Question

Paul and Anita value consumption in period 0 (C₀) and in period 1 (C₁) using the same utility function u = ln(C₀)+0.8In(C₁). Paul's income in period 0 (y₀) is 102 while that in period 1 (y₁) is 132. Anita's income is 132 in period 0 and 99 in period 1. Both Paul and Anita pay 22 in taxes in period 0 and in period 1 (i.e. t₀ =t₁ = 22). Anita can borrow or save at the interest rater. However, everybody knows that Paul is dishonest; As a result, nobody is willing to lend to him. Of course, Paul can still save at the interest rate r. Suppose that r=0.1.

a) Determine how much money Paul would consume in period 0 and in period 1 if he was able to borrow. Determine his actual consumption in period 0 and in period 1. Illustrate both allocations on a graph. What is the cost of this credit constraint?

b) Determine Anita's optimal consumption plan. Find the value of so which allows Anita to achieve this plan.

Answer 1

Let us first consider the general budget constraint of Paul and Anita.

Y0 is the income of period 0 and Y1 is income of period 1.

C0 is the consumption of period 1 and C1 is the consumption of period 2.

t0 and t1 is the tax in period0 and period1 respectively.

r be the interest rate.

Hence the budget constraint becomes -

a)

Given Paul's income in period 0 : Y0=102

Paul's income in period 1 : Y1=132

Paul pays tax at period 0: t0=22

Paul pays at period 1: t1 = 22

Rate of interest : r=0.1

Hence with this information we can calculate Paul's equilibrium without borrowing constraint as follows -

But this is not the actual consumption bundle of Paul, because Paul is constrained by borrowing constraint. So the actual consumption bundle of Paul can be calculated as follows -

Without borrowing constraint Paul will consume 100 units in period 0. But due to the boroowing constraint Paul can consume maximum amount of (Y0-t0) = 102 - 22 =80 unit. Hence paul can't consume 100 unit under borrowing constraint.

So the optimal consumption of Paul in period 0 is 80 unit. (Since paul is willing to consume more than 80 unit but he can't consume more, hence he will consume the maximum he can attain under this constraint which is given by 80)

Since here under the optimal under borrowing constraint Paul is not maximizing utility. So we will calculate here the optimal of period 2 from the budget constraint.

given, C0=80, Y0=102, Y1=132, t0=22, t1=22, r=0.1

The optimal of period 2:

The graphical representation can be given as follows -

where the downward sloping line with vertical intercept of 198 and horizontal intercept of 180 denotes the budget line without borrowing constraint and the bold line with vertical portion at 80 is the budget line with borrowing constraint (since with borrowing constraint Paul can't consume more than 80 unit in period 0, so the budget line takes that peculiar shape)

E denotes the equilibrium without borrowing constraint and E1 represents the equilibrium with borrowing constraint.

It seems paul with borrowing constraint attains equilibrium at a lower indifference curve, hence it can be stated that Paul is worse off under the borrowing constraint.

b) Anita don't have any borrowing constraint, so we can get optimal of Anita by using the optimal we have derived through the Lagrangian.

Anita's Income in period 0: Y0=132

Anita's Income in period 1: Y1=99

Tax at both period, t0=t1=22

Rate of interest, r= 0.1

Hence optimal consumption bundle of Anita is -