Question

Consider the two-period Neoclassical consumption model seen in class. Suppose that income is measured in dollars. Let the utility function take the logarithmic form U(C)=ln C, and the representative consumer maximize her lifetime utility subject to her budget constraint. Suppose that income in period 1 is $50,000, income in period 2 is $30,000, ?=1 and ?=5%.

Do you think that the consumption profile of the agent in this numerical example is going to be smoother than her income? If so, why? If not, why not? Please explain as clearly as you can.

Answer 1

Let us state the problem first.

The utility of the consumer is

U(C) = ln(C)

Let the consumption in period 1 is C1 and consumption in period 2 is C2. And, discount rate is β. Hence, the intertemporal utility function is

U(C1, C2) = ln(C1) + β.ln(C2)

We are told that, β = 1

Hence,

U(C1, C1) = ln(C1) + ln(C2).........(1)

Her income in period 1 is Y1=$50000 and income in period 2 is Y2=$30000.

The rate of interest is given as

R = 5% or R = 0.05

The consumer consumes C1 in period 1.

Hence, residual income for period 2 is (Y1 - C1).

Now, interest rate R is applied on (Y1 - C1) and she gets (Y1 - C1).(1 + R) along with the income Y2 in period 2.

Hence,

C2 = Y2 + (1+R).(Y1 - C1)

or, C1 + C2/(1+R) = Y1 + Y2/(1+R)........(2)

Here, Y1 = $50000 and Y2 = $30000 and R=0.05

Hence,

C1 + C2/(1+0.05) = 50000 + 30000/(1+0.05)

or, C1 + 0.95C2 = 78571.43........(3)

This is the intertemporal budget constraint.

The slope of the budget constraint is

Slope = -(1/0.95)

Now, from the utility function

U(C1, C2) = lnC1 + lnC2

We will calculate the MRS or Marginal Rate of Substitution. Hence,

Marginal Utility of C1 is

MU1 = dU/dC1 = 1/C1

And, Marginal Utility of C2 is

MU2 = dU/dC2 = 1/C2

Hence, MRS = -(MU1/MU2)

or, MRS = -(C2/C1)

Hence, at utility maximizing situation, MRS equals the slope of the budget line.

Hence,

MRS = Slope

or, -(C2/C1) = -(1/0.95)

or, C1 = 0.95.C2........(4)

Putting C1 in equation (3), we get

0.95.C2 + 0.95.C2 = 78571.43

or, C2* = $41353.38

And, from (4) we get

C1* = 0.95.C2* = 0.95×41353.38

or, C1* = $39285.71

Hence, the consumption profile of the agent is

(C1*, C2*) = ($39285.71, $41353.38)

Now, the consumer is smoothing consumption, when he consumes almost equal amount in both period. Also, the consumer is smoothimg income, when she earns almost equal amount in both period.

Here, we can see, Y1=$50000 and Y2=$30000.

The incomes are not equal in both period. In period 1 she earns almost double than in period 2. Hence, income is not smooth between two periods.

But, if we look at her consumption profile, we see

C1* = $39285.71 and C2* = $41353.38

Hence, the consumptions are almost equal in both periods. Hence the consumption is smooth between two periods.

Hence, the consumption profile is going to be smoother than her income.

Reason: There is equality in consumption between two periods but her income is unequal between two periods.

Hope the solution is clear to you my friend.