Q1 [Inflation in a monetary OLG model] Consider an overlapping generations model in which individuals live for two periods, young and old. The economy begins in period 1, when there are NO numbers of the initial old. In each period t2 1, Nt young individuals are born, where Nt = nNt-1 and n > 1. There is only one good in this economy. The good cannot be stored from one period to the next. In each period, each young individual is endowed with y units of the consumption good and old individuals have zero endowment. The young generations' preference can be expressed use the following utility function: u(c1,t, c2,t+1) = log(c1,t) + B log(c2,t+1) Members of the initial old generation only live for one period and have utility u(c0,1) = logc0,1. In this economy, individuals can use fiat money to facilitate trades between different generations. Assume the stock of fiat money M grows at a constant rate y, i.e., Mt = yMt-1, where y > 1. The money created each period is used to finance a lump- sum subsidy of at+1 goods to each old individual. (a) Set up the central planner's problem and solve for the (stationary) Pareto efficient allocation (cPE,cPE). (b) Write the government's budget constraint in period t + 1. (c) Define a monetary competitive equilibrium for this economy.
Q1 [Inflation in a monetary OLG model] Consider an overlapping generations model in which individuals live for two periods, young and old. The economy begins in period 1, when there are NO numbers of the initial old. In each period t2 1, Nt young individuals are born, where Nt = nNt-1 and n > 1. There is only one good in this economy. The good cannot be stored from one period to the next. In each period, each young individual is endowed with y units of the consumption good and old individuals have zero endowment. The young generations' preference can be expressed use the following utility function: u(c1,t, c2,t+1) = log(c1,t) + B log(c2,t+1) Members of the initial old generation only live for one period and have utility u(c0,1) = logc0,1. In this economy, individuals can use fiat money to facilitate trades between different generations. Assume the stock of fiat money M grows at a constant rate y, i.e., Mt = yMt-1, where y > 1. The money created each period is used to finance a lump- sum subsidy of at+1 goods to each old individual. (a) Set up the central planner's problem and solve for the (stationary) Pareto efficient allocation (cPE,cPE). (b) Write the government's budget constraint in period t + 1. (c) Define a monetary competitive equilibrium for this economy.
Chapter27: Issues In Macroeconomic Theory And Policy
Section: Chapter Questions
Problem 6P
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![Q1 [Inflation in a monetary OLG model] Consider an overlapping generations model in which individuals live for two periods, young and old. The economy begins in period 1,
when there are NO numbers of the initial old. In each period t 2 1, Nt young individuals are born, where Nt = nNt-1 and n>1. There is only one good in this economy. The good
cannot be stored from one period to the next. In each period, each young individual is endowed with y units of the consumption good and old individuals have zero
endowment. The young generations' preference can be expressed use the following utility function:
u(c1,t, c2,t+1) = log(c1,t) + B log(c2,t+1)
Members of the initial old generation only live for one period and have utility u(co,1) = logc0,1. In this economy, individuals can use fiat money to facilitate trades between
different generations. Assume the stock of fiat money M grows at a constant rate y, i.e., Mt = yMt-1, where y >1. The money created each period is used to finance a lump-
sum subsidy of at+1 goods to each old individual.
(a) Set up the central planner's problem and solve for the (stationary) Pareto efficient allocation (cPE,cPE).
(b) Write the government's budget constraint in period t+ 1.
(c) Define a monetary competitive equilibrium for this economy.
(d) Solve for the rate of return of money ot+1/ot and the growth rate of the price level pt+1/pt in a stationary equilibrium.
(e) Solve for the consumption allocation (c*1,c*2) and a lump-sum subsidy a* in a stationary competitive equilibrium.
(f) Draw a graph to show the Pareto Optimal allocation and the competitive equilibrium allocation with inflation, and verify that the former achieves a higher utility level.
Now, suppose the government does not provide subsidy to the individuals. Instead, the government has to finance its consumption through both money creation and taxes
G) Department of Economics Suppose the government tax each young individual's endowment at a rate tk. Set up a young individual's utility maximization problem and
solve for the optimal consumption allocation in the competitive equilibrium.
H) Suppose the government imposes a consumption tax on each young individual at a rate tc. Set up a young individual's utility maximization problem and solve for the
optimal consumption allocation in the competitive equilibrium.
I) Suppose the government imposes a lump-sum tax of t units of goods on each young individual in the economy. Set up a young individual's utility max- imitation problem
and solve for the optimal consumption allocation in the competitive equilibrium.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8816abe-4bfb-4b22-87b8-94e500bd3691%2F3cb698c1-dd0d-494b-9f39-ab9601e7f1f4%2Fsqa446_processed.png&w=3840&q=75)
Transcribed Image Text:Q1 [Inflation in a monetary OLG model] Consider an overlapping generations model in which individuals live for two periods, young and old. The economy begins in period 1,
when there are NO numbers of the initial old. In each period t 2 1, Nt young individuals are born, where Nt = nNt-1 and n>1. There is only one good in this economy. The good
cannot be stored from one period to the next. In each period, each young individual is endowed with y units of the consumption good and old individuals have zero
endowment. The young generations' preference can be expressed use the following utility function:
u(c1,t, c2,t+1) = log(c1,t) + B log(c2,t+1)
Members of the initial old generation only live for one period and have utility u(co,1) = logc0,1. In this economy, individuals can use fiat money to facilitate trades between
different generations. Assume the stock of fiat money M grows at a constant rate y, i.e., Mt = yMt-1, where y >1. The money created each period is used to finance a lump-
sum subsidy of at+1 goods to each old individual.
(a) Set up the central planner's problem and solve for the (stationary) Pareto efficient allocation (cPE,cPE).
(b) Write the government's budget constraint in period t+ 1.
(c) Define a monetary competitive equilibrium for this economy.
(d) Solve for the rate of return of money ot+1/ot and the growth rate of the price level pt+1/pt in a stationary equilibrium.
(e) Solve for the consumption allocation (c*1,c*2) and a lump-sum subsidy a* in a stationary competitive equilibrium.
(f) Draw a graph to show the Pareto Optimal allocation and the competitive equilibrium allocation with inflation, and verify that the former achieves a higher utility level.
Now, suppose the government does not provide subsidy to the individuals. Instead, the government has to finance its consumption through both money creation and taxes
G) Department of Economics Suppose the government tax each young individual's endowment at a rate tk. Set up a young individual's utility maximization problem and
solve for the optimal consumption allocation in the competitive equilibrium.
H) Suppose the government imposes a consumption tax on each young individual at a rate tc. Set up a young individual's utility maximization problem and solve for the
optimal consumption allocation in the competitive equilibrium.
I) Suppose the government imposes a lump-sum tax of t units of goods on each young individual in the economy. Set up a young individual's utility max- imitation problem
and solve for the optimal consumption allocation in the competitive equilibrium.
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