Questions for Review – General Equilibrium
1. Bob has an initial endowment consisting of 10 units of food and 10 units of
clothing. Eric has an initial endowment of 10 units of food and 20 units of
clothing. Draw an Edgeworth Exchange Box for these consumers.
We did this one in class. The box (EEB) should be 20 units high by 30 units
wide, presuming that you put food on the vertical axis and clothing on the horizontal.
Let’s assume that you decide to put Bob’s origin at the bottom left and Eric’s origin at
the top right. Mark amounts of food and clothing on the axes so that Bob measures
increasing amounts of food in an upward direction and increasing amounts of
clothing in a rightward direction. Eric will then measure increasing amounts of food
in a downward direction and increasing amounts of clothing in a leftward direction.
2. Bob regards food and clothing as perfect one-for-one substitutes, while
Eric regards them as perfect complements but in a ratio of 3 units of
clothing for 2 units of food. What is the set of allocations that are paretopreferred or pareto superior to the original endowment given above?
Where is the contract curve in the EEB? What set of trades could lead to
the contract curve? What price ratio would be required to ensure that a
competitive equilibrium reaches the contract curve from the initial
endowment?
Perfect one-for-one substitutes means that Bob’s indifference curves are straight lines
joining equal amounts of food and clothing along the axes (i.e., one indifference curve
is a straight line joining the point representing 10 units of food and no clothing to the
point representing 10 units of clothing and no food). This means that Bob gets equal
amounts of utility from 10 units of food or 10 units of clothing or any linear
combination of these two. In contrast, Eric’s indifference curves are L-shaped
(upside-down L-shaped since his origin is in the northeast corner of the EEB). The
vertex of the L always has an allocation of food and clothing to Eric in the ratio of 3
units of clothing to 2 units of food. Any other ratio involves either excess clothing or
excess food, in the sense that Eric gets no additional utility from the excess food which
is above a ratio of 2 to 3, and no additional utility from the excess clothing which is
above a ratio of 3 to 2.
The initial endowment given in Question #1 gives Eric 10 units of food and 20 units
of clothing. This implies that Eric has 5 units of excess clothing at this initial
endowment. He is therefore willing to trade away clothing in exchange for food. On
the other hand, Bob gets equal utility from a unit of food or a unit of clothing (they
are perfect substitutes). So, for instance, if Eric were to trade 2 units of clothing and
get 2 units of food in return, he would be better off and Bob would be no worse off, so
this would be a pareto superior move.
In fact, any point on the lens formed by their two indifference curves that pass
through the initial endowment will be pareto superior (except for the one exception of
the allocation at the extreme opposite end of the lens). Further, since this trade would
give Eric 18 units of clothing and 12 units of food, it would give him an allocation
that was pareto-optimal (Eric’s allocation could not be further improved without
making Bob worse off). A competitive price ratio of 1-1 (i.e., PF/PC = 1) would allow
this pareto-optimal trade to take place.
3. Consider a simple economy with only food and clothing and two
consumers, Ann and Bill. Both have regularly shaped convex indifference
mappings. For a given initial endowment, when the ratio of food to
clothing prices in the economy is 3/1, Ann wants to buy 6 units of
clothing, while Bill wants to sell 2 units of food. Is the current price ratio
an equilibrium price ratio? If not, in what direction will this price ratio
change?
This price ratio cannot be an equilibrium one, because both Ann and Bill want to buy
clothing and sell food at this price ratio. There will be excess demand for clothing and
excess supply of food at this price ratio and their desired demands and supplies will
therefore not clear the market. In order to bring their demands and supplies into
balance (consistency with one another), the price of clothing would have to rise
(reducing quantity demanded) and the price of food would have to fall (reducing
quantity supplied). In other words, the ratio of the price of food to the price of
clothing would have to fall.
4. How will the answer to #3 change if Ann wants to sell 3 units of clothing
and Bill wants to sell 2 units of food?
This would mean that Ann would sell 3 units of clothing and buy 2 units of food. At
the current price ratio (PF/PC = 3/1) , Ann would get $3 for selling her 3 units of
clothing, but then would buy only 1 unit of food with this $3. The market does not
clear at this price ratio. Bill wants to sell more food than Ann wants to buy. The
price of food will therefore have to fall (or, what is the same thing, the ratio of the
price of food to the price of clothing will have to fall).
5. Given the current allocation of productive inputs, the marginal rate of
transformation of food for clothing in a simple two-good economy is
equal to 2. At the current allocation of goods for consumption, each
consumer’s marginal rate of substitution of food for clothing is 1.5. Is this
economy efficient in its product mix? If so, explain why. If not, describe a
reallocation that would lead to a pareto improvement.
For product-mix efficiency, the common MRS of each of the consumers (along their
indifference curves) must be equal to the MRT (measured along the PPF). In this
case, the MRS’s are equal across consumers, but are 1.5, whereas the MRT is 2.0.
