Question

Assume a consumer lives for two periods. His income and consumption in the two periods are Y₁ and C₁, and Y₂ and C₂ respectively. Ignore price level changes and further assume that this consumer saves income (S) in the first period and this saving earns interest. With consumption in two periods being constrained by income in the two periods derive the intertemporal budget constraint.

Answer 1

In the first period saving (s) is the difference between income and consumption-

S = Y₁ – C₁ ---------- (1)

In the second period, consumption equals the accumulated saving (which includes the interest earned on that saving), plus second-period income

C₂ = (1 + r)S + Y₂ ---------- ( 2)

WHERE 'r ' IS REAL INTEREST RATE.

We can now derive the consumer’s budget constraint by combining equations (1) and (2). If we substitute the first equation for S into the second equation we get

For deriving intertemporal Budget Constraint, we need to combine both the equations.

Subsituting the value of S from equation 1 in equation 2 we get,

C₂ = (1 + r) (Y₁ – C₁) + Y₂-------------- (3)

C₂ = (1 + r) Y₁  – (1 + r)C₁  + Y₂

(1 + r) C₁ + C₂ = (1 + r)Y₁ + Y₂-------------- (4)

Dividing both sides of equation (4) by 1 + r we get Intertemporal Budget Constraint

C₁ + C₂ / 1 + r = Y₁ + Y₂ / 1 + r -------------- (5)

Since this equation relates consumption in two periods to income in both the periods it expresses the consumers’ intertemporal budget constraint.