Question

Homer Simpson does not abide by the life cycle theory of consumption. Homer has a “let’s live life like it’s our last day” mentality and thus, he prefers to consume more today, relative to the future. In particular, Homer prefers to consume exactly twice as much today (c), relative to consumption next period (c^f). Homer’s current income equals $160K and his future expected income = $160K. He has no wealth (neither current nor expected) since he lives like today is his last! Homer faces a real interest rate of 0.03, answer the following questions.

a) (5 points) Solve for Homer’s optimal consumption basket today (C*) and his optimal consumption basket next period (C^f*). Please provide a completely labeled graph depicting these results and label this point as C*_A. (10 points for a completely labeled graph – be sure to label the no lending / no borrowing point = NL/NB)

b) (5 points) Now Ben Bernanke and the Fed are tired of being criticized for keeping interest rates “too low” and thus, make a big move so that real interest rates rise to 10% (use 0.10). Recalculate the optimal bundle for Homer and add this point to your graph and label as point C*_B.  

c) (10 points) Is Homer better or worse off due to the rise in the real rate of interest? Be sure to define the income effect and the implications of the income effect on Homer’s current consumption. Then define the substitution effect and the implications of the substitution effect on Homer’s current consumption. Comment on whether these income and substitution effects work in the same or opposite direction (i.e., is it a tug of war or do they work in the same direction?) in this particular case?

d) (5 points) How would your answer to d) above change in Homer was a saver rather than a borrower?

e) (10 points total, 5 for correct (and completely labeled) diagram and 5 for discussion) Now derive and draw Homer’s desired savings function. Describe three reasons why Homer’s savings would shift to the right – that is, three reasons why Homer would increase his desired savings at any given real rate of interest.

f) (5 points total) Intuition question: What exactly does the slope of the budget constraint represent and why is it (the slope) so important in determining consumption behavior.

g) (5 points total) Name 3 reasons why Homer's budget constraint would shift inward (to the left).

Answer 1

(a) The consumer's budget constraint is:

Since it is given that consumption in period one is twice the consumption in period two, therefore,

Substituting this in the constraint:

Plotting the constraint and the optimal choice:

(b) At the new rate of interest, the constraint is:

Again using the fact that :

Plotting the new constraint and the new optimal bundle:

(c) The consumer is worse off after the decrease in real interest rate as now he is consuming a bundle which he did not prefer with the higher interest rate This bundle lied in the consumer's budget set with the original rate of interest and the consumer chose the bundle (212.28,106.14) over (209.99,104.99).

To calculate the income and substitution effects, we need to compute the intermediate bundle. This bundle is the bundle chosen with enough income to purchase the old bundle at new interest rate and the consumer is facing the new interest rate.

Intermediate constraint:

Substitution effect = 212.27 - 212.28 = (-0.01)

Income effect = 209.99 - 212.21 = (-2.22)

Both income and substitution effects work in the same direction.