EFFM Economics 5005/In-course exam/2022

There are two questions.

●  Each question is worth 100 points.

● Complete only one question.

● Time allotted: One hour.

● Open books and notes.

1   1.1 Solve the following integral:                                             (10 points)

〇尸

3 e t dt

尸

ANswER.

尸〇尸 3 e t dt = 3  尸〇尸  e t dt = 3  et │〇尸  一 3  et │尸  = 3  e 〇尸  一 3  

= 6(e5 一 1) = 6(148.41 一 1) = 6 × 147.41 = 884.46

1.2  One-good model. Consider the utility function

1 一 e-yc1

γ

A consumer solves

max u(c)

c

subject to the constraint

y = cp

y = 35, 000       γ = 2.7       p = 12

i. As a preliminary step, write down the formula for u\ (c).

points)

ANswER.

u\ (c) = e-yc

ii. Write down the Lagrangian.

ANswER.

1 一 e-yc γ

(20 points)

+ λ (y 一 pc)

iii. Solve for optimal consumption c as a function of the Lagrange multiplier. [HINT: Start by writing down the first order condition for c. Then take the logarithm.]                                (25 points)

ANswER.

e-yc  = pλ

Now take logs

ln(pλ)

γ

iv. Solve for the Lagrange multiplier. [HINT: Substitute the answer from the previous question into the budget constraint.]

ANswER.     Substitute the solution for c into the budget con-

straint:

ln(pλ)

γ

Take the exponential:

y                        ln(pλ)                                  1

e p   = e-    γ         = (pλ)- γ

Now take the exponent 一γ on both sides and solve.

e-y  

p

(25 points)

2  Deterministic optimal growth model. Let

1 一 e-yc

γ

1 一 e-ak

α

A representative individual solves

o

max       e-ptu(ct )dt

尸

subject to

k˙t  = f (kt ) 一 ct           k尸  = k*

(a) Write the appropriate Hamiltonian

ANswER.

H = 1 一γ(e) -yc  + ξ ╱  1 一α(e) -ak  一 c、

(b) State the solution for consumption as a function of the costate vari-

able. [HINT: You might need to take a logarithm.]                      (20 points)

ANswER.

e-yc  = ξt  → c = 一

(c) State the equation for the costate variable.                      (10 points)

ANswER.

ξ˙t  = ξt (ρ 一 f\ (kt )) = ξt (ρ 一 e-ak )

(d) State the steady-state capital.                                         (20 points)

ANswER.   At the steady state

f\ (k) = ρ → e-ak  = ρ → k = 一

(e) State the steady-state interest rate.                                 (20 points)

ANswER.

r = f\ (k) = ρ

(f) State the flow of profit in the steady state.                      (20 points)

ANswER.

1 一α(e) -ak  一 ρe-ak  =  一 ╱  + ρ、e-ak

(g) State the price of an equity share in the representative firm in the

steady state.                                                                    (20 points)

ANswER.

 ╱  1 一α(e) -ak

一 ρe-ak\