pset4sols - 14.05 Intermediate Applied Macroeconomics...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 01/12/2011 for the course ECO 010023 taught by Professor Mrraggillpol during the Fall '09 term at Paris Tech.

Page1 / 10

pset4sols - 14.05 Intermediate Applied Macroeconomics...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online