Economic Growth: Theory and Empirics (2012) Problem set I
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1 Economic Growth: Theory and Empirics (2012) Problem set I Due date: April 27, 2012 Problem 1 Consider a Solow model with given saving/investment rate s. Assume: Y t = K α t (A tl t ) 1 α 2) a constant growth rate of labour L denoted by n, 3) a constant growth rate of technologyadenoted by g, and 4) a constant rate of depreciation of capital denoted by δ. 1. Derive the capital accumulation equation; 2. prove the existence of a not trivial (locally) stable equilibrium; 3. calculate the level of equilibrium of capital, income and consumption per capita in efficiency units; 4. suppose that the growth rate of population n increases. Which are the implications for the level and the growth of income per capita in the short and in the long run? 5. Derive the Golden Rule condition for capital accumulation; 6. Suppose that the equilibrium level of capital is lower of the gold-rule level. Discuss a fiscal policy which could allow to get it. 1
2 Problem 2 Consider two countries R and P. Their capital follows the Solow model. In particular, assume for each country i, withi {R,P}: Yt i = ( Kt) i α ( ) A i t L i 1 α t 2) a constant growth rate of labour L i denoted by n i, 3) a constant growth rate of technologya i denoted by g, 4) a constant rate of depreciation of capital denoted by δ i, and 5) a constant saving/investment rate s i. 1. derive the capital accumulation equation for the country i; 2. calculate the level of equilibrium of capital, income and consumption per capita in efficiency units for the country i; 3. discuss the conditions under which we observe absolute convergence in the income of two countries under the assumption that k P < k R. 4. suppose that n P > n R. Which are the implications for the convergence of the growth of income per worker in the short and in the long run? 5. Suppose that capital can flow from one country to the other in relation to differences in the return on capital. Discuss the implications for the convergence of incomes between the two countries. 2
3 Problem 3 Consider a Solow model with given saving/investment rate s. Assume: Y t = K α t (A t L) 1 α β R β t 2) a constant supply of labour L, 3) a constant growth rate of technologyadenoted by g, 4) a constant rate of depreciation of capital denoted by δ, 5) a stock of natural resources equal to Z t, which are used in production in each period for a fractiond. All natural resources employed in production are destroyed, so that d is also the rate of depletion of natural resources. No regeneration of natural resources is present. 1. Derive the (de)accumulation equation for natural resources, 2. derive the accumulation equation for physical capital; 3. calculate the equilibrium growth rate of income per worker; 4. calculate the maximum level of depletion rate compatible with a positive growth rate of income per worker; and 5. discuss the criteria to be used to choosed. 3
4 Problem 4 Consider a Solow model with given saving/investment rate s. Assume: Y t = K t (L) 1 α 2) a constant supply of labour L, 3) a constant rate of depreciation of capital denoted by δ, 1. derive the accumulation equation for physical capital; 2. calculate the equilibrium growth rate of income per worker; 3. discuss the main determinants of the equilibrium growth rate in light of the standard Solow model; and 4. discuss the possible absolute and conditional convergence in this framework. 4
5 Problem 5 Consider a Solow model. Assume: Y t = K α t (A tl t ) 1 α 2) a constant growth rate of labour L denoted by n, 3) a constant growth rate of technologyadenoted by g, and 4) a constant rate of depreciation of capital denoted by δ. 5) a saving/investment rate s depending on the level of capital k: s = s for k < k and s = s for k k withs < s. 1. Discuss why the saving rate could effectively depend on the level of capital; 2. derive the capital accumulation equation; 3. derive under which conditions on the values of parameters there exist two not-trivial (locally) stable equilibria (i.e. poverty trap); 4. discuss the implications of the existence of two non-trivial equilibria for the absolute and conditional convergence of income per worker of countries with different initial levels of capital; 5. discuss if a transfer of capital from rich to poor countries is effective in the long run. 6. discuss why it is not possible from an empirical point of view to distinguish the outcome of this model in equilibrium from the outcome of a model with constant but heterogeneous saving rate across countries. 5
6 Problem 6 Consider a Solow model with given saving/investment rate s. Assume: 1) the following CES technology: Y = [ αk ρ +(1 α)(al) ρ] 1/ρ, where ρ 1 (for the concavity). The elasticity of substitution between factors is given by 1/(1+ρ). 2) a constant growth rate of labour L denoted by n, 3) a constant growth rate of technologyadenoted by g, and 4) a constant rate of depreciation of capital denoted by δ. 1. Derive the capital accumulation equation; 2. determine the conditions for the existence of a not trivial (locally) stable equilibrium; 3. discuss the implications for the dynamics of the model and the equilibrium of a change inρ; 4. determine the conditions under which even if with g = 0 economy is growing in the long run; 5. discuss the dynamics of the marginal return on capital, i.e. the interest rate, if the conditions at the previous point are satisfied. 6
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