In Graph 9.4, we illustrated how you can derive the labor supply curve from a consumer model in which workers choose between leisure and consumption.
A: In end-of-chapter exercise 3.1, you were asked to illustrate a budget constraint with labor rather than leisure on the horizontal axis. Do so again, assuming that the most you can work per week is 60 hours.
(a) Now add to this graph an indifference curve that would make working 40 hours per week optimal.
(b) Beginning with the graph you have just drawn, illustrate the same wealth and substitution effects as drawn in the top panel of Graph 9.4a for an increase in the wage.
(c) Then, on a second graph right below it, put weekly labor hours on the horizontal axis and wage on the vertical, and derive the labor supply curve directly from your work in the graph above. Compare the resulting graph to the lowest panel in Graph 9.4a.
(d) Repeat this for the case where wealth and substitution effects look as they do in Graph 9.4b.
(e) Repeat this again for the case in Graph 9.4c.
(f) True or False: We can model the choices of workers either using our 5 standard assumptions about tastes defined over leisure and consumption, or we can model these choices using tastes defined over labor and consumption. Either way, we get the same answers so long as we let go of the monotonicity assumption in the latter type of model.
B: Now suppose that a worker's tastes over consumption and leisure can be defined by the utility function u(c,ℓ) = cαℓ(1−α) (and again assume that the worker has a leisure endowment of 60 hours per week).
(a) Derive the labor supply function by first deriving the leisure demand function.
(b) How would you define a utility function over consumption and labor (rather than consumption and leisure) such that the underlying tastes would be the same.
(c) Which of our usual assumptions about tastes do not hold for tastes represented by the utility function you have just derived?
(d) Using the utility function you have just given, illustrate that you can derive the same labor supply curve as before by making labor (rather than leisure) a choice variable in the optimization problem.