Exercises on chapter 4

Transcription

1 Exercises on chapter 4 Exercise : OLG model with a CES production function This exercise studies the dynamics of the standard OLG model with a utility function given by: and a CES production function: U(c t ; d t+ ) = ( a) ln c t + a ln d t+ F (K; L) = A K + L All other assumptions are as in the course. Find the dynamics of the variable k t = K t =N t : Prove that if aa > 4( + n); there exists three steady states (0 and two positive steady states). Show that there exists a poverty trap for k 0 to small. Exercise 2: OLG model with agents living 3 periods A model with overlapping generations is considered, in which N t agents are born at each period t. N t increases at a constant rate n : N t = ( + n)n t : Each agent is living during 3 periods. He supplies one unit of labor during his two rst periods of life, and is retired during the last period. The utility function of a generation t agent depends on his consumptions during the three periods, denoted by c t ; d t+ and e t+2 ; and is de ned by: U (c t ; d t+ ; e t+2 ) = ln c t + 2 ln d t+ + 3 ln e t+2 The parameters i are positive and such that: = : s t is the amount of savings (the stock) held by the agent at the end of period t; and u t+ this amount at the end of period t + : The production technology is Cobb-Douglas: Y t = F (K t ; L t ) = K t L There is full depreciation of capital in one period. w t is the wage in period t, R t the factor of interest.. Write the 3 budget constraints of an agent of generation t, and the intertemporal budget constraint. Find the consumptions. Show that savings are equal to: s t = ( )w t w t+ R t+ u t+ = 3 R t+ w t + 3 w t+ t

2 2. Write the rm program at period t: Give the optimality conditions. What is the total amount of labor supply at period t? The variable k t is de ned as k t = K t =(N t +N t ): Assuming an equilibrium on the labor market, give the equilibrium value of the wage w t and of the interest factor R t with respect to k t. 3. Write the equation expressing the equilibrium of the capital market. Show that k t follows a recurrence equation of order 2: an equation between k t+ ; k t and k t : Explain why the initial conditions need to make precise the values of both k 0 and k : 4. A change of variable is introduced: x t = k t =kt : Is x t a backward or forward variable? Show that the dynamics of x t follows: 2 + n + x t+ = ( )( ) + 3 ( ) + xt + n Study this dynamics with a graph. Show that x t converges toward a stationary state with oscillations. The stationary state is denoted by x (it is not asked to nd its value). 5. Write the program allowing to de ne the optimal stationary state of the economy. How is modi ed the standard golden rule? What is the value of x at the golden rule? Find the condition on the parameters of the economy ensuring that the stationary state of the competitive economy is optimal? Exercise 3: Growth in an OLG model with a PAYG pension system A model with overlapping generations is considered, in which N t agents are born at each period t. N t increases at a constant rate n : N t = ( + n)n t : Each agent is living during 2 periods. He supplies one unit of labor during his rst period of life, and is retired during the second period. The utility function of a generation t agent depends on his consumptions during the two periods, denoted by c t and d t+ ; and is de ned by: U (c t ; d t+ ) = ( a) ln c t + a ln d t+ The parameter a is such that: 0 < a < : s t is the amount of savings held by the agent at the end of period t: The production technology is Cobb-Douglas: Y t = K t (B t L t ) K t is the capital stock, L t the quantity of labor. B t represents the technical progress. Its expression will be de ned after. There is full 2

3 depreciation of capital in one period. w t is the wage in period t, R t the factor of interest.. Write the budget constraints of an agent of generation t, and his intertemporal budget constraint. Find the optimal levels of consumptions and savings. 2. Write the rm program at period t: Give the optimality conditions. 3. It is assumed that the technical progress B t is de ned by B t = K t =N t. Can you interpret this assumption? Under this assumption, and using the equilibrium condition on the labor market, give the expressions of w t and R t with respect to k t = K t =N t : 4. Write the equation expressing the equilibrium of the capital market. Show that k t increases at a constant rate. Explain the impact of the di erent parameters on this growth rate. 5. A PAYG pension system is introduced in the economy. Young agents income is taxed at a constant rate : This tax nances a pension for old agents. p t denotes the amount of the pension received by old agents in period t (agents of generation t ). 5.a Write the new budget constraints and solve the consumer program, nd the new amount of savings. 5.b Write the equation ensuring that the budget constraint of the social security system is balanced at each period t. 5.c Find the new expression of the dynamics of k t : What is the impact of on the growth rate of k t? How can you interpret this result? Exercise 4: Growth in an OLG model with transfers A model with overlapping generations is considered, in which N t agents are born at each period t. N t increases at a constant rate n : N t = ( + n)n t : Each agent is living during 2 periods. He supplies one unit of labor during his rst period of life, and is retired during the second period. The utility function of a generation t agent depends on his consumptions during the two periods, denoted by c t and d t+ ; and is de ned by: U (c t ; d t+ ) = ( a) ln c t + a ln d t+ The parameter a is such that: 0 < a < : s t is the amount of savings held by the agent at the end of period t: The production technology is: Y t = AK t + K t (L t ) K t is the capital stock, L t the quantity of labor. A is a positive parameter. There is full depreciation of capital in one period. w t is the wage in period t, R t the factor of interest. 3

