ→ϕ is only possible if we have knowledge of the value function x → U (x ; ϕ ) or, equivalently, of the optimal choice function of the Ramsey agent which corresponds to function x → c * 0ϕ (x ). To the best of our knowledge, we have an analytical formula for the value function only under the two joint hypotheses that the production function is Cobb-Douglas and that the utility function is logarithmic. In this case, for any technique ϕ, the value function corresponding to the problem (1) is logarithmic in the level of capital, so there exist constants M ϕ and N ϕ such that:

∀x > 0, U x ; ϕ = M ϕ + N ϕ ln x x * ϕ
We identify constants M ϕ and N ϕ through equations ( 3) and ( 5):

M ϕ = U x * ϕ ; ϕ = 1 + θ θ ln c * ϕ = θ 1 + θ ln A * ϕ -x * ϕ N ϕ = x * ϕ ∂ U ∂ x x * ϕ ;ϕ = (1 + θ )x * ϕ ln c * ϕ = (1 + θ ) x * ϕ A * ϕ -x * ϕ
Thus -under the Cobb-Douglas-logarithmic assumption -for any technique ϕ, the corresponding value function U ( • , ϕ) is:

U x ; ϕ = 1 + θ θ ln A * ϕ -x * ϕ + θ x * ϕ A * ϕ -x * ϕ ln x x * ϕ
Consequently, the passage from technique ϕ to technique ϕ while initially at the ϕsteady state brings an increase in utility equal to:

∆U | ϕ→ϕ ≡ U x * ϕ ; ϕ -U x * ϕ ; ϕ = 1 + θ θ ln A * ϕ -x * ϕ A * ϕ -x * ϕ -θ x * ϕ A * ϕ -x * ϕ ln x * ϕ x * ϕ (42)
We treat the problem of the endogenous direction of technological change through a steady state-to-steady state approach: that is, we suppose that the agent is always at the steady state corresponding to some technique ϕ n when he switches to technique ϕ n +1 , and he does not expect technological change to happen again after that.

We suppose that the agent has the following technological menu:

α +,n = α 1-χ n
where χ ∈ (-µ, λ) with λ ∈ (0, 1) and µ > 0 like in subsection 3.1.2. From proposition 1 we know that the agent will always whoose χ = -µ or χ = λ; we will verify it by calculus below.

For any χ ∈ (-µ, λ), if the agent chooses χ at the n th opportunity, like in subsection 3.1.2 we have:

α n +1 = α 1-χ n A n+1 = A n α -χ α 1-χ n 1-α 1-χ n n x n+1 = x n α -χ 1 1-α 1-χ n n
From equation (42), the resulting increase in subjective welfare is, then:

∆U (α n , χ) = 1 + θ θ χ α 1-χ n 1 + θ -α 1-χ n ln 1 α n + ln 1 + θ -α 1-χ n 1 + θ -α n
We call Γ the function:

Γ : (0, 1) × (-∞, 1) -→ + (α, χ) -→ χ α 1-χ 1+θ -α 1-χ ln 1 α + ln 1+θ -α 1-χ 1+θ -α
Γ is the value of the χ option at the α state, it takes positive values in virtue of theorem 1.

The derivative of Γ with respect to χ is:

∂ Γ ∂ χ = ln 1 α 2 α 1-χ 1 -α 1-χ (1 + θ )χ ≷ 0 when χ ≷ 0
which conforms to proposition 1. Γ has also well-defined limits as χ tends to extreme values:

lim χ→-∞ Γ α, χ = ln 1 + θ 1 + θ -α ≡ L C (α) > 0 lim χ→1 Γ α, χ = 1 θ ln 1 α + ln θ 1 + θ -α ≡ L S (α) > 0 L C (α)
and L S (α) respectively denote the increase in subjective welfare in case of 'total complementation' (χ → -∞) and in case of 'complete substitution' (χ → 1). Remark that L C increases with α while L S decreases with α.

