```MGEC02: Topics in Price Theory
General Equilibrium with Production
James Campbell
UTSC
September 14, 2014
1
Problems
1.1
Figure 1: 1.1 a)
a)
b)
∆lyrics
∆music
= 32 : if he produces 2 fewer lyrics he can produce 3 extra music, and vice versa.
Note it’s OK to write MRT the opposite way ( ∆music
) as long as you’re clear about which
∆lyrics
you’re doing.
c)
d)
∆lyrics
∆music
= 4: if she produces 4 fewer lyrics she can get 1 extra music.
1
Figure 2: 1.1 c)
e) Kanye has a comparative advantage in music production, Nicki in lyrics.
Figure 3: 1.1 f)
f)
g) In this case the most profitable production plan is to have Kanye produce 21 units of music
and Nicki produce 28 units of lyrics.
1.2
a) M RT =
1
√
2 x
b) 1 coconut.
2
Ux
c) M RS = − M
=
M Uy
√
y
12−x
x
12−x
=
d) U (x = 12, y = 0) < U (x = 11, y = 1)
e) M RS = M RT at
f ) x = 4, y = 2,
px
py
1
√
2 x
√
x
:
12−x
=
x = 4, y = 2.
1
4
=
1.3
a) M RT =
√1
x
b) Decreasing returns to scale.
Ux
c) M RS = − M
= − −2
1 = 2y
M Uy
y
d) MRS is positive because the consumer dislikes x: from a given point she is willing to give up
some amount of y for less of x. This contrasts with the usual situation in which the consumer
likes each of two goods and so is willing to give up some amount of one for more of the other.
√
e) M RS = M RT at √1x = 2y = 4 x: x = 14 , y = 1.
px
py
f ) x = 41 , y = 1,
=2
1.4
√
a) π = y − lw = 2 l − lw
b)
dπ
dl
=
1
√
l
− w = 0 ⇒ lU∗ =
p
c) π(lU∗ ) = 2 lU∗ − lU∗ w =
d) Since π(lU∗ ) =
1
w
1
w2
1
w
and py ≡ 1, he can consume
e) Since he earns w per hour, he can consume
income from l hours.
1
w
f ) U = y − 2 ln l =
g)
dU
dl
=w−
2
l
+ wl − 2 ln l
= 0 ⇒ lJ∗ =
h) lU∗ = lJ∗ when
i) y = 4, x = 4,
1
w2
pl
py
1
w
=
2
,
w
2
w
which is true at w = 12 .
= w = 21 .
3
1
w
units of economics by spending the profit.
+ wl units of economics with profit plus earned
1.5
a) M RT =
dy
dx
=
√1
x
b) y = 0 when x = 0 but y = 2 when x = 1: he can produce 2 questions in the first hour.
c) If he doesn’t work, U = (3 ∗ 0) − 0 = 0; if he works for an hour, U = (3 ∗ 2) − 1 = 5.
Ux
=
d) M RS = − M
M Uy
1
3
e) This means that Jim is always just willing to do three hours of labor to get one extra exam
question. It is positive because Jim dislikes labor and likes exam questions, so to maintain
his utility level while changing his bundle he must get more of both goods, or less of both
goods.
f ) M RS = M RT when
g) x = 9, y = 6,
px
py
√1
x
= 13 ; that is, when x = 9.
= 13 .
4
```