4 . Write the budget constraints of an agent of generation t, and his intertemporal budget constraint. Find the optimal levels of consumptions and savings. 2. Write the rm s program at period t: Give the optimality conditions. 3. Using the equilibrium condition on the labor market, give the expressions of w t and R t with respect to k t = K t =N t : 4. Write the equation expressing the equilibrium of the capital market. Make a drawing to describe the dynamics of k t. Show that there is convergence toward a steady state k : 5. Write the program that de nes the optimal steady state of this economy. Show that this steady state only exists if A < + n: Under this inequality, what is the condition ensuring that the competitive steady state is in overaccumulation? How can you interpret the case A > +n? 6. In this question, a tax rate ; < ; is introduced at each period on capital income. It means that a share of the second period income of old agents is taken by the government. It remains a share ( ) for their second period consumption. At each period t; this tax on old agents is used by the government to nance an additional income for young agents. The government s budget is balanced at each period. 6.a Show that this assumption implies that each young agent in period t receives a transfer t such that: R t K t = t N t Write the new budget constraints at each period for a generation t agent, and solve the consumer program. Find the new amount of savings. 6.b Find the new expression of the dynamics of k t : 6.c In this question, it is assumed that aa=( + n) > : Show that if is high enough, k t increases in the long run with a positive growth rate. Can you interpret the result? 6.d In this question, it is assumed that aa=( + n) < : What can you say on the long run dynamics of k t? Exercise 5: Growth in an OLG model with a public good A model with overlapping generations is considered, in which N agents are born at each period t. N is constant. Each agent is living during 2 periods. He supplies one unit of labor during his rst period of life, and is retired during the second period. The utility function of a generation t agent depends on his consumptions during the two periods, denoted by c t and d t+ ; and is de ned by: U (c t ; d t+ ) = ( a) ln c t + a ln d t+ The parameter a is such that: 0 < a < : s t is the amount of savings held by the agent at the end of period t: 4

5 The production technology is: Y t = K t L t Gt K t is the capital stock, L t the quantity of labor. There is full depreciation of capital in one period. w t is the wage in period t, R t the factor of interest. G t is the amount of public good that is provided by the government. This amount G t is nanced by a tax on the income w t of young agents at period t; at a constant rate (w t is paid by each generation t agent). The government does not issue debts: the amount of public good corresponds to the amount of taxes. G t is an externality for rms.. How can you interpret the property that output Y t depends on G t? 2. Write the budget constraints of an agent of generation t, and his intertemporal budget constraint. Find the optimal levels of consumptions and savings. 3. Write the rm s program at period t: Give the optimality conditions. Using the equilibrium condition on the labor market, give the expressions of w t and R t with respect to K t, N and G t : 4. Give the expression of G t with respect to w t ; taking into account that the government budget is balanced. Using question 3, nd the value of w t with respect to K t ; N and : 5. Using the equilibrium condition on the capital martket, nd the dynamics of K t : Show that K t increases with a constant rate. Find the value of that maximizes the growth rate. Exercise 6: pension systems Part A: a PAYG pension system A model with overlapping generations is considered, in which N new agents are born at each period t, N being a constant parameter. Each agent is living during 2 periods. He/she supplies one unit of labor during his rst period of life, and is retired during the second period. The utility function of a generation t agent depends on his consumptions during the two periods, denoted by c t and d t+ ; and is de ned by: U (c t ; d t+ ) = ( a) ln c t + a ln d t+ The parameter a is such that: 0 < a < : s t is the amount of savings held by the agent at the end of period t: The production technology is Cobb-Douglas: Y t = K t L K t is the capital stock, L t the quantity of labor. There is full depreciation of capital in one period. w t is the wage in period t, R t the factor of interest. All markets are perfectly competitive. 5 t

