Question

(1) P.J. is trying to decide how much to consume in time periods 1 and 2, (c, c2). He will get $100 in period 1 and $100 in period 2 . He can save and borrow as much as he wants between the two time periods at an interest rate of 20%. If PJ.'s utility function is U(G, сг) = c1c2, what level of consumption does he choose in period 2 if the inflation rate is also 10%? Assume that pi-Si. 9 (a) 100 (b) 140 (c) 180 (d) 80 (e) None of the above

Answer 1

Solution:

Correct option is (a)100. This is how:

U(c1, c2) = c1*c2

Income in period 1, m1 = $100, income in period 2, m2 = $100

Price in period 1, p1 = $1

With inflation of 10%, price in period 2, p2 = (1 + inflation rate)*p1 = (1 + 0.1)*p1 = 1.1*p1

Then, our P.J.'s budget line in period 1 is: p1*c1 + s = m1; where s is savings (also note that savings is negative of borrowings, so if we represent borrowings by b, then s = -b)

Period 1 budget line: 1*c1 + s = 100 ... (1)

This gives us, s = 100 - c1

Period 2 budget line: p2*c2 = m2 + (1 + r)*s; where r is the interest rate

(Why? Since, the amount saved (borrowed) in period 1 will get returns in period 2, so we receive (pay) the amount saved (borrowed) with returns in period 2; this acts as an additional income (expenditure) in period 2)

Period 2 budget line: 1.1*c2 = 100 + (1 + 0.2)*s ... (2)

Substituting the value of s from (1), in (2), we get

1.1*c2 = 100 + (1.2)*(100 - c1)

1.1*c2 + 1.2*c1 = 100 + 1.2*100

1.2*c1 + 1.1*c2 = 220

OR, c1 + (1.1/1.2)*c2 = 220/(1.2)

This is our inter-temporal budget line, with income = $220, price in period 1, P1 = $1.2, and price in period 2, P2 = $1.1

Now, P.J. aims at maximizing own utility. We know that utility is maximized where the Marginal Rate of Substitution, MRS (of consumption in period 1 and period 2) equals the price ratio:

MRS_c1,c2 = P1/P2

MRS_c1,c2 = Marginal utility from consumption in period 1 (MUc1)/ Marginal utility from consumption in perriod 2 (MUc2)

MUc1 = $\partial U/\partial c1$ = c2

MUc2 = $\partial U/\partial c2$ = c1

So, MRS_c1,c2 = c2/c1

Also, P1/P2 = 1.2/1.1

Then, using the utility maximizing condition: c2/c1 = 1.2/1.1 or 1.1*c2 = 1.2*c1

Substituting this in the budget line, we get

1.1*c2 + 1.1*c2 = 220

2.2*c2 = 220

c2 = 220/2.2 = 100

So, consumption level in period 2 chosen by P.J is 100. So, correct answer here is (a) 100.