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EconS 301, Microeconomics with Calculus
Spring 2015, Homework 11
1. Explain why universal percentage taxes do not make an economy Pareto inefficient, but universal per unit taxes do.
(Hint: Start with the Pareto efficient conditions that MRS=MRT=price ratio between good. Think about how
percentage taxes and per unit taxes impact the price ratio.) (3 points)
Universal percentage taxes do not change the price ratio hence the consumers remain on the optimality condition (MRS
= MRT = P1/P2). For the per unit taxes, the price ratio will change and the economy will stop operating at optimality.
MRS will not equal MRT in such a scenario.
Percentage Tax: MRS = P1(1+t)/P2(1+t) = P1/P2
Firms price ratio = MRT = P1/P2 . Since MRT = MRS, economy is Pareto efficient.
Per unit tax: MRS = (P1+ T)/(P2+ T)P1/P2 (firms price ratio). Hence per unit taxes have introduced inefficiency in the
economy.
2. Currently about 16-17% of GDP in the US is spent on health care. Many economists argue that health insurance and
subsidized health care artificially increase the demand for health care by lowering the effective price consumers must
pay, leaving the allocation of resources between health care and other goods Pareto inefficient. Using an Edgeworth
box diagram that includes the PPF, analyze this problem from the perspective of general equilibrium and welfare
economics. Specifically discuss:
a. How health insurance causes too much of our economy to be devoted to health care; (2 points)
b. Health insurance and subsidized health care means that there is an equivalent to an income transfer from
health people to sicker people. Assume in your (two-person Edgeworth box economy) the initial endowment
is equal, so both start in the middle of the Edgeworth box. Suppose one person is sickly and the other healthy,
but both must buy health insurance. If the two goods in your economy are “healthcare” and “other goods”
explain what how an insurance requirement will redistribute income and leave the economy Pareto
inefficient. (3 points)
a) This is a case where the MRT > MRS and we see that there is "too much" health care. Because of the subsidy and
insurance, the price of health care paid by consumers is lower than the price for producers. Let H be health care and G
stand for other goods. Also, let Ph* be the (subsidized) price consumers pay for health care, Ph be the true price of
health care, and Pg be the price of other goods. We have MRS(HforG)=Ph*/Pg<Ph/Pg=MRT(HforG). Graphically we have
firms maximizing profits putting the economy on the blue tradeoff while consumers maximizing utility show the red
tradeoff.
sickly
healthy
b) In this case, the insurance transfers buying power from the healthy person – shown in the lower left corner – the the
sickly person – in the upper right corner. Because of the subsidized price for health care thir MRS is still different to the
MRT. Therefore we still have MRSMRT
3. Analyze this question from the perspective of Pareto efficiency: Simpson has found $2000 and decides to divide it
between his two children, Sara and Bill. Consider what is necessary (in terms of their marginal utility of wealth) for
each of the following distributions to be Pareto efficient.
a. Giving each $1000. (1 point)
b. Allocating it so each child’s share is inversely proportionate to his or her wealth. (2 points)
A) This is actually a subtle question. We have to think about the MU of wealth against the MU of goods. If wealth
is measured in currency, the price of wealth is just 1. So if g stands for goods, we need MUg/MUw=Pg. Before
the gift, if both were optimizing and solving for MUw we find MUw=Mug/Pg and if would be the same for both
children and so it would be Pareto efficient to give them the same amounts if their MUw change at the same
rate.
B) All the arguments still hold. However, we expect having more of something means that its MU goes down, so
the wealthier child would consume both more of goods (to lower its Mug) and have more wealth (since MUw is
lower with more wealth). Because we might expect the MUw of the wealthier child to drop more quickly (from
diminishing marginal utility), this might be a good scheme.
4. Consider a pure endowment economy (no production) with two goods, A and B, and two people, Sue and Bill. There
are a total of 4 units of good A and 8 units of good B.
a. Draw the Edgeworth Box for this economy. (1 point)
b. Suppose Bill has 5 units of B and 1 unit of A. Show this allocation on your graph? (1 point)
c. Both Bill and Sue have fixed proportion utility functions, so U=min(2A,B). What is the ratio of A and B they
each want to consume? (1 point)
d. Find the contract curve for this economy, and mark it on your graph? (2 points)
e. Show the part of the contract curve that would improve both Bill’s and Sue’s utility. Explain why this is the
relevant portion. (2 points)
f. Suppose Bill still has the fixed proportion utility function, but Sue has a simple Cobb-Douglas production utility
function of the form U=A*B. Now find the contract curve. Why is it unchanged? What will be Sue’s MRS? (3
points)
Parts a, b and c.
A is the point where Bs = 4, As = 2 and for Bill, Bb = 4 and Ab = 2.
d and e) The contract curve is a region where the utility curves are tangent. The initial endowment is E and it is better for
both of them to move towards the point A. The red indifference curves are for Bill and green are for Sue. Bill wants to
trade B for A and Sue wants to trade A for B.
f) MRS for Sue = MUA/MUB = B/A. But Bill still requires B/A=2. We know on the contract curve the indifference curves
are tangent. Hence, because Bill’s requires B/A=2, Sue will be forced to that same ratio.
5. Consider a two-person economy (Al and Sue) with two goods (A and B) where both individuals have a utility function
U=A*B. The PPF for A and B is a straight line with a constant MRT. With the resources in the economy, at most 20
units of B can be produced, and at most 10 units of A.
a. What is the MRT in this economy? (1 point)
b. Find the formula for the MRS for each person. (1 point)
c. How much A and how much B will be produced if the economy is Pareto efficient? Explain. (Hint, remember
we need the MRS=MRT.) (2 points)
a) Using the two point formula for slope: (x-y)/(x'-y') = (0-20)/(10-0) = -2. The negative of this is the MRT.
b) MRS = B/A for both individuals because of symmetric preferences.
c) We need MRS=MRT, but we know the MRT=2. Hence we need the MRS=2 as well. Efficient production where B/A = 2
--> B = 2A. We also know that B=20-2A from the PPF. Hence 2A=20-2A implying 4A=20 or A=5. Then B=10. This is the
only point on the PPF where B=2A.
XC1. Add production in question 4. If the economy is Pareto efficient, what will be MRT. Explain. (3 points)
Bill always requires B=2A and Sue has an MRS=B/A. Exchange equilibrium thus required Sue’s MRS=B/A=2A/A=2. Hence
whatever formula accounts for the MRT, the value of the MRT=2.
XC2: In question 5, suppose Al can produce only A and Sue can produce only B.
a. Draw the Edgeworth box of question 5 that illustrates efficient production and the initial allocation. (2 points)
b. Show the exact point at which they will end up. Explain why they end up there. (3 points)
Let's assume, because it has to end up there, that the economy starts with the efficient amount of production that we
found in question 5, so B=10 and A=5. Al produces 5 A and Sue produces 10 B. Since Al produces all of the A and Sue all
of the B, the initial allocation is at the purple dot. We know both have an MRS=2 and B=2A for both Sue and Al, which
means they must lie along the blue dotted line, meaning they end up right in the middle, at the black dot, where each
consumes 2.5 A and 5 B.