Question

There is a consumer who lives for two periods. His income is given by Y1 and Y2. He has access to the credit market with the interest rate r.

The government collects lump-sum taxes T1 and T2 (note that T1 and T2 might be negative meaning that the government makes a transfer). The government can run a surplus or a deficit, but must borrow (or save) in the credit market at the interest rate r.

Assume that the government is not constraint by the balanced budget and can have deficit in both periods. Now the government is being generous and pays transfers −T1 > 0 and −T2 > 0. Show the new consumption choice.

Answer 1

Consider the given problem here consumer having income “Y1” in period1 and “Y2” in period2. So, the sum of “present consumption” and “savings” should be equal to “present disposable income”, => C1 + S = Y1+T1, where “T1” be the present transfer to the consumer. Now, the future consumption should be sum of future disposable income and the “future value of savings”, => C2 = S*(1+r) + (Y2+T2). The value of “S” is “Y1+T1-C1”, where “T2” be the future transfer to the consumer.

=> C2 = S*(1+r) + Y2+T2, => C2 = (Y1-T1-C1)*(1+r) + Y2+T2.

=> C2/(1+r) = (Y1+T1-C1) + (Y2+T2)/(1+r),

=> C1 + C2/(1+r) = (Y1+T1) + (Y2+T2)/(1+r), be the consumer’s intertemporal budget constraint. The following fig shows the intertemporal budget constraint of the consumer, where “w1” be the endowment point where “C1” is equal to “Y1+T1” and “C2” is equal to “Y2+T2”.

Future Consumption (C2) E1 U2 W1 c*1 yi Y1+T1 A2 A4 Present Consumption (CI) C3

So, the new budget line is given by “A3A4” and the new equilibrium point is “E2” where the budget line “A3A4” is tangent to “U2”. So, here the both the present and futute consumption increases.