Paul and Anita value consumpti
Paul and Anita value consumption in period 0 (C0) andin 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) is132. Anita’s income is 132 in period 0 and 99 in period 1. BothPaul and Anita pay 22 in taxes in period 0 and in period 1 (i.e.t0 =t1 = 22). Anita can borrow or save at theinterest rater. However, everybody knows that Paul is dishonest; Asa result, nobody is willing to lend to him. Of course, Paul canstill save at the interest rate r. Suppose that r=0.1.
a) Determine how much money Paul would consume in period 0 andin period 1 if he was able to borrow. Determine his actualconsumption in period 0 and in period 1. Illustrate bothallocations on a graph. What is the cost of this creditconstraint?
b) Determine Anita’s optimal consumption plan. Find the value ofso which allows Anita to achieve this plan.
Answer:
Let us first consider the general budget constraint of Paul andAnita.
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 ofperiod 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 equilibriumwithout borrowing constraint as follows –
But this is not the actual consumption bundle of Paul, becausePaul is constrained by borrowing constraint. So the actualconsumption bundle of Paul can be calculated as follows –
Without borrowing constraint Paul will consume 100 units inperiod 0. But due to the boroowing constraint Paul can consumemaximum amount of (Y0-t0) = 102 – 22 =80 unit. Hence paul can’tconsume 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’tconsume more, hence he will consume the maximum he can attain underthis constraint which is given by 80)
Since here under the optimal under borrowing constraint Paul isnot maximizing utility. So we will calculate here the optimal ofperiod 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 –
wherethe downward sloping line with vertical intercept of 198 andhorizontal intercept of 180 denotes the budget line withoutborrowing constraint and the bold line with vertical portion at 80is the budget line with borrowing constraint (since with borrowingconstraint Paul can’t consume more than 80 unit in period 0, so thebudget line takes that peculiar shape)
E denotes the equilibrium without borrowing constraint and E1represents the equilibrium with borrowing constraint.
It seems paul with borrowing constraint attains equilibrium at alower indifference curve, hence it can be stated that Paul is worseoff under the borrowing constraint.
b) Anita don’t have any borrowing constraint, so we can getoptimal of Anita by using the optimal we have derived through theLagrangian.
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 –
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