Dr. Donna Feir
Economics 313
Problem Set 4: Solutions
General Equilibrium: Allocative efficiency
1. There are two producers of goods x and y. Firm 1’s PPF is y = 30 – x and firm 2’s PPF is y = 30 –
½x. Firm 1 is producing no units of x and 30 units of y. Firm 2 is producing 40 units of x and 10
units of y. There are two consumers of goods x and y. Consumer A’s utility function is UA = x + 2y
and consumer B’s utility function is UB = xy. Consumer A has the consumption bundle consisting
of 10 units of x and 25 units of y. Consumer B has the consumption bundle consisting of 30 units
of x and 15 units of y.
a. Is the definition of productive efficiency satisfied in this economy? Explain.
At any production choice, MRT1 = 1 & MRT2 = ½, so the MRTs are not equal. But this does not
necessarily mean that the allocation of production is inefficient. For this, we need to check
whether the production choice is on the aggregate PPF.
y
60
40
30
40 60
90
x
Using the tools from Topic AI part (ii), you can easily
see that xT = 40 & yT = 40 is on the aggregate PPF,
and so this production point satisfies productive
efficiency. All of the x is being produced by firm 2,
the low cost producer of good x. Note that firm 2 is
also producing a positive amount of good x. Firm 1,
in contrast, is specializing in the production, namely
the production of good y, the good for which it is the
low cost producer.
b. Is the definition of distributive efficiency satisfied in this economy? Explain.
MRSA = ½ no matter what consumption bundle A has. MRSB = yB/xB. yB/xB = ½ at yB = 15 and xB
= 30 so MRSA = MRSB. So we have distributive efficiency.
c. Is the definition of allocative efficiency satisfied in this economy? Explain.
MRSA = MRSB = MRT at the aggregate production point, so we have allocative efficiency.
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Dr. Donna Feir
Economics 313
2. Suppose there is only one producer in the economy, with PPF yT = 100 – 2xT. There are two
consumers, each with identical Cobb-Douglas preferences, where UA = xy and UB = xy.
a) Draw a diagram of the PPF. What is the MRT equal to in this economy?
y
100
MRT = 2.
50
x
b) Derive an expression for the MRS for each consumer in this economy.
MRSA = yA/xA & MRSB = yB/xB.
c) Use your answers to (a) and (b) to solve for allocative efficiency in this economy
(you should be able to solve for an (xT, yT) production pair).
We know that we will want MRSA = MRSB = MRT, where that MRT always equals 2. Thus each
consumers’ MRSs must also equal 2 in order for efficiency in all three contexts. The MRSs for
the consumers are as follows: MRSA = yA/xA and MRSB = yB/xB. In order for each of these to
equal 2 (the MRT), it must be the case that each consumer has twice as many units of y as of x.
Thus in the economy as a while, there must be twice as many units of y as of x. Thus, the
efficient mix of goods x and y in this economy must satisfy yT = 2xT. The only point on the
aggregate PPF where this is true is where xT = 25 & yT = 50.
d) Use your answer to c) to draw an Edgeworth Box diagram of this economy. Show
the contract curve in your diagram.
100
50
Allocative efficiency is the point on the
agg. PPF where we have:
MRSA = MRSB = MRT.
B
CC: yA = 2xA: MRSA = MRSB.
A
25
50
x
2
Dr. Donna Feir
Economics 313
3. Now suppose there are two producers in the economy, with PPFs as given below.
Firm 1’s PPF:
Firm 2’s PPF:
y = 100 – 0.01x2
y = 50 – 0.02x2
a) Derive an expression for the aggregate PPF in this economy (that is, yT as a function
of xT). Also derive an expression for the MRT.
We have seen in class that these PPFs give rise to the following aggregate PPF: y T = 150 –
(1/150)xT2. The MRT is therefore (1/75)xT.
There are two consumers, each with identical Cobb-Douglas preferences, where UA = xy and UB
= xy.
b) Derive an expression for the MRS for each consumer in this economy.
