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 :

c_{1} + s = y_{1} where s = saving in period
1

=> s = y_{1} - c_{1}

Period 2 :

c_{2} = s + rs + y_{2} where r = interest rate =
8% = 0.08

=> c_{2} = (1 +r)(y_{1} - c_{1}) +
y_{2}

=> c_{2} + 1.08c_{1} = 1080 + 1500*1.08 =
2700

Thus we have to maximize : U = ln(c_{1}) +
ln(c_{2})

Subject to : c_{2} + 1.08c_{1} = 2700
------------(1)

Legrange is given by :

ln(c_{1}) + ln(c_{2}) + u(2700 - c_{2} -
1.08c_{1}) where u = Legrange multiplier

First order condition :

Dividing (2) from (3) we get :

c_{2}/c_{1} = 1.08 => c_{2} =
1.08c_{1}

Putting this in (1) we get :

c_{2} + 1.08c_{1} = 2700 => 1.08c_{1}
+ 1.08c_{1} = 2700

=> c_{1} = 1250 => c_{2} =
1.08c_{1} = 1.08*1250 = 1350

Hence, the optimal values are : **c _{1} = 1250 and
c_{2} = 1350.**

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