Endogenous growth Sophia Kazinnik University of Houston Economics Department

A bit of background: The neoclassical growth model that we studies previously (Solow- Swan) provides important insights about growth, but also has serious limitations. One of the limitations is that the long run rate of growth is determined outside of the model - exogenously. As we have seen before, in neoclassical growth model, the steady state rate of growth will be driven by the rate of technological progress. But what if we want to know what drives technological progress itself? Neoclassical Growth Model vs. “New Growth Theory” 2

A “new growth theory” (endogenous growth) was developed to extend neoclassical growth theory (exogenous growth). It extends the neoclassical growth model to allow for endogenously driven growth (Romer, Lucas). Today we will look at couple of core models underlying this research. 3 Neoclassical Growth Model vs. “New Growth Theory”

4 AK model - preliminaries Consider the following model: Closed economy, no technological progress, population size is fixed. Production function of the form: Depreciation is enclosed in A. Production function is CRS at the firm level. Capital can be transformed into consumption with no cost. 4

The infinitely lived representative consumer-manager has preferences given by: As usual, we can derive the Euler equation, in which equilibrium per capita consumption obeys: Now lets determine the equilibrium in this model. 5 AK model

We know that in each period t firms invest up to the point where the net marginal product of capital equals the interest rate: Finally, the model is closed by the goods-market equilibrium condition: Equilibrium? 6 AK model - equilibrium

We can see on the graph how the equilibrium is determined: 7 AK model - equilibrium

Results? We can see that the equilibrium growth is constant through time and is determined by the intersection of the two curves we’ve seen in the previous slide. This intersection yields: This is the rate of consumption growth in equilibrium. We must assume the following condition : Otherwise, in equilibrium, we will gradually “eat” our capital away. 8 AK model - solution

In conclusion: The rate of consumption growth is “g bar”. In equilibrium, per capita investment is: And since we can see that the level of consumption in equilibrium will be: We can see that in this model (as opposed to the neoclassical growth model) a change in the saving rate (beta) now has a permanent effect on the rate of growth of the economy (the higher the beta, the higher the growth rate). 9 AK model - conclusion

Consider now “learning by doing” variant of the model: Suppose that consumption is still characterized by the Euler equation from the previous model (meaning utility function and BC are same). However, each firm’s j’s output is given by: where the “first” k is the individual firm’s level of capital per worker, and “second” k is the economy-wide average level of capital per worker. 10 “Learning by doing” model

11 “Learning by doing” model 11 - We can see that in this model each individual firm faces diminishing returns to it’s own investment, but production function is CRS in “first” and “second” k together. Reason for this: production process generated knowledge externalities. The higher the average level of capital intensity (k) in the economy, the greater the incidence of technological spillovers that raise the marginal productivity of capital throughout the economy.

Individual firm views the marginal product of its own investment as: OR (Given the fact that in equilibrium : and that net marginal product of capital must equal interest rate ) An individual’s inter-temporal optimal consumption allocation is still characterized by the Euler equation: Equilibrium? 12 “Learning by doing” model

We can see on the graph how the equilibrium is determined: 13 “Learning by doing” model: equilibrium

Results? The steady state rate of growth of the economy is: As before, the economy adjusts immediately to its steady-state equilibrium growth path. Once again, we impose the restriction: 14 “Learning by doing” model: conclusion

15 AK vs. “learning by doing” AK vs. “learning by doing”: Notice that the equilibrium growth rate and the market interest rate (r) are lower here than in the AK model. Why? Because individual firms do not internalize the “learning by doing” externality their investment produces for other firms. Hence, this equilibrium is not Pareto Optimal. 15

International capital market integration can raise the level of world output by allowing capital to migrate toward its most productive global uses. Here, we will use a stochastic version of the AK model to illustrate how world capital market integration can raise steady-state growth. We will start with the closed economy in investment autarky. 16 International Capital Market Integration

Suppose that the representative agent has an infinite-horizon Expected utility function: And we normalize the population to 1. As in AK model, we assume linear technology; CRS in capital at the firm level. What is different? 17 International Capital Market Integration

18 International Capital Market Integration But in this model there are two types of capital instead of one: 1) The first type offers constant risk-less return (1+A) per unit invested (by the previous logic, the gross risk-less interest rate is r). 2) The second type offers a risky return (1+ “r tilde”) per unit of capital invested on date t. “r tilde” is i.i.d random variable, such that it’s expected return is bigger than r - the gross risk-less interest rate. 18

Let K denote the total amount of capital, both safe and risky, accumulated by end of period t-1. Capital is the only source of income in the model, so the representative agent’s period budget constraint is: where x denotes the end-of-period t-1 share of capital invested in the risky asset. How is this share determined? 19 International Capital Market Integration

The first order conditions for this problem will yield two Euler equations: And we know that the level of consumption is defined by: We can linearize the second Euler equation and combine it with first to yield: 20 Optimal Consumption and Portfolio Shares

Now, we want to get rid of K’s and express consumption: Substituting this into: Yields: 21 Optimal Consumption and Portfolio Shares

22 And solving for x: Naturally, the share of risky capital is positively related to expected return differential and negatively related to the variance of the risky return. Optimal Consumption and Portfolio Shares 22

Having solved for x, we can find the economy’s expected growth rate of consumption growth: Expected consumption growth rate is a decreasing function of the variance of the risky return. 23 Optimal Consumption and Portfolio Shares

24 Closed Economy vs. Open Economy Now, lets extend the analysis to an open - economy setting: -assume all countries have the same preferences and technologies, but the returns to risky projects are imperfectly correlated internationally. -individuals will hold the same portfolio (same log preferences) -risky capital has the same mean rate of return for all countries n: 24

Following the same steps that we just did, we find: Since the world portfolio of risky capital is globally diversified: It follow immediately that expected consumption growth under capital market integration (first equation in this slide) will be higher than under autarky. Logic: the opportunity to diversify their portfolios induces people to allocate a larger share of wealth to risky assets. Hence, expected growth rises. 25 Closed Economy vs. Open Economy

Comments? Questions? 26