6 There exists a PAYG pension system in the economy. Young agents income w t is taxed at a constant rate : This tax nances a pension for old agents. p t denotes the amount of the pension received by each old agent in period t (agent of generation t ).. Write the budget constraints of an agent of generation t, and his intertemporal budget constraint. Find the optimal levels of consumptions and savings. 2. It is assumed that the budget constraint of the pension system is balanced at each period t: What is the value of p t with respect to w t? 3. Write the rm s program at period t: Give the optimality conditions. Using the equilibrium condition on the labor market, give the expressions of w t and R t with respect to k t = K t =N: 4. Write the equation expressing the equilibrium of the capital market. Show that k t follows a dynamics given by the equation: k t+ = a( )k t () Gives the expression of the function : Show that this function decreases with and interpret the result. 5. Show (on a gure) that k t converges towards a steady state k (): Give the value of k (): 6. De ne the optimal stationary state of the economy. Show that it is possible to choose a value of = ^ such that the stationary state of the competitive economy corresponds to the optimal stationary state. Discuss the sign of ^: How can you interpret the result? If = ^, can you explain why the consumptions c and d for the competitive equilibrium at steady state are equal to the optimal levels? Part B: a fully funded pension system The economy is the same as in part A except that the pension system is a fully funded system. A generation t agent pays a tax at the rate on his wage w t : The amount w t is invested by the pension system in capital. The pension received in t + is p t+ = R t+ w t :. Find the new value of s t. What is the total amount of savings in this economy. Characterize the intertemporal equilibrium of the economy. What is the impact of such a system on the economy? 2. In this question, it is assumed that nancial markets are imperfect: private savings are constrained to be non-negative, s t 0: What is the amount of savings s t for a generation t agent? What is the total amount of savings in the economy? How do these results depend on the value of? Find the new steady state value of k t : How does this value depend on : Explain the result. Is it possible to reach the optimal stationary state for an appropriate choice of? Exercise 7: Pollution and accumulation (60% of marks) 6

7 I The competitive economy An Overlapping Generations Model is considered in which at each period t, agent is born. Each new born agent lives during 2 periods, supplies one unit of labor during the rst period of life, and is retired in the second period. Population is then constant and equal to 2 ( young + old). The utility function of a generation t agent is given by a function U (c t ; d t+ ; P t+ ) = 2 ln c t + 2 ln d t+ P t+ c t is the quantity of the aggregate good of the economy that is consumed in the rst period of life, and d t+ is the quantity for the second period. P t+ is the level of pollution in period t + : It is an externality for the agent (taken as given). Savings of the agent are denoted by s t. The production of the aggregate good is obtained with a production technology Y t = F (K t ; L t ) = AK =2 t L =2 t that uses two production factors: labor and capital. Capital fully depreciates at each period. w t is the real wage in period t, R t the real interest factor. Pollution is a by-product of production and has the following evolution: P t+ = ( )P t + Y t with such that 0 < < : It is also an externality for the rm (taken as given). ) Interpret the evolution law of pollution. 2) Write the budget constraints of a generation t agent and the intertemporal constraint. Write the consumer program and nd the expressions of c t ; d t+ and s t : 3) Write the rm s program at period t and nd the optimality conditions. Express the equilibrium condition on the labor market and nd the equilibrium values of prices w t and R t with respect to K t : 4) Using the equation expressing the equilibrium on the capital market, nd the dynamics of K t. Show on a gure that this dynamics has a positive stable steady state. Find the value of this steady state. 5) Find the long run value of the stock of pollution P: q 8 3 6) Calculate for A = steady state: K; c; d; P. II The optimal stationary state ) The productivist social planner the values of all variables at the competitive 7

8 In this question, the social planner has no concern for environment. He determines the optimal stationary state of the economy, taking into account the objective function: 2 ln c + 2 ln d a) Write the program that de nes the optimal productivist stationary state (for a general value of A). b) Solve this program (for a general value of A). Then, take A = p 8=3 and compare the results with the competitive solution. How can you explain the result? 2) The green social planer In this question, the social planner cares about environment. He determines the optimal stationary state of the economy, taking into account the externality of pollution induced by production. To determine the optimal stationary state, he uses the objective function: 2 ln c + 2 ln d P 2a) Write the program that de nes the green optimal stationary state of the economy (for a general value of A). In eliminating k; show that this program can be expressed only with the variables c; d and P: 2b) Take A = p 8=3 and solve this program (using a Lagrangian). The easiest way to solve is to calculate rst the value of =; with the shadow price associated with the constraint. Give the values of P; c; d and K and compare these values with the competitive steady state and with the productivist optimal state. Can you interpret the results? 2c) Assume that the government wants to decentralize the green optimal stationary state in the competitive economy. What could be the appropriate instrument(s) for this decentralization? 8