Question

Let Jennifer's utility function be U(C,L) = log(C)+4log(L). Let T denote time available to split between leisure and work, w denote the wage rate and V ≥ 0 denote non-labor income (as in class). Derive her optimal choice as a function of w, T , and V . Calculate the ?rst derivative of her optimal leisure choice with respect to w. Does the substitution e?ect dominate the income e?ect? (Hint: hours worked = T − L so the change in hours has the opposite sign as the derivative you calculate.) Is this also true if V = 0?

Answer 1

(6, L) = Log(C) + 4 log(L), Tə time; Wewage rate and vyo Now, hint given Hz T-L org L+H=T sos we need to derive Budget Constraint. c=v+wH C= V+W(TL) or c=v+WT-WL or K+WL = V+WII-a. Again, Lagrange is given by (2) 2= log C+ 4 Log L + 2(V+WT-WL-C) sos ad=0, so we get 22=0, so we get Ź= 2 to 4 = wd this From (i) and (ii) we have or 4 = w: I or L= 48 from eqra) we have C+ WL = V+WT C+ w. 4c = ut wT 5c=v+wT or c* = 1 [v+wT] → optimal consumption

دہی - =444 et Crows] = WowT). optimal Leisure Choice Now, derivative of Lt. So, this means as the mage Increase, then amount of Leisure the decreases Again we have AL = {substitution Effect} + { Income Effect} AW Here, substitution effect is awe and Powme Effect is tve. Yes, the substitution effect dominates the Income Effect becaurse? - o related to me Effect substitution Inversely with Income Effect we have as mage (wi reases cases with respect to L due to Income Increases, but we have 2 getting eased due to substitection effect. all from above two L decreases, which that Substitution Effect is greater Effect Increases decreased due to So, overall from implies that s. than Income Effect.