Again, MRSA = yA/xA & MRSB = yB/xB.
c) Use your answers to a) and b) to solve for allocative efficiency in this economy
(again, you should be able to solve for an (xT, yT) production pair. In contrast to
question 1, however, you will not get “nice” whole numbers).
The math in this one gets a little messy, but make sure you try to follow along with all the
steps. Keep the economics in mind and this will be fairly straightforward.
Just like in question 2, we want MRSA = MRSB = MRT. MRSA = MRSB  yA/xA = yB/xB, where yB =
yT – yA & xB = xT – xA (B’s consumption of both goods is the total amount available less A’s
consumption), Substituting and simplifying yields yA/xA = yT/xT, which tells us that, when we
equate the MRSs for each consumer, that common MRS will equal the ratio of the total
amount of good y to the total amount of good x. For allocative efficiency then, we need the
MRS (which equals = yT/xT) to equal the MRT (which equals (1/75)xT). So MRSA = MRSB = MRT
 yT/xT = (1/75)xT  yT = (1/75)xT2. From the equation for the PPF we also know that yT = 150
– (1/150)xT2, so we know (1/75)xT2 = 150 – (1/150)xT2  (3/150)xT2 = 150  xT = 150/(3)½.
When xT = 150/(3)½, yT = 100. So xT = 150/(3)½, yT = 100 is the efficient production pair in this
economy. Check to that the MRT will equal the equalized MRS, given this production point.
d) Use your answer to c) to draw an Edgeworth Box diagram of this economy. Show
the contract curve in your diagram.
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Dr. Donna Feir
Economics 313
Agg. PPF (productive efficiency): MRT1 = MRT2.
yT
150
B
100
A
50(3)½
Allocative efficiency:
MRSA = MRSB = MRT.
xT
150
CC (distributive efficiency, given the
production choice): MRSA = MRSB.
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MRSA = MRSB at every
point on CC. Equalized
MRSs = MRT at the
production point in the
economy.
Dr. Donna Feir
Economics 313
4. There are two individuals in the economy, each of whom is both a producer and a
consumer. Each has the same PPF, given by y = 50 – (1/5)x2. Individual A’s preferences are
given by UA = 20 ln xA + yA and individual B’s preferences are given by UB = yB.
a) Solve for the aggregate supply function for good x.
Each producer sets its MRT = price ratio  for A we have (2/5)xA = px/py  xSA = (5/2)px/py.
Because B’s PPF is identical, we also know that B’s supply of x is given xSB = (5/2)px/py. So
aggregate supply is A’s supply plus B’s supply, which is xS = 5px/py.
b) Solve for the aggregate demand function for good x.
Consumer A sets her MRS = price ratio  20/xA = px/py  xDA = 20/(px/py). Because B
consumes no x (it gives him no utility), the aggregate demand for x is just A’s demand for x. So
we have xD = 20/(px/py).
c) Use your answers to part a) and b) to solve for the equilibrium price ratio.
xS = xD  5px/py= 20/(px/py) px/py = 2.
d) Given your answer to c), how much of each good will A and B produce?
From the individual producer supply curves we know that if px/py = 2, then each firm will
produce (5/2)(2) = 5 units of x. When x = 5, y = 45 for each producer.
e) Given your answer to c), how much of each good will A and B consume?
We know that B does not consume any good x. A’s demand for x is xDA = 20/(px/py), so A
consumes 10 when px/py = 2. You can use the equation for each consumer’s BL to solve for
their consumption of y. A consumes 35 units of y and B consumes 55 units of y. (Remember –
the endowments in the budget lines are given by the quantities of goods x and y produced by
each).
f)
Why does individual B produce positive amounts good of x, given seeing that in part
e) you showed that her optimal consumption of good x is zero?
Even though B does not consume x, it is still worthwhile for him to produce x, as he can then
sell it to A and buy y in return. This enables him to consume more y than he would otherwise
be able to, were he just producing on his PPF.
