Additional problems 2, solutions

1. Suppose a consumer consumes commodities x1, . . . , xL and any con-sumption bundle x ∈ RL

≥0. Given consumer’s income m > 0 and pricevector p� 0, her (differentiable) Walrasian demands are

x(p,m) = (x1(p,m), . . . , xL(p,m)).

You are told that all the income elasticities of the Walrasian demands arethe same. Is it correct to conclude that the elasticities are unitary?

solution: 1. From the Walras law p · x(p,m) = m we have

L∑i=1

pi∂xi(p,m)

∂m= 1

L∑i=1

pi∂xi(p,m)

∂m

m

xi(p,m)

xi(p,m)

m= 1

L∑i=1

εimpi · xi(p,m)

m= 1

ε·m

L∑i=1

pi · xi(p,m)

m= 1

ε·m = 1

2. Consider a consumer who consumes food and clothing (there are noother commodities). During a war, food and clothing are rationed. Inaddition to money price, certain number of ‘rationing points’ must be paidto obtain a good. The consumer has certain ‘budget’ of rationing points shemay use to obtain either good. In addition, she has an income. When herincome increases, she buys/demands more food and less clothing. Does itfollow that clothing is an inferior good?

solution: 2. Not necessarily. Referring to the figure below, x and x′

are the consumption bundles the consumer would choose in the absence ofrationing. Since the demand for clothing increases in income, clothing is anormal good.

During a rationing period, the consumer faces a constraint which is anintersection of the budget constraint and of the rationing constraint (the greyregions in the figure). Before the income increase, the consumer demands yand after the increase, she demands y′.

1

Additional problems 2, solutions

clothing

food

x

x′

y y′

budget lines rationing line

3. A consumer ‘moonlights’ . . . takes on a second job since she is limited inthe number of hours she can work on her primary job. If leisure is a normalgood, is it correct to state that an increase in the wage rate offered on theprimary job will reduce the number of hours the consumer moonlights?

solution: 3. The situation is depicted below. c− l̄ is the maximum numberof hours the consumer can work on her primary job. The slopes of the budgetconstrain reflect the fact that the wage rate on the primary job is higher(otherwise the consumer would quit her primary job and would only workin the second job).

Before the change the consumer chooses bundle x and works c− l hours.After the wage rate change, the consumer can be either at x′ or y′ (wherex′ and y′ lie on the new budget constraint such that lx ∈ [l, l̄] and ly ∈ [l̄, c]respectively) . If at x′, then because the increase in the wage rate on theprimary job involves essentially only an income effect and leisure is normal.In this case she works c − lx < c − l hours. Change to y′ is also possible,but then c− ly < c− l.

other goods

leisurel̄ c

x

l

x′

lx

y′

ly

2

Additional problems 2, solutions

4. Suppose there are two goods x ≥ 0 and y ≥ 0 a consumer can purchaseat prices px > 0 and py > 0. Show that x and y are net substitutes.

solution: 4. Let hx(p, u) and hy(p, u) be the Hicksian demands. At somepx > 0 and py > 0 consider the following three price vectors

po = (px, py)

p′ = (p′x, py)

p′′ =

(p′xpxp′x, py

pxp′x

)=

(px, py

pxp′x

)=(px, p

′y

)where p′x > px and hence p′y = py

pxp′x< py. Since own price effects are non-

positive, hx(po, u) ≥ hx(p′, u). By homogeneity of degree zero hx(p′, u) =hx(p′′, u). Thus hx(po, u) ≥ hx(p′′, u).

3