intertemporal utility function usage : calculating consumption

Question

I have encountered this a lot in my exams and can not seem to understand how to use these functions here is an easy exemple :

A consumer who will only live 2 periods receives 1000€ in the first period and 5000€ in the second period, if the interest is at 0% how much will he consume in both periods knowing that his utility function is :

U(C¹,C²) = 3×C¹×C²

Where C¹ is the consommation of first period and C² of second.

How do I make use of this utility function to calculate the consumption in both periods ?

Answer 1

Now that the OP has provided his own answer, let's also give the standard treatment of this problem.

There is no production, the consumer receives windfall endowments in each period, $Y_1, Y_2$, and he can borrow (or lend) during the first period at an exogenous non-negative interest rate $r$.

What is the two-period budget constraint of the consumer? It is more intuitive to write it as

$$C_2 = Y_2 + (1+r)(Y_1-C_1) \tag{1}$$

The consumer can consume his second period endowment adjusted by the results of its borrowing or lending activities in the first: If his endowment is larger than his consumption in the first period, $Y_1-C_1 >0$, it means that the consumer acted as a creditor, and in the second period he will receive the principal plus interest, to consume in addition to his 2nd-period endowment.

If $Y_1-C_1 <0$ it means that the consumer acted as a borrower, and in the second period he will have to return the loan with its interest. So $(1)$ covers both cases.

Then the utility maximization problem is stated as

$$\max_{C_1,C_2} U(C_1,C_2) \\ s.t. C_2 = Y_2 + (1+r)(Y_1-C_1) \tag{2}$$

We can insert the constraint into the objective function and maximize only with respect to $C_1$. So the first-order condition is

$$\frac {\partial U(C_1,C_2(C_1))}{\partial C_1} = 0 \tag {3}$$

and the second-order condition is

$$\frac {\partial^2 U(C_1,C_2(C_1))}{\partial C_1^2} < 0 \tag {4}$$

at the critical point.

Using the specific functional form of the utility function of the question, $ U= 3C_1C_2$ we have

$$\frac {\partial U(C_1,C_2(C_1))}{\partial C_1} = 3C_2 - 3C_1\cdot(1+r) $$

$$ = 3[Y_2 + (1+r)(Y_1-C_1)] - 3(1+r)C_1$$

$$ = 3Y_2+ 3(1+r)Y_1 - 6(1+r)C_1 \tag{5}$$

Note that

$$\frac {\partial^2 U(C_1,C_2(C_1))}{\partial C_1^2} = -6(1+r) <0$$ so the second-order condition for a maximum is satisfied.

Setting $(5)$ equal to zero we obtain $$ C_1^* = \frac {Y_2}{2(1+r)}+ \frac 12 Y_1 \tag{6}$$

From $(6)$ we conclude that consumption in the first period will never fall below half of the period's endowment, and that it is a negative function of the interest. Moreover the consumer will find it optimal to borrow when

$$C_1^* > Y_1 \implies \frac {Y_2}{2(1+r)}+ \frac 12 Y_1 > Y_1$$

$$\implies Y_2 > (1+r)Y_1$$

Using the specific numerical assumptions of the question, $Y_1 = 1000, Y_2 = 5000, r=0$, we obtain $$C_1^* = \frac {5000}{2}+ \frac 12 1000 = 3000 $$

Answer 2

This is the two-period budget constraint:

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

Derivation is straightforward. On the LHS, you have the present value of consumption (considered during period 1), and on the RHS you have the present value of income.

Intuitively, think about 1/(1+r) on the LHS as the price ratio between Good 1 and Good 2. Now you can solve the u-max problem as per usual.

Answer 3

I finally figured out how to use the utility function so I will detail so anyone who needs a detailed answer as it was quite unclear even though I was well aware of the equation :

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

But did not know how to use it so I will explain ,

Y1 = 1000 Y2 = 5000 r = 0%

So we have

C1+C2/(1+0)=Y1+Y2/(1+0 C1+C2=Y1+Y2 So C1=Y1+Y2-C2 with Y1+Y2=6000

Replace the value of C1 in the utility function :

U=3×(6000-C2)×C2

We know that utility is max when

dU/dC2 = 0 = -6C2+18000 C2=3000

To get C1 we just have to put C2 in the two period budget constraint to get

C1=3000

And there it is, if i did something wrong feel free to comment