Thus, the agent has the choice between choosing χ = -µ and getting an increase in utility equal to ∆U | comp below, and choosing χ = λ and getting an increase in utility equal to ∆U | subs , where:

∆U | comp = 1 + θ θ -µ α 1+µ 1 + θ -α 1+µ ln 1 α + ln 1 + θ -α 1+µ 1 + θ -α ∆U | subs = 1 + θ θ λ α 1-λ 1 + θ -α 1-λ ln 1 α + ln 1 + θ -α 1-λ 1 + θ -α
Let's define function ψ by:

ψ = θ 1 + θ ∆U | subs -∆U | comp so: ψ(α) = λ α 1-λ 1 + θ -α 1-λ + µ α 1+µ 1 + θ -α 1+µ ln 1 α -ln 1 + θ -α 1+µ 1 + θ -α 1-λ (43)
ψ(α) measures the relative benefits of substitution over complementation when the capital share is originally at α. If ψ(α) > 0, then the agent prefers substitution (at rate λ) over complementation (at rate µ). Function ψ is the guide to the dynamics of the model. If ψ lies everywhere above one, then the agent always chooses substitution and output, capital and consumption grow asymptotically at the gross rate of e λ 1-λ > 1 like in subsection 3.1.2b. If ψ lies everywhere below one, then the agent undertakes continual complementation and ends up consuming A ∞ defined in equation [START_REF] Gollin | Getting Income Shares Right[END_REF]. If ψ crosses the abscissae axis from above, then there exists an interior steady state, and the capital share fluctuates around some long-run value α ∞ that depends on θ , µ and λ.

First, remark that: lim

α→0 ψ = lim α→1 ψ = 0
Secondly, the derivative of ψ is:

ψ (α) = ln 1 α λ(1 -λ) α -λ 1 + θ -α 1-λ 2 + µ(1 + µ) α µ 1 + θ -α 1+µ 2 + α µ 1 + θ -α 1+µ - α -λ 1 + θ -α 1-λ so: lim α→0 ψ = +∞ ; lim α→1 ψ = 0
Thus, function ψ is locally strictly increasing around 0, and since lim α→0 ψ = 0 we conclude that ψ is locally strictly positive near 0. When the capital share approaches 0, the agent always reacts by undertaking substitution at some date. This property reflects the Habakkuk hypothesis.

The limit of ψ around one does not allow to conclude on the form of ψ near 1. The formula for ψ is: Thus, when µ > λ, 1 is a local maximum of ψ. Since ψ has limit 0 near 1, this means that ψ is locally negative at the left of 1: in this case, the agent is always chooses to undertake complementation at some point and does not let capital share diverge to one like in the puresubstitution case. We could not prove formally that in this case the (interior) steady state was unique, although simulations strongly suggest that this is the case.

ψ (α) = ln 1 α λ(λ -1) α -1-λ λ(1 + θ -α 1-λ ) -2α 1-λ 1 + θ -α 1-λ 3 +µ(µ + 1) α -1+µ µ(1 + θ -α 1+µ ) + 2α 1+µ 1 + θ -α 1+µ 3 - 1 α λ(1 -λ) α -λ 1 + θ -α 1-λ 2 + µ(1 + µ) α µ 1 + θ -α 1+µ 2 + α -1+µ µ(1 + θ ) + α 1+µ 1 + θ -α 1+µ 2 + α -1-λ λ(1 + θ ) -α 1-λ 1 + θ -α 1-λ 2 so: lim α→1 ψ (α) = (1 + θ )(λ + µ) θ 2 (λ -µ)
In the contrary, when µ < λ, lim α→1 ψ (α) < 0, so 1 is a local minimum of ψ, which means that ψ is locally strictly positive near 1: the agent then undertakes substitution, and make the capital share diverge to one. We have been unable to prove formally that in this case no interior equilibrium exists 13 , but simulations for a wide range of parameters strongly suggest that this is the case.

Figures 10 and11 present the two cases. The first one assumes θ = 1, µ = 1 and λ = 0.5 while the second assumes µ to 0.3. In the first case, the interior steady state is around α ∞ = 0.36, in the second case ψ lies everywhere above zero.

So when µ > λ, the agent chooses substitution for α < α ∞ and complementation for α > α ∞ .

When (1 -λ)(1 + µ) = 1 -for example if µ = 1 and λ = 0.5 -, the capital share only takes two values asymptotically, since in this case α 1+µ 1-λ = α for all α, with α ∞ standing somewhere between these two values.

If (1 -λ)(1 + µ) > 1 (which is equivalent to µ 1+µ > λ), then α 1+µ 1-λ < α for all α. Then, if the agent always alternated complementation and substitution, the capital share would tend to zero; consequently, the agent sometimes undertakes substitution twice in a row. Between two such episodes, the economy alternates complementation and substitution and the capital share has a downward trend.

