Question

Decomposition of the effect of a wage decrease on labour supply

Bob’s preferences over consumption (c) and leisure (r) are represented by the following utility function:

u(c, r) = c^(0.6)r^(0.4).

Suppose that Bob is endowed with $100 and 50 hours (per week) to allocate between leisure and

work. Denote the price of the consumption good by p and the wage by w.

1. Express mathematically all the constraints faced by Bob.

2. Draw Bob’s feasible set.

3. Derive Bob’s (gross Marshallian) demands for the consumption good and leisure as functions of p, w, and the (dollar) value of Bob’s endowment (mω).

Suppose that p = 1 and w = 2.

4. How much of the consumption good does Bob buy, how many hours does he allocate to

   leisure, and how many hours is he working?

5. Indicate the chosen bundle on your graph from part (b) and label it O.

Bob is worried that he might get fired, in which case he would only be able to find a job that pays a wage w′ =1.

f. In such a scenario, how many hours will Bob work?
It turns out that Bob is not fired but demoted. As a result, his wage drops to w = 1.6.

7. After the demotion, how much of the consumption good does Bob buy, how many hours does he allocate to leisure, and how many hours is he working?

8. Indicate the new feasible set and the chosen bundle (F) on your graph from part (b).

9. Decompose the effect of the wage drop into a price effect and an endowment effect.

10. Show the decomposition on your graph from part (b).

11. Derive the indirect utility function.

12. Using duality, derive the expenditure function.

13. Using Shephard’s lemma, derive the Hicksian demands for the consumption good and leisure.

14. Decompose the price effect from part (i) into and an income effect and a substitution effect.

15. Draw a separate graph showing this decomposition.

Answer 1