Question

Bob learns that his income in the second period of this two period existence will now be $100 greater than he previously expected. Assuming he perfectly smooths his consumption (so that consumption will be the same in both his periods), he isn't borrowing constrained, and the interest rate is 16%:

a) What will happen to his consumption in the future period?

b) what will happen to his savings in the current period?

Answer 1

Solution:

Say initially the consumption levels for two periods = C1 = C2 = x

Income in first period is M1 and in second period is M2, interest rate as given is, i = 16%. Let's denote savings in first period by s.

Now, he expects in period 2 income, M2' = M2 + 100

Initially, the budget line for first period was: C1 + s = M1, so s = M1 - C1

Budget line in second period: x = M2 + s(1 + i)

C2 = M2 + (M1 - C1)*(1+ 0.16)

So, the inter-temporal budget line becomes:

1.16*C1 + C2 = 1.16*M1 + M2

Since, we have consumption smoothing, with C1 = C2 = x, we have 1.16*x + x = 1.16*M1 + M2

x = (1/2.16)*(1.16*M1 + M2)

So, initial consumption in future period = x = (1/2.16)*(1.16*M1 + M2)

And savings in current period, s = M1 - x

s = M1 - (1/2.16)*(1.16*M1 + M2) = (1/2.16)*(M1 - M2)

Now with M2' = M2 + 100, there are three ways to see what could possibly happen to future consumption and current savings:

First way: Intuitive

a) Since, now the income in second period has increased, and so the overall income has increased, one shall expect Bob to increase the consumption as well (taking good to be a normal good). So, consumption in both periods (and so, consumption in second period) will increase, under consumption smoothing principle.

b) Since, the consumption in period 1 has also increased, while income in period 1 has remained constant, we can claim that savings in current or first period will decrease (as more part of current income is spent on consumption).

Second way: Derivative

a) We have already found the optimum values in introductory part of solution. We have to now see how x (= C2) change as M2 changes, and the partial differentiation of x with respect to M2 shall help us know that. Since,

x = (1/2.16)*(1.16*M1 + M2)

= (1/2.16) which is positive. So, with increase in M2, x or consumption in future period will increase as well.

To find the actual value: Change in x = *change in M2

Change in x = (1/2.16)*100 = 100/2.16

So, new consumption in future period, new X = (1/2.16)*(1.16*M1 + M2 + 100)

b) Similarly, how savings in current period changes can be found by finding

s = (1/2.16)*(M1 - M2)

= -(1/2.16) which is negative, so we can say as M2 increases, s decreases that is as income in future increase savings in current period decreases.

Again the exact amount of new savings can be found as: Change in s = *change in M2

Change in s = -(1/2.16)*100 = -100/2.16

So, new savings in current period, new s = (1/2.16)*(M1 - M2 - 100)

Third way: Solving

You could also solve for the optimum using the income in future period now as (M2 + 100), as we have done in the introductory part. You shall reach the same answers as above.