Question

4. Let a person's utility function over consumption, X, and leisure, L, be given by U = XL2, SO MUx = L2 and MUL = 2xL.The individual may work up to 24 hours per day at wage rate, w = $10 per hour, and he has non-labor income of $50 per day. The price of x, px, is $5. (a) Find the utility-maximizing x and L. (b) Show that at the utility- maximizing quantities of x and L, the consumer's marginal rate of substitution of leisure for consumption equals the real wage, i.e., MRSLX = wipx. Now suppose a lump-sum tax equal to $50 is imposed on the consumer. (c) Find the utility-maxim- izing values of x and L. (d) Show that the marginal rate of substitution of leisure for consumption is still equal to the real wage.

Answer 1

solution Given that given information, According to the a) Given U= XL? mux=6² MU₂= 2XL mrs= mux mul = 22 74 equation MRS: px Py la solo Since (2-x) + Budget =) 5X+ IOL=290 ->) SLF101=290 => 151=290 ») L:19.333]

L-X-19.333 Therefore, utility maximizing (2-x) = 19.333 6 ) u suppose , A lump- sum tar $ 50 and Now, Budget line is 5x+10L=240 =) 5 (flot : 240 ») 15L-240 > L=167 [ L=X=16 7 х: Both C) and (d). * MRS = mox MUL 22 2

Since OR Because at equation MRs will be always equal to the real wage. Thus, Hence showed.