Research

In this short note, we show how the space of elasticity of substitution functions maps into the space of 2-input neoclassical production functions. In doing so, we derive a general analytical formula for every 2-input neoclassical production function of class C². We present a simple set of sufficient conditions for the Inada conditions to hold; and prove that the Solow model under capital-augmenting (or investment-specific) technical change is asymptotically balanced if and only if the capital share converges to a non-degenerated limit as the capital–labor ratio tends to infinity.

Textbook growth models typically fail to account for the absence of global convergence in income per capita, for the growth effects of the investment rate, and for the existence of large swings in the labor share over the medium run. In this paper, I set a Solowmodel animated by a formof localized technical change. I assume that productivity growth is an increasing function of the capital-output ratio. I prove that the model has a globally stable balanced growth path. If technical change is locally biased, then the growth rate is a strictly increasing function of the saving rate, and the labor share slowly tends to zero along any balanced growth path.

We present an alternative form of technical change within the traditional two-input framework. The aggregate production function is the convex hull of an increasing, finite number of Leontief production functions. At each date, each of these local production functions mutates into two Leontief production functions: one featuring exogenously increased labor-augmenting productivity, the other featuring exogenously increased capital-augmenting productivity. We embed this model of technical change into an otherwise standard, discrete-time Solow model. We do not specify technical change as purely labor-augmenting; still, it comes out that this modified Solow model has a globally stable balanced growth path. Along this path, technical change jointly determines the growth rate, capital-output ratio, and marginal productivity of capital and the competitive factor shares.

For each production or utility function, we can define the corresponding elasticities of substitution functions; but is the reverse true? This paper shows that yes, and that this link is fruitful. By inverting the system of partial differential equations defining the elasticities of substitution functions,we uncover an analytical formula which encompasses all production and utility functions that are admissible in Arrow-Debreu equilibria. We highlight the "Constant Elasticities of Substitution Matrix" (CESM) class of functions which, unlike the CES functions, does not assume uniform substitutability among all pairs of goods. A shortcoming of our method is that it permits only to control for local concavity while it is difficult to control for global concavity.

We present an alternative to growth accounting à la Solow, on the same set of variables, that provides a metric for labor-saving technical change (‘λ’) and capital-saving technical change (‘μ’). These two components are identified through the variations of the factor shares, which we assume to reflect marginal productivities. We run our algorithm using BEA data from 1948 to 2015, and compare the predictive power of our time series of (λ,μ) with the one of the Solow residual. Through simple regressions, we find: (i) that λ and μ are as good predictors of the growth rate of GDP per capita as the Solow residual, and (ii) that λ and μ, together with capital accumulation, are  strong predictors of the variation of the factor shares, while the Solow residual is not. We conclude that a bi-dimensional representation of productivity has a stronger empirical relevance than the usual linear representation; however the former carries some different theoretical properties than the latter – notably on the consequences of capital accumulation.

We present a new way to picture technological change in an otherwise standard Ramsey framework. Technological change takes the form of alterations of the production function itself, rather than changes in total factor productivity. These changes can take two directions that we dub respectively ‘complementation’ and ‘substitution’. Complementation results in a production function that is superior for lower values of capital, while substitution results in a production function that is superior for higher values of capital. Under the most general conditions, when the agent is initially at steady state, both options bring strictly positive utility gains to the agent. We analyze sequence of steady states with exogenous and endogenous direction of technological change. With exogenous growth, we prove that when the production functions are Cobb-Douglas or CES (with the same elasticity of substitution), output and consumption grow asymptotically at a common rate and the capital share tends to one under continual substitution; while continual complementation makes output and consumption converge to a common limit and the capital share tend to nil. With endogenous direction of technological change and under the most general conditions, the agent has a bias towards complementation which brings quicker gains than substitution. We assume that the production functions are Cobb-Douglas and that utility is logarithmic. Then, when the potential rate of complementation is strictly greater than the potential rate of substitution, the labor share oscillates around some endogenous long-run value, determined by the rates of complementation/substitution and by the impatience rate. This growth regime reproduces the Kaldor facts.

The relative price of capital (or equipment) goods with respect to consumption goods is strongly, negatively correlated with income per capita in cross-sections of countries. This stylized fact suggests that economic growth takeoffs are associated with changes in the direction of technical change. It also suggests that increases in productivity that are embodied in capital goods lead to relatively quicker growth. The goal of this paper is to explore the message of the discrete-time Ramsey model with logarithmic utility, augmented with endogenous direction of technical change. We suppose that the representative agent, while initially at steady state, is offered the possibility to increase either labor-augmenting productivity or investment-specific productivity. We derive the marginal increase in utility from each option. We find that when the elasticity of substitution, the capital share and the rate of impatience lie within the usual ranges, investment-specific technological change is relatively undervalued, because its fruits take relatively more time to materialize. This approach reflects some interesting ideas on the macroeconomics of structural change. However, its predictions stand at odds with cross-country evidence as well as with the early British growth experience (~1770–1913). We argue that the fixity of the production function constitutes a major obstacle for a consistent theory of the direction of technological changes on neoclassical bases.

What type of technical progress is able to increase income per capita, instead of merely translating into higher fertility? To investigate this question, this paper first sets up an OLG growth model with capital, land and endogenous fertility. Children compete with capital as a means of saving for the young. This framework is then put into motion by continuous neutral and investment-specific technical change. Neutral technical change leads to well-known Malthusian dynamics and cannot make the wage rate grow asymptotically. On the contrary, investment technology alters the relative price of capital and children and so also affects the households’ accumulation/fertility decisions. If capital and labor are strict substitutes in the production function, continuous investment-specific technical change results in long-term growth of per capita income. The theory is used to interpret some evidence on the first steps of the Industrial Revolution.

We contribute to the literature on optimal growth in two-sector models by solving a Ramsey problem with a concave utility function. The unique possible steady-state is independent of initial conditions and of the instantaneous utility function, but not of the discount rate, and is characterized by a wage-rental ratio depending solely on the technology of the capital sector. For an initially low-capital economy, we show that the wage-rental ratio increasingly converges to its balanced value during transition. If the consumption sector is relatively capital-intensive, the relative price of capital increases during transition. If the investment sector is relatively more capital-intensive, it decreases. We also prove that a negative shock on the subjective rate of impatience, that makes the social planner more patient, leads to an immediate positive jump in asset prices.

We contribute to the literature on optimal growth in two-sector models by solving a Ram-sey problem with a concave utility function. The unique possible steady-state is independentof initial conditions and of the instantaneous utility function, but not of the discount rate, andis characterized by a wage-rental ratio depending solely on the technology of the capital sector.For an initially low-capital economy, we show that the wage-rental ratio increasingly convergesto its balanced value during transition. If the consumption sector is relatively capital-intensive,the relative price of capital increases during transition. If the investment sector is relativelymore capital-intensive, it decreases. We also prove that a negative shock on the subjectiverate of impatience, that makes the social planner more patient, leads to an immediate positivejump in asset prices.