Answer to Question #188622 in Microeconomics for Lani

Question

Terry’s utility function over leisure (L) and other goods (Y ) is U(L, Y ) = Y + LY. The associated marginal utilities are MU_Y = 1 + L and MU_L = Y. He purchases other goods at a price of $1, out of the income he earns from working. Show that, no matter what Terry’s wage rate, the optimal number of hours of leisure that he consumes is always the same. 

Answer 1

Solution:

U(y,l) = y+l.y

MU y=1+l

MU l =y

a. (1-l) w=y

L as a percentage of time available

Using Lagrangian form of equation:

U= y+ly+A((1-l)w-y)

U1= 1+L-A=0

U2=Y-AW=0

U3= (1-l)w-y=0

Solving, A=y/w

Substituting L in U3,

w=y

L=y/w-1

L=w/w-1=0

b. He would like to always have ‘0’ leisure hours .

c. Total effect= Substitution effect + Income effect

Income effect is the change in demand resulting from change in purchasing power.

For an increase in the value of leisure time (wage), the income effect dominates the substitution effect. Terry would like to work more to get paid extra.