Question

# A consumer’s consumption-utility function for a two-period horizon (t = 1, 2) is given by U(c1,c2)...

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

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.

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