5. There are two individuals in the economy, each of whom has identical Cobb-Douglas
preferences such that UA = xAyA and UB = xByB. Individual A can produce both goods according to
the PPF given by y = 100 – x. Individual B cannot produce either good, but is endowed with 20
units of good x.
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Dr. Donna Feir
Economics 313
a) Suppose the price ratio in this economy is greater than 1. Show that there is an
excess supply of good x in this case.
If px/py > 1, then A will produce all x and no y. That is, A will produce 100 units of x and 0
units of y. B is endowed with 20 units of x and no y. Aggregate supply of x is therefore 120,
while aggregate supply of y equals 0. Both the consumers have Cobb-Douglas preferences,
which means that each will always want to consume positive amounts of each good. We
do not have any y, however, meaning that there will be excess demand for good y. If there
is excess demand for y, there must be excess supply of x.
b) Suppose the price ratio in this economy is less than 1. Show that there is an excess
demand for good x in this case.
If px/py < 1, then A will produce all y and no x. That is, A will produce 100 units of y and 0
units of x. B is endowed with 20 units of x and no y. Aggregate supply of x is therefore 20,
while aggregate supply of y equals 100.
On the demand side, both the consumers choose the level x where their MRSs equal the
price ratio. For A we have MRSA = yA/xA = px/py  yA= (px/py)xA. A’s BL is given by
(px/py)wAX + wAy = (px/py)xA + yA , where we know wAX = 0, wAy =100, and – from the
tangency condition – yA= (px/py)xA. Substituting these into the equation for the BL yields
100 = 2(px/py)xA  xA= 50/(px/py). We know (px/py) < 1, so we know A’s demand for x > 50,
while supply is only 20. There is excess demand for good x when we just look at A’s
demand, so there will be even greater excess demand in aggregate (since we know B
always wants to consume positive x as well).
c) Use your answers to a) and b) to deduce the equilibrium price ratio in this economy.
If px/py > 1  excess supply of x and px/py < 1  excess demand for x, the only possibility
for an equilibrium price ratio is px/py = 1.
d) Given your answer to c), how much of each good will B consume?
For B we have MRSB = yB/xB = px/py = 1  yB = xB. B’s BL is given by (px/py)wBX + wBy =
(px/py)xB + yB , where we know wBX = 20, wBy = 0, and yB = xB. Substituting these into the
equation for the BL yields 20 = 2xB  xB = 10. We know xB = yB , so yB = 10 also. So B sells
10 of his x and buys 10 y in return.
e) Given your answer to c), how much of each good will A produce and consume?
We know when px/py = 1, A is indifferent among producing at any point on her PPF.
This means we cannot solve for her production of x and y just yet. But think about her
consumption. We know that she will consume where MRSA = yA/xA = px/py = 1  yA =
xA at her optimal consumption. Using the equation for her BL, we have (px/py)wAX +
wAy = (px/py)xA + yA , where we know px/py = 1 and yA= (px/py)xA. Substituting these into
the equation for the BL yields wAX + wAy = 2xA. We also know that wAX = whatever x she
produces (which we’ll call xAS) and wAy = whatever y she produces (which we’ll call yAS).
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Dr. Donna Feir
Economics 313
So we know (from the equation for the PPF) that wAy = yAS = 100 - xAS , wAx = xAS, so that
wAX + wAy = xAS + (100 - xAS ) = 100. Substituting this into the equation for the BL yields
100 = 2xA, so xA = 50. This is the x that A consumes. We know that her consumption of
x = her consumption of y given the price ratio of 1, so we also know that she consumes
50 units of y.
Now we can figure out how much of each good she produces. We know that she sells
10 units of y to B, and consumes 50 herself, so she must be producing 60 units. We
also know she buys 10 units of x from B and consumes 50 in total, so she must be
producing 40.
f)
Draw an aggregate PPF/Edgeworth Box diagram illustrating the equilibrium in this
economy.
See lecture slides to get an idea of what this diagram looks like.
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