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Unformatted text preview: 14.05 Intermediate Applied Macroeconomics Problem Set 4 Solutions Distributed: October 27, 2005 TA: Jos´ e Tessada Due: November 3, 2005 Frantisek Ricka Question 1 Diamond Overlapping Generations Model Consider the Diamond overlapping generations model. L t individuals are born in period t and live for two periods, working and saving in the first and living off capital in the second period. Assume population is growing at a constant rate, n , and technological progress occurs at exogenous rate g . Markets are competitive and labor and capital are paid their marginal products. There is no capital depreciation. Utility is logarithmic with individual discount rate > ρ > − 1. 1 U = log( c t ) + log( c t +1 ) . 1 + ρ The production function in per capita terms is α y t = f ( k t ) = k t . (a) Determine the intertemporal budget constraint for each individual. Set up the consumer’s c t +1 utility maximization problem and derive the equilibrium condition for c t (Euler equa- tion). Use that condition and the budget constraint to solve for first period consumption, c t , and the savings rate (ie, the fraction of income saved), s t . Answer. The intertemporal budget constraint of each individual is c t +1 c t + = A t w t . 1 + r t +1 The consumer’s utility maximization problem is 1 max log( c t ) + log( c t +1 ) c t ,c t +1 1 + ρ subject to c t + c t +1 = A t w t . 1 + r t +1 We can solve this problem using the Langrangean and taking first order conditions with respect to c t and c t +1 . The Lagrangean is given by 1 c t +1 L = log( c t ) + 1 + ρ log( c t +1 ) + λ w t A t − c t − 1 + r t +1 . The first order conditions are 1 = λ c t 1 1 1 = λ . 1 + ρ c t +1 1 + r t +1 1 Combining the two equations we obtain the Euler equation c t +1 1 + r t +1 = c t 1 + ρ And using the Euler equation in the budget constraint we obtain c t c t + = A t w t 1 + ρ 1 + ρ c t = w t A t 2 + ρ The savings rate, s t is given by c t 1 + ρ 1 s t = w t A t − c t = 1 − = 1 − = w t A t w t A t 2 + ρ 2 + ρ . (b) Using the saving rate derived in part (a) and the production function determine the re- lationship between k t +1 and k t , and show it in a graph. Write down the expression that implicitly defines the equilibrium capital stock k ∗ . Is the equilibrium stable? Answer. The capital stock is given by K t +1 = s t w t A t L t Therefore, capital per effective unit of labor is K t +1 s t w t 1 w t k t +1 = = = A t +1 L t +1 (1 + n )(1 + g ) 2 + ρ (1 + n )(1 + g ) Using the fact that labor earns its marginal product, we have w t = f ( k t ) − k t f ( k t ) = (1 − α ) k α t Substituting above 1 1 k t +1 = 2 + ρ (1 + n )(1 + g ) (1 − α ) k t α (1) k t +1 = m ( k t ) In equilibrium, k t +1 = k t = k ∗ . So, k ∗ is implicitly defined by 1 1 k ∗ = 2 + ρ (1 + n )(1 + g ) (1 − α )( k ∗ ) α The equilibrium level is then given by 1 − α 1 / (1 − α ) 1 k ∗ = 2 + ρ (1 + n )(1 + g ) Figure 1 shows the dynamic system described by equation...
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- Fall '09
- Macroeconomics, Stock and flow, Capital accumulation, Saving