Question

Assume the representative consumer lives in two periods and his preferences can be described

by U(c, c')=c^(1/2)+β(c')^(1/2)

where c is the current consumption, c' is next period consumption, and β = 0.95. Let’s assume that the consumer can borrow or lend at the interest rate r = 10%. The consumer receives an income y = 100 in the current period and y' = 110 in the next period. The government wants

to spend G = 30 in the current period and G' = 35 in the future period.

1. Solve the consumer’s problem by finding the optimal allocations c^* and c^(l*). [10 points]

2. Is the economy at the equilibrium? Explain. [05 points]

3. What are the equilibrium values of c and c'? [05 points]

4. What is the equilibrium interest rate? [05 points]

5. How will the equilibrium interest rate respond to an increase in G? [05 points]

6. How will the equilibrium interest rate respond to an increase in G'? [05 points]

Answer 1

U(c, c')=c^(1/2)+β(c')^(1/2)

β = 0.95, y = 100, y' = 110, G = 30, G' = 35, r = 10%

so U(c, c')=c^(1/2)+0.95(c')^(1/2)

A) solving for c and c' maximise U(c,c') subject to c + c'/(1+r) = y + y'/(1+r) that is the present value of total consumption must be equal to the present value of income at r = 10%

U(c, c')=c^(1/2)+0.95(c')^(1/2) where c + c'/(1.1) = 100+ 110/(1.1) = 100 + 100 = 200

U(c, c')=c^(1/2)+0.95(c')^(1/2) where c + c'/(1.1) = 200

or U(c, c')=c^(1/2)+0.95(c')^(1/2) where c = 200 - c'/(1.1)

putting the value of c in utility function, we get

U(c') = [200 - c'/(1.1)]^(1/2) + 0.95(c')^(1/2)

first order derivative

solving for c'

c' = 109.75

c =  200 - c'/(1.1) = 200 - 109.75/1.1 = 200- 99.77 = 100.22

B) total expenditure in period 1 = c+g = 100.22 + 30 = 130.22

total expenditure in pd = c'+g' = 109.75+35 = 144.75

present value of total expenditure = 130.22 + 144.75/1.1 = 130.22 + 131.59 = 261.81

however, present value of total income is 100 + 110/1.1 = 200

since total expenditure in present value is more that present value of total income, economy is not in equilibrium.

C) equilibrium value of c,c' will be such that

c+g + (c'+g')/(1.1) = 200

c + 30 + c'/1.1 + 35/1.1 = 200

c+c'/1.1 = 200-30-35/1.1 = 138.18

c+c'/1.1 = 138.18 has to be the constraint on utility maximisation.

solving for c and c' same as in part a)

we will get

c' = 150.89/1.99 = 75.82

c = 138.18 - c'/(1.1) =138.18 - 75.82/1.1 = 138.18 - 68.92= 69.25