On the Invariance of the Rate of Return to Convex Adjustment Costs∗

The Modified Golden Rule, which relates the rate of return on capital and the growth rate of the capital stock along long-run growth paths that maximize the utility of a representative infinitely-lived consumer, is invariant to the introduction of convex capital adjustment costs. Therefore, along balanced growth paths in neoclassical optimal growth models with an exogenous long-run growth rate of capital, the rate of return is invariant to the introduction of convex adjustment costs, though the capital-labor ratio is reduced along such paths. In AK models, convex adjustment costs reduce the growth rate and rate of return on capital. ∗This paper supercedes ”The Golden Rule and Convex Adjustment Costs.” †I thank Bill Dupor, Joao Gomes, Urban Jermann, Amir Yaron and the Penn Macro Lunch Group for helpful discussion, and an anonymous referee for helpful comments. Convex adjustment costs were introduced in general equilibrium models to help smooth the behavior of aggregate investment in these models and to allow the equilibrium price of capital to be determined endogenously. Although the introduction of convex adjustment costs has interesting effects on the price and accumulation of capital in the long run, as well as in the short run, the long-run rate of return on capital is invariant to the introduction of adjustment costs in neoclassical growth models in which the utility of a representative consumer is maximized, either by a social planner or by individuals in competitive markets. Though this invariance is a robust and easily-derived result in a widely-analyzed class of neoclassical growth models, it has apparently gone unnoticed. For instance, Abel and Blanchard (1983) concluded that convex adjustment costs require a modification to the Modified Golden Rule relating the marginal product of capital and the growth rate of the economy. However, here I focus on the rate of return to capital, rather than on the marginal product of capital, and show that there is no need to modify the Modified Golden Rule relationship between the rate of return and the growth rate of the economy. The invariance of the Modified Golden Rule relationship between the rate of return and the growth rate has several important implications. First, as noted above, in neoclassical growth models in which the long-run growth rate of the economy is exogenous, the long-run rate of return is invariant to adjustment costs, either in competitive economies with a representative consumer or in allocations chosen by a social planner. Second, by letting the social discount rate equal zero, the Modified Golden Rule delivers the Golden Rule. Therefore, the introduction of convex adjustment costs has no effect on the equality of the rate of return and the growth rate of capital in the Golden Rule. Third, Phelps (1961, 1965) and Swan (1964) have shown that in neoclassical growth models without adjustment costs, the Golden Rule is characterized by the equality of aggregate consumption and aggregate wage income. I show that this characterization also holds in the presence of convex adjustment costs, and I go on to generalize this result to characterize optimal consumption when the social discount rate is positive and there are convex adjustment costs: in the absence of productivity growth, aggregate consumption equals aggregate wage income plus the value of the end-ofperiod capital stock multiplied by the discount rate. Fourth, in endogenous See, for example, Abel and Blanchard (1983) and Baxter and Crucini (1993). See, for example, Basu (1987), Jermann (1998), and Abel (2000).