modern monetary economics

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QUESTION 1. (12 points) Consider an Overlapping Generation (OLG) model with individuals

living for two periods; young and old. Let Nt denote the population of the young at time t

and assume the population is growing at, Nt = n Nt-1. Suppose each individual receives an

endowment of a perishable good y when young and nothing when old.

Let the utility function of an individual be:

U (c1,t , c2,t+1) = !

!"# 𝑐!,& !"# + !

!"# 𝑐(,&)! !"# ; 0 < 𝛽 < 1

where c1,t and c2,t are consumption when young and consumption when old, respectively. The

consumption function is strictly increasing and concave in both the arguments.

a. Find the feasible set line in a stationary environment?

b. Derive the first order condition characterizing the Golden Rule allocations and find

the optimal solution (c1*, c2*). [If you find your optimal choices are not easy to

summarize, just leave them as it is.]

c. Depict the Golden Rule allocations in a diagram.

QUESTION 2. (17 points) Now consider the same environment as in Question 1 and suppose

that the government prints new fiat money in each period at a constant rate z, Mt = z Mt-1 ;

z>1. The newly printed money is then distributed equally among the current old as a lump-

sum transfer at .

a. Use the money market equilibrium to derive the gross real rate of return of fiat money

in the stationary environment.

b. Find the life-time budget constraint.

c. Derive the first order condition characterizing the optimal choices of a young

individual born in time t. Derive the allocations c*1, c*2 where q(=vtmt) is the real

demand for money by an individual. [you do not need to summarize your findings of

c*1, c*2]

d. Now assume 𝛽 = 0.5. Fin the real demand for money by an individual at the optimal solution. [you do not need to summarize your finding]

e. Depict the result of the in the monetary equilibrium in the same diagram in part c of

Question 1.

f. Is monetary equilibrium as efficient as Golden Rule allocation? Explain.

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QUESTION 3. (8 points) Suppose the economy has the same environment as in Question 2.

Now assume that the newly printed fiat money does not distribute as lump sum transfer.

Instead, the government uses the newly printed fiat money as its seigniorage revenue. Also,

government collects a non-distortionary tax τ from old generation.

a. What is government budget constraint? b. What is the feasible set line? c. Without solving the problem, compare the results of competitive equilibrium with

golden rule allocation. Which one is more efficient? Explain. [Hint: you could compare

the lifetime budget constraint and feasible set line together]

Question 4. (13 points) Consider a two-period Overlapping Generations (OLG) model with

individuals living for two periods where population is growing at a rate of n, Nt=n Nt-1. Let the

utility function of an individual born at time t be

U (c1,t , c2,t) = ! a 𝑐!,&3 +

! a 𝑐(,&3 ; 0 < 𝛼 < 1

Each young individual receives y units of a perishable good as an endowment when young and

nothing when old. In this environment, each individual young possess a technology that

converts kt units of capital goods saved in period t into k b t units of goods into the next period.

a. Find the feasible set line.

b. Set up the social planner problem. Derive the first order conditions of the social

planner problem and find the optimal level of capital investment k*.

c. Write down the intertemporal budget constraint of an individual.

d. Derive the first order conditions associated with the optimal choices of an

individual (You do not need to find the optimal choices of k*,c1*, c2* but the first

order conditions).

e. Would the optimal capital in the competitive equilibrium be above or under

accumulation in compare to Golden rule allocations?