Consider a consumer with preferences for consumption of a composite good (C) and leisure (L) given by the following utility function:

Question

U(C,L) = 2C1/2 + L
Denote the consumer's wage rate by w and total time available for labor and leisure is normalized to one. The price of consumption is one. Denote the amount of labor supplied as N, so that
N + L = 1. The consumer also earns non-labor income ("allowance") of 0.
a. Write out the budget constraint determining feasible allocations of leisure and consumption.
b. Compute the optimal bundle of leisure and optimal bundle of consumption.
c. Derive the consumer's labor supply function: N*(w, ).
d. Determine the effect of increasing non-labor income on the supply of labor (that is, compute the relevant partial derivative).
e. How does non-labor income affect the consumption of the composite good, C?
f. Compute the effects of an increase in wage on consumption and labor supply. Is leisure a normal good?

Answer 1

a. π + w = C + wL
In words, the amount of consumption goods (in $) equals the allowance plus earnings from work. The reformulation is just rearrangement to look more like our "typical" BC where income (maximum earnings from work plus allowance) equals the consumption of goods plus the consumption of leisure (where w is the "implicit price" of leisure).
b. Optimization using a Lagrangian:
L = 2C1/2 + L + [ π + (1 – L)w – C]
The derivatives are
LC = C-1/2 – π = 0
LL = 1 – w = 0
L = π + (1 – L)w – C = 0
The first two conditions imply
C* = w2
Substitute in to the third condition to get
L* = 1 + π/w – w
Notice that L < 1 requires:
π < w2.
Alternatively, the optimal bundle is L* = 0 and C* = w + π (a corner solution).
c. N* = 1 – L* = w – π/w
d. dN/dπ = -1/w < 0
e. It doesn't.
f. dN/dw = 1 + π/w2 > 0; dC/dπ = 2w > 0. Neither of these derivatives tells us whether leisure is normal. To find out if leisure is normal, we simply need to isolate the income effect. This is seen by looking at the effect of non-labor income on leisure, which is positive. Leisure is therefore normal.