Question 1.
General Monsters Corporation has two plants for producing juggernauts, one in Flint and one in
Inkster. The Flint plant produces according to 𝑓𝐹 (𝑥1 , 𝑥2 ) = min{𝑥1 , 2𝑥2 } and the Inkster plant
produces according to 𝑓𝐼 (𝑥1 , 𝑥2 ) = min{2𝑥1 , 𝑥2 }, where 𝑥1 and 𝑥2 are the inputs.
a. Draw isoquants for 40 juggernauts at the Flint plant and the same for Inkster plant.
b. Suppose that the firm wishes to produce 20 juggernauts at each plant. How much of each
input will the firm need to produce 20 juggernauts at the Flint plant? How much of each
input will the firm need to produce 20 juggernauts at the Inkster plant. Label with an 𝑎 on
the graph, the point representing the total amount of each of the two inputs that the firm
needs to produce a total of 40 juggernauts, 20 at the Flint plant and 20 at the Inkster plant.
c. Label with a 𝑏 on your graph the point that shows how much of each of the two inputs is
needed in toto if the firm is to produce 10 juggernauts in the Flint plant and 30 juggernauts
in the Inkster plant. Label with