Finally, If (1 -λ)(1 + µ) > 1 (or equivalently µ 1+µ < λ), then α 1+µ 1-λ > α so the alternation of complementation and substitution leads to an upward trend in capital share, so the agent sometimes has to undertake complementation twice in a row.

Figure 12 shows the first terms of the sequence of steady states with endogenous direction of technological change, for α 0 = 0.3, A 0 = 1, θ = 0.1, λ = 0.1 and µ = 0.115. This case pertains to the second case above, where (1 -λ)(1 + µ) > 1 (and µ > λ). The capital share oscillates around a long term value in the neighborhood of 0.405, between 0.366 and 0.444. This economic growth regime thus quits the Kaldor facts: GDP per capita fluctuates around some exponential pattern and the capital share is stationary. But, in contrast to neoclassical theory based on exogenous production functions, this latter conclusion is endogenous (and only valid in the case λ < µ) and is consistent with the observation that the labor share tends to peak during recessions during the whole US postwar history. Another empirical regularity emerges from figure 12: consumption is pro cyclical and less volatile than investment (x ) and than overall GDP (A); interestingly the mechanism at work here is not through demand but through supply.

Conclusion

In this paper, we have investigated a simple Ramsey model with an unfamiliar way of introducing technical change. In doing so, we have depicted capital-labor substitution (and its opposite, labor-capital substitution) as a technological change move, rather than an accident of the production function.

We have done so for several reasons. First, the traditional productivity-linked picture of technology is a limit that the theory should get rid of, for the production function can credibly evolve in ways that cannot be represented by an increase of any productivity termindeed, our economic growth regime reproduces the same Kaldor facts as the neoclassical model under labor-augmenting technical change, but also explains the constancy of the capital share. Secondly, to contribute to the strand of research already cited in the introduction that sees -either through theory or through empirics -the production function as being written dynamically, rather than given. Thirdly, to argue that a simple Ramsey framework might not be such a bad guide to the analysis of long-run growth 14 . Fourth and finally, to make sense of Habakkuk's hypothesis under minimal hypotheses -in this framework, highwage share countries have more chance to undertake capital-labor substitution, and so have also more chance to experience rapid GDP growth 15 .

The approach taken here predicts that poor countries should display a lower capital-output ratio, and a lower labor share. The former point seemingly enters in contradiction with the often seen empirical result that the correlation between the capital-output ratio and GDP per capita is weak 16 , the relevance of cross-country data on relative values of capital stocks (prices and quantities) is highly doubtful. Data on labor shares are more reliable as they do not rely on estimates related to capital, and we have shown in Senouci (2014) 4 the relationship between GDP per capita and the labor among OECD countries and in the OECD database clearly displays an inverted-U pattern.

Although in closed economy, the model also carries an answer to the question: why do not all the countries adopt the more advanced techniques? Imagine that some country has undertaken complementation for a long time so that savings are approximately nil there and GDP per capita is very low. Now suppose that this country starts to observe some rich country, which has undertaken, say, only substitution in the past, so GDP is very high and the capital-output ratio is also very high. This would be very costly in terms of present and near future consumption if the poor country adopted the rich country's more capital-intensive method. Moreover, if the poor country has initially a small tradable sector and is creditconstrained in the international markets, he cannot import foreign capital goods up to the point where the rich country's technology is more profitable than his own 17 . The poor country would then rather wait that the rich country operates some complementation, making the sacrifice less painful.

Indeed, there exists some empirical support for this view. Comin and Hobijn (2004), Comin et al. (2008) and Comin et al. (2010) investigated the cross-country pattern of technology adoption and found significant and variable lags of adoption, and also detected some form of temporal complementarity: quick past adoption predicts quick future adoption. Comin and Hobijn (2010) investigated the level of use of technologies (instead of only mere qualitative 'adoption' variable) and have shown that although adoption lags have shortened all over the twentieth century, the asymptotic of use of technologies (once adopted) is much less than in the part. This result is understandable in our terms: today's modern technologies (mobile phones, etc.) are much less expensive than yesterday's modern technologies (factories, etc.) so countries tend to adopt them more easily; however, in developed countries these technologies come to complement some big stock of capital, while in developing countries they are not part of such a great capital stock -so the use rates are not as high as in developed countries.

