Solution of EC904 Resit Exam 1. [20 marks] Answer both parts (a) and (b) of this question. Dynamic efficiency in the OLG economy. Consider the two period OLG economy discussed in the lectures with positive population growth n > 0 and zero depreciation δ = 0 . Assume that the household utility function is log c y t + β log c o t , where c y t , c o t represent consumption when young and old, respectively. Denote the wage as w t and fix the labor supply as 1. (a) Consider the simple partial equilibrium where wage and interest rate are fixed as w and r , respectively. In this simple setup, we can ignore capital k t . i. [5 marks] What is the decentralized consumption allocation for each generation, given wage w ? (Hint: solve for c y t , c o t as functions of w . No need to go further to discuss k ) Answer: c y t = w 1+ β , c y t = β (1+ r ) w 1+ β . ii. [5 marks] In this partial equilibrium, if the interest rate is low: r < n , is the decen- tralized allocation dynamically inefficient, i.e., is it possible to obtain an consump- tion allocation that is better than the decentralized allocation, for all generations? If it is possible, give an example. If it is not possible, find the conditions such that the decentralized allocation is dynamically inefficient. (Hint: try to find an alternative consumption allocation c y * t c y t , c o * t c o t for all t while c o * 0 > c o 0 for the first generation t = 0 .) Answer: Yes, it is dynamically inefficient. A better allocation: Increase c o 0 by (1 + n ) > 0 unit; decrease c y 0 by , increase c o 1 by (1 + n ) ; decrease c y 1 by , increase c o 2 by (1 + n ) ; etc. (b) [10 marks] If we consider the general equilibrium as in the lectures, where w t and r t are determined by k t , what is the condition for dynamical inefficiency at the steady state? If the economy is dynamically inefficient, what government policies can be used to restore the economy to efficiency? Give at least two examples and explain how these policies change capital and consumption allocations and restore the efficiency. Answer: The condition for dynamic inefficiency is r < n at the steady state. Government policies including pension, government debt and money issuing can be implemented to restore the efficiency. Pension: lumpsum tax on the young and lumpsum pension pay- ment to the old, so that household savings and capital decrease and r increases above n ; Government debt: issue positive debt so that part of household savings become govern- ment debt, and the rest of savings become capital, which is low enough and r is higher than n ; Money: issue money so that the part of household savings that are transformed into capital becomes low enough and r > n . 2. [40 marks] Answer all parts (a) - (e) of this question. Consider a standard Diamond-Mortensen-Pissarides model covered in the lecture. The econ- omy is populated by a unit measure of infinitely lived firms and workers who discount the 1
future at e - rt and are risk neutral. Firms post vacancies at a flow cost equal to c . Income of an unemployed worker is z . Market tightness is θ = v/u , where u denotes the unemployment rate and v denotes the vacancy rate. The aggregate matching function M ( u, v ) is homoge- neous of degree one. Each firm employs only one worker. A filled job generates a flow output x . Jobs are destroyed at an exogenous rate δ . The bargaining weights are β for the worker and 1 - β for the firm. The equilibrium can be characterized by the following six equations: rU = z + λ ( θ )[ E ( w ) - U ] , rE ( w ) = w + δ [ U - E ( w )] , rV = - c + λ ( θ ) θ [ J ( w ) - V ] , rJ ( w ) = x - w + δ [ V - J ( w )] , V = 0 , w ( θ ) = βx + (1 - β ) rU. where U, E, V, J are the value functions and w is the equilibrium wage. Let us use λ ( θ ) to denote job finding rate of unemployed workers as a function of θ , which comes from the matching function M ( u, v ) . (a) [5 marks] Pick at least 2 equations from the above 6 equations that characterize the equilibrium and explain their economic intuition. Answer: Equation 1: value of unemployment for worker; 2: value of employment with wage w for worker; 3: value of a vacancy for firm; 4: value of a filled job with wage w for firm; 5: free entry condition; 6: determination of wage, derived from Nash bargaining between worker and firm. (b) [10 marks] Show that the equilibrium conditions can be reduced to one equation (1 - β )( x - z ) = [ r + δ + λ ( θ ) β ] c λ ( θ ) , with one unknown variable - labor market tightness θ . Answer: follow the lecture notes. (c) [10 marks] Suppose the government sets a minimal wage w m . What is the impact of the minimum wage on market tightness θ and unemployment rate u , if the minimal wage is higher than the wage determined by the competitive equilibrium described above? Answer: The minimal wage reduces θ and increases u . (d) [5 marks] What is the impact of w m on the aggregate flow of workers who leave the unemployment pool at the steady state? Answer: w m reduces the flow out of the unemployment pool. Employment 1 - u de- creases, the flow into the unemployment pool δ (1 - u ) decreases, and at the steady state, the outflow and inflow of the unemployment pool are the same. (e) [10 marks] If there is a positive productivity shock, i.e., x increases, how does the new steady-state equilibrium, especially θ and u , change? Answer: θ increases and u decreases. 3. [40 marks] Answer both parts (a) and (b) of this question. Consider a continuous time growth model. The representative household supplies labor in- 2
elastically, and chooses the consumption path to maximize utility: U 0 = Z 0 log ( c t ) · e - ρt dt, subject to the budget constraint: ˙ a t = (1 - ξ t ) r t a t + w t - c t - τ t . The initial asset is given by a 0 . ξ t is the tax rate on asset incomes and τ t is the lumpsum tax at time t. The household takes the return on asset r t , the real wage w t , asset-income tax rates and lumpsum tax as given. Factor markets are competitive. The production technology is given by: y t = k α t , where y t is output per capita and k t is the capital-labor ratio. Assume that α < 1 and there is neither population growth nor capital depreciation. The initial capital stock is given by k 0 . Government expenditure g t is can be financed using lumpsum and asset-income taxes, and its budget balances in each period: g t = ξ t r t a t + τ t . Wages and rental rates for capital are determined in a competitive fashion. The other market equilibrium conditions are given by: y t = c t + g t + ˙ k t , k t = a t . (a) Assume that government expenditure is exogenously given as a constant and financed using only lumpsum tax: g t = g > 0 , τ t > 0 and ξ t = 0 . i. [10 marks] Derive the dynamic equations which characterize the equilibrium in this economy. Answer: Two equivalent approaches: (1) Write the social planner's problem and derive FOCs; (2) Derive household Euler equation and use k to substitute a . Even- tually, the dynamic equations are: ˙ c c = αk α - 1 - ρ, ˙ k = k α - c - g. ii. [5 marks] Explain in a diagram how the economy moves from the initial state to the steady state, assuming that the initial capital-labor ratio is lower than the steady- state level. Answer: First, draw the curves ˙ c = 0 and ˙ k = 0 . The intersection pins down the steady state. Starting from a initially low capital level, c and k increase towards the steady state levels. iii. [5 marks] Assume that the economy is at the steady-state at time t 0 . Surprisingly, the government expenditure g increases to g 0 > g from time t 0 on, permanently. How does the economy respond to the increase of government expenditure? Illustrate in a diagram and provide an intuitive explanation. Answer: From time t 0 on, the curve ˙ k = 0 shifts downward, and the curve ˙ c = 0 3
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