13.7 a. By changing the utility of household 1 to
0.2
0.6 0.2 1
1 1 1 1 ,U x y l l
we increased household 1’s relative preference for x as opposed to y. By running
the simulation we obtain:
0.3744,
0.2377,
0.1239,
0.2641.
x
y
k
p
p
p
p
The utility-maximizing choices for household 1 are
118.10,x
19.10,y
and
11
8.55,ll
giving
113.69.U
The utility-maximizing choices for household 2
are
28.1,x
212.75,y
and
22
5.74,ll
giving
29.06.U
intensive good, the relative price of capital will also decrease. Since x is the labor-
intensive good, the increased demand for x will cause the price of labor to
increase and this will lead to more labor-hours being supplied, thus reducing the
leisure demanded by each household.
b. After reversing the production functions, we obtain the following equilibrium
0.4
0.4 0.2 1
1 1 1 1
0.4
0.3 0.3 2
2 2 2 2
,
,
U x y l l
U x y l l
the following equilibrium is obtained:
0.4040,
0.1912,
0.0904,
0.3143.
x
y
k
l
p
p
p
p
The utility-maximizing choices for household 1 are
112.27,x
16.48,y
and
11
15.78,ll
giving
19.91.U
The utility-maximizing choices for household 2
are
26.27,x
213.25,y
and
22
10.75,ll
giving
29.74.U
The greater utility gained from the labor-hours not sold on the market will
cause households to sell less labor. This will cause the relative price of labor to
increase and the relative price of the other input (capital) to decrease. Thus, the
labor-intensive good, good x, will become relatively more expensive and the
capital-intensive good, good y, will become relatively cheaper.
Analytical Problems
13.8 Tax equivalence theorem
0.3989,
0.2667,
0.1131,
0.2213.
x
y
k
l
p
p
p
p
13.9 Returns to scale and the production possibility frontier
We have never succeeded in deriving an analytical expression for all these cases. We
13.10 The trade theorems
For all of these proofs, draw the PPF and its underlying Edgeworth Box Diagram. The
world price ratio determines where production will occur on the PPF and where it will
occur in the Edgeworth Box. Given the assumption about factor intensities, the contract
curve in the Edgeworth Box will be concave (i.e., bowed upward).
a. Factor price equalization theorem: Because productive technology is
*
*,
xx
yy
Up
MRS
Up
and profit-maximizing firms to choose
*
*.
x x x
y y y
T MC p
RPT
T MC p
13.13 Initial endowments, equilibrium prices, and the first theorem of welfare economics
a. The value of A’s initial endowment is
.
AA
px y
Hence, his or her demand for
d. Part (b) shows that increase in the endowment of either good for person A will
raise the relative price of good x because that good is favored by this person.