To constitute a more satisfying framework of analysis of medium to long-run growth, the skills part of technology obviously misses from the approach. The interaction between skills and tools is probably of considerable importance to understand the dynamic patterns of economies, and one can legitimately object that our approach completely neglects labor. The evolution of skills might even be as important as the evolution of tools to understand the dynamics of factor shares and growth 18 . If the word 'technology' refers to some set of capital goods, and if a technique is the sum of a technology and some set of skills, then this study investigated technological change but not technical change. In future research, we hope to analyze the relationships between skills and tools along this framework.

often creates a conflict between the objective of achieving competitiveness by acquiring advanced technology (which invariably tends to be capital-intensive) and the objective of creating decent jobs in sufficient quantity. (...) The LDCs' manufacturing export sector is included in regional and global value chains, and it must accordingly apply the international standards of quality and production processes in which those chains operate. Still, the segments of these chains that are located in LDCs are mainly the labour-intensive ones (...) Clearly, the LDCs cannot afford to ignore the fact that they need foreign exchange to import capital goods, technology and other inputs required to build their productive capacities. They must also bear in mind the need to maintain or increase their export capacity. To be able to export, they may need to attract FDI, which typically chooses capital-intensive technologies that do not generate much employment. " [UNCTAD (2013, chapter 4)]. 18 Among many studies focusing on the role of human capital in long-run growth, see [START_REF] Bessen | Productivity Adjustments and Learning-by-Doing as Human Capital[END_REF][START_REF] Bessen | Technology and Learning by Factory Workers: The Stretch-out at Lowell, 1842[END_REF][START_REF] Bessen | Was Mechanization De-Skilling? The Origins of Task-Biased Technical Change[END_REF][START_REF] Bessen | More Machines, Better Machines... or Better Workers?[END_REF], [START_REF] Caselli | Technological Revolutions[END_REF],

Appendix: Proof of proposition 1

When f ϕ 1 and f ϕ 2 meet only at x * ϕ , two cases are conceivable: either [f ϕ 2 (x ) > f ϕ 1 (x ) for x ∈ (0, x * ϕ ) and f ϕ 2 (x ) < f ϕ 2 (x ) for x ∈ (x * ϕ , +∞)] or [f ϕ 2 (x ) < f ϕ 1 (x ) for x ∈ (0, x * ϕ ) and f ϕ 2 (x ) > f ϕ 2 (x ) for x ∈ (x * ϕ , +∞)]. Let's denote by c * ϕ 1 ,0 , c * ϕ 1 ,1 , • • • the optimal consumption path from t = 0 on if the agent incurs technical change from technique ϕ to technique ϕ 1 at date t = 0 (before consumption decision) while initially at the steady state corresponding to ϕ. Let's also denote by c * ϕ 2 ,0 , c * ϕ 2 ,1 , • • • the corresponding consumption path resulting from technical change from ϕ to ϕ 2 .

-When f ϕ 2 (x * ϕ ) < f ϕ 1 (x * ϕ ) < 1 + θ (complementation case), suppose that the agent undergoes ϕ 2 technical change but, instead of choosing the optimal c * ϕ 2 ,0 , he consumes c * ϕ 1 ,0 at date t = 0. Then, at the beginning of date t = 1 he holds x1 = f ϕ 2 (x * ϕ )

= f ϕ 1 (x * ϕ )=A * ϕ -c * ϕ 1 ,0 = x * ϕ 1 > x * ϕ units of capital.
Suppose now that at date t = 1 he chooses to consume c *

Figure 1 :

 1 Figure 1: The first technological opportunity -complementation f ϕ (x * ϕ ) < 1 + θ .

Figure 2 :

 2 Figure 2: The second technological opportunity -substitution f ϕ (x * ϕ ) > 1 + θ .

Figure 3 :

 3 Figure 3: Ranking technological opportunities -complementation.

Figure 4 :

 4 Figure 4: Ranking technological opportunities -substitution.

Figure 6 :

 6 Figure 6: Post-substitution shock dynamics (Assumptions: α 0 = 0.3; A * ϕ = 1; θ = 0.1; α ϕ = (α ϕ ) 0.5 ).

Figure 7 :

 7 Figure 7: Post-complementation shock dynamics in a case where steady-state consumption decreases c * ϕ < c * ϕ

Figure 8 :

 8 Figure 8: Successive steady states with steady complementation (Assumptions: α 0 = 0.3; A 0 = 1; θ = 0.1; µ = 0.1).

Figure 9 :

 9 Figure 9: Successive steady states with steady substitution, log scale (Assumptions: α 0 = 0.3; A 0 = 1; θ = 0.1; λ = 0.1).