This means that, at the current product-mix, the economy can produce 2 units more
of food by giving up 1 unit of clothing. However, at the current product-mix,
consumers are willing to give up 1.5 units of clothing to get 1 more unit of X. In
other words, consumers don’t value the extra X as highly as the cost of the extra X in
terms of units of Y given up (along the PPF). In other words, this economy is
producing too much X, and not enough Y. That means that additional Y should be
produced. When more Y is produced, the MRT (the opportunity cost of X) will fall
from 2.0 to a lower number (because the PPF is an increasing cost function). This
product mix with more Y and less X will mean that the MRS of consumers changes
(it will become greater than 1.5, because as more Y is consumed, X becomes relatively
more valuable to consumers). Somewhere in the middle, the MRS and the MRT
would be equal, with more Y and less X in the product mix.
6. A castaway on a desert island (Robin) can make 5 units of food per day or
10 units of clothing, using all his labour resources in producing either one
or the other. There is no capital equipment. If he divides his time
between the two, his output of each is proportional to the amount of time
spent on each. This castaway is fortunate to have been left on the island
with his girlfriend, whose name is Tuesday. Tuesday can make either 10
units of clothing or 15 units of food or any proportional combination.
Given this information, describe the production possibilities frontier for
their economy.
Assuming that clothing is measured on the vertical axis and food on the horizontal,
when Robin and Tuesday combine their efforts they could, at most, produce 20 units
of clothing on the vertical axis, or 20 units of food on the horizontal axis.
The PPF is not a straight line joining these two points because Robin and Tuesday
are differentially productive in producing food and clothing. As you can see from the
numbers given, Robin does not give up much food when he produces clothing, but
Tuesday gives up a lot of food when she produces clothing. You can, if you like,
calculate the value of opportunity cost of food and of clothing produced by each one of
these producers. The result is that Robin’s opportunity cost of clothing is low, while
Tuesday’s is high. Correspondingly, Robin’s opportunity cost of food is high (a lot of
clothing given up for each unit of food produced), whereas Tuesday’s opportunity cost
of food is not so high. The implication is that Tuesday should be the first one to
switch from producing clothing to producing food as we move down to the right
along the PPF.
The PPF is therefore composed of a straight line from 20 units of clothing and no food
to a point where 15 units of food and 10 units of clothing are produced (i.e., Tuesday
is producing only food and Robin only clothing at this point). From there, the PPF
follows a straight line directly to the point on the horizontal intercept representing 20
units of food and no clothing. In other words, the PPF is not regular, but is piecewise
linear, with two pieces.
7. If Robin and Tuesday both regard food and clothing as perfect one-for-one
substitutes, how much should each produce?
If food and clothing are one-for-one substitutes, then the indifference curves have a slope
of –1. There are two linear parts that make up the PPF. The top part (representing
Tuesday’s production) has a slope of -2/3 (10 clothing to 15 food). The bottom part
(representing Robin’s production) has a slope of –2 (10 clothing to 5 food). Therefore, the
indifference curves touch the PPF right at the point where Tuesday’s production ends
and Robin’s production begins (at 10 units of clothing and 15 units of food). This is the
highest indifference curve that can be reached, so Tuesday should produce 15 units of
food and Robin should produce 10 units of food.
8. Now, suppose a trading ship visits the island each day and offers to buy
and sell food and clothing at a price ratio of 4/1 (price of food divided by
price of clothing). Unfortunately, the ship will not take them off the
island, but it is willing to trade. Will this ship’s willingness to trade
change the production and consumption decisions of Robin and Tuesday?
How?
Goods can be exported or imported at the price ratio of $4 for food and $1 for clothing.
The slope of the price line is –4, which is quite steep. Therefore, the highest point at
which this price line touches the PPF is on the X axis, where Robin and Tuesday are
together producing 20 units of food. These 20 units of food could be sold for $80.
Then Robin and Tuesday can consume any amount they want between 20 units of
food and no clothing and 80 units of clothing (at $1 each) and no food. Since, Robin
and Tuesday regard food and clothing as one-for-one substitutes, they will prefer to
trade all 20 units of food in exchange for 80 units of clothing.
9. Ann and Bill are price-taking consumers in a two-person, two-good,
exchange economy with a fixed endowment of 40 units of good X and 80
units of good Y. Ann has 12 units of X and 30 units of Y as initial
endowment. Ann’s utility function is given by U = X2Y and Bill’s is given
by U = XY. Find the equation that expresses the contract curve.
The contract curve will join together all points where Bill and Ann’s MRS are equal. For
Ann, the MRS = 2XY/X2= 2YA/XA. For Bill, the MRS = YB/XB. However, this is
equivalent to (80 – YA)/(40 – XA). The two MRS’s will be equal when 2YA/XA = (80 –
YA)/(40 – XA) or when 80YA –2YAXA = (80XA – YAXA) or when YA = XA + (1/80)YAXA.
10. Using the information in Question #9, calculate Ann and Bill’s MRS’s at
the initial endowment point and explain both why this initial endowment
is not pareto-efficient and who will want to sell X and who will want to
sell Y.
At the initial endowment point, Ann’s MRS = 60/12 = 5. At the initial endowment
point, Bill’s MRS = 50/28 = 25/14 (or a little bit less than 2). Therefore, Ann is
willing to give up 5 Y to get 1 X at the initial endowment and Bill is willing to give
up about 2 Y to get 1 X. Relative to Bill, Ann regards X as valuable, so she will want
to buy X and sell Y and Bill will want to do the opposite.