Question

A consumer’s consumption-utility function for a two-period horizon (t = 1, 2) is given by U(c1,c2) = ln(c1)+ln(c2). The consumer’s income stream is y1 = $1500 and y2 = $1080, and the market rate of interest is 8%. Calculate the optimal values for c1 and c2 that maximize the consumer’s utility

Answer 1

Lets first form intertemporal budget constraint.

Period 1 :

c₁ + s = y₁ where s = saving in period 1

=> s = y₁ - c₁

Period 2 :

c₂ = s + rs + y₂ where r = interest rate = 8% = 0.08

=> c₂ = (1 +r)(y₁ - c₁) + y₂

=> c₂ + 1.08c₁ = 1080 + 1500*1.08 = 2700

Thus we have to maximize : U = ln(c₁) + ln(c₂)

Subject to : c₂ + 1.08c₁ = 2700 ------------(1)

Legrange is given by :

ln(c₁) + ln(c₂) + u(2700 - c₂ - 1.08c₁) where u = Legrange multiplier

First order condition :

Dividing (2) from (3) we get :

c₂/c₁ = 1.08 => c₂ = 1.08c₁

Putting this in (1) we get :

c₂ + 1.08c₁ = 2700 => 1.08c₁ + 1.08c₁ = 2700

=> c₁ = 1250 => c₂ = 1.08c₁ = 1.08*1250 = 1350

Hence, the optimal values are : c₁ = 1250 and c₂ = 1350.