Figure 10 :

 10 Figure 10: Function ψ for θ = 1, µ = 1 and λ = 0.5.

Figure 11 :

 11 Figure 11: Function ψ for θ = 1, µ = 0.3 and λ = 0.5.

Figure 12 :

 12 Figure 12: The growth path with endogenous direction of technological change (Assumptions: α 0 = 0.3; A 0 = 1; θ = 0.1; λ = 0.1; µ = 0.115).

Of course, these results depend on the specific assumptions we make on the amount of technological change -especially on the amount of substitution that we allow for at each period. But we believe that these assumptions deserve special attentions, since they bring asymptotically steady growth.

An interior steady state exists as long as f ϕ (0) > 1 + θ and f ϕ (∞) < 1 + θ , which we assume all through the paper for any technique ϕ.

The presence of growth bottlenecks in the pure-complementation case above is much more general and holds beyond the Cobb-Douglas assumption, since A n always decreases.

This could happen if function ψ crossed the abcissae axis twice over (0, 1).

We have found some conditions under which the impatient agent would still enjoy some technological change that makes his steadystate consumption decrease. Without assuming any externality (e.g. of Malthusian form), we have seen that a Ramsey agent can rationally impoverish himself

.[START_REF] Champernowne | A Dynamic Growth Model Involving a Production Function[END_REF] We have departed from the original[START_REF] Habakkuk | American & British Technology in the 19 th Century: The Search for Labour-Saving Inventions[END_REF] argument by assuming that wages were equal to the marginal product of labor, rather than depending on some exogenous

option.[START_REF] Durlauf | Multiple Regimes and Cross-Country Growth Behaviour[END_REF] See Easterly and Levine (2001).

This view is reflected in the United Nations Conference on Trade And Development 2013 report on LDCs: "The choice of technology

Caselli & Coleman (2001, 2006) and Beaudry, Doms & Lewis (2010).

ϕ 1 ,1 ; then, he ends up with x2 units of capital at the beginning of date t = 1, with x2= f ϕ 2 ( x1 )c * ϕ 1 ,1 > f ϕ 1 ( x1 )c * ϕ 1 ,1 = x * ϕ 1 ,2 > x * ϕ . So he can throw away x 2x * ϕ 1 ,2 units of capital, produce f ϕ 2 (x * ϕ 1 ,2 ) > f ϕ 1 (x * ϕ 1 ,2 ), consume c * ϕ 1 ,2 : he will end up holding x3 = f ϕ 2 (x * ϕ 1 ,2 )c * ϕ 1 ,2 > f ϕ 1 (x * ϕ 1 ,2 )c * ϕ 1 ,2 = x * ϕ 1 ,[START_REF] Allen | Technology and the Great Divergence: Global Economic Development Since 1820[END_REF] . By a clear recurrence, this shows that the consumption pathc * ϕ 1 ,0 , c * ϕ 1 ,1 , • • • is feasible through technique ϕ 2 . So U c * ϕ 2 ,0 , c * ϕ 2 ,1 , • • • ≥ U c * ϕ 1 ,0 , c * ϕ 1 ,1 , • • • .And since there is some strictly positive waste in the proof, we also have:U c * ϕ 2 ,0 , c * ϕ 2 ,1 , • • • > U c * ϕ 1 ,0 , c * ϕ 1 ,1 , • • • , which amounts to ∆U ϕ→ϕ 2 > ∆U ϕ→ϕ 1 . -When f ϕ 2 (x * ϕ ) > f ϕ 1 (x * ϕ ) > 1 + θ (substitution case), a symmetric reasoning shows that, also here, the consumption path c * ϕ 1 ,0 , c * ϕ 1 ,1 , • • • is feasible through technique ϕ 2 , and then that ∆U ϕ→ϕ 2 > ∆U ϕ→ϕ 1 . In this case, the production function f ϕ 2 happens to be strictly above f ϕ 1 for levels of capital above x * ϕ , which are the levels of capital which are reached after both ϕ 1 and ϕ 2 shocks.