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.

Write down the government intertemporal budget constraint. Note that the government also has the access to the credit market. Write down the consumer’s budget constraint. Show the consumption choice graphically.

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. 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”.

=> 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) Y2-T2 Y1-T1 A2 Present Consumption (C1)

Here the tax be the government’s income, => in period 1 government’s income is “T1” and in period 2 government’s income is “T2”. Now, “G1” be the current government spending, “G2” be the future government spending and “B” be the government’s budget.

=> B = T1-G1, and “G2 = T2 + B*(1+r)”.

=> G2 = T2 + (1+r)*(T1-G1), => G2/(1+r) = T2/(1+r) + (T1-G1).

=> G1 + G2/(1+r) = T1 + T2/(1+r), be the government’s intertemporal budget constraints.

Future Spending (G2) W2 11 A4 Present Spending (G1)

Here in the above fig “A3A4” be the government’s intertemporal budget constraints and “W2” be the endowment point where “G1” is equal to “T1” and “G2” is equal to “T2”.