Question

Woodstock has preferences for leisure time and birdseed defined by the utility function U(l,B)= l^.75B^.75, where l is hours of leisure time and B is the number of birdseeds Woodstock consumes. Suppose Woodstock has 24 hours a day for leisure time and for gathering birdseed. He gathers 20 birdseeds each hour he works (e.g., 20 birdseed is Woodstock’s wage). Let H be the hours Woodstock gathers birdseed. Woodstock chooses leisure time and birdseed according to the optimization problem max_l,_b l^.75B^.75 st l+H=24 and B=20H.

(a) Rewrite Woodstock’s optimization problem by substituting his constraints into his utility function so that he maximizes his utility with respect to H.

(b) Solve Woodstock’s optimization problem for the hours he spends gathering birdseed. (Hint: You will have to use the chain rule when you find the FOC)

(c) Woodstock is considering migrating to a neighborhood where birdseed is more prevalent and consequently his wages would be higher. If Woodstock moved to the new neighbor- hood, would he work more, fewer, or the same number of hours?

Answer 1

Utility function is U(l,B)= l^0.75B^0.75. Suppose Woodstock has a total of 24 hours a day for leisure time and for gathering birdseed. Wage rate = 20. We have the optimization problem maxU (l,b) such that Z = l^0.75B^0.75 with a constraint l+H=24 and B=20H.

(a) We see that B = H*20 = (24 - l)*20. This is because (24 - l) is the number of hours not lesiured and worked and 20 is each hours wage. Hence we have

B = 480 - 20l

20l + B = 480

This is the actual budget constraint

max U (l,b) such that Z = l^0.75B^0.75 + λ(480 - 20l - B)

(b) Find the partial derivative of Z wrt I and B and keep them equal to 0

Z'(I) = 0

0.75l^-0.25B^0.75 - 20λ = 0

Z'(B) = 0

0.75l^0.75B^-0.25 - λ = 0

Solve them to get l^-0.25B^0.75/I^0.75B^-0.25 = 20 or B/I = 20 This gives B = 20I

The third FOC is Z'(λ) = 0

20l + B = 480

Substiture B = 20I

B + B = 480

B = 240 and I = 240/20 = 4 hours.

(c) Woodstock is considering migrating to a neighborhood where birdseed is more prevalent and consequently his wages would be higher. If Woodstock moved to the new neighborhood, he would probabily work for more because both consumption and leisure are normal goods. Income effect for normal goods is positive so he will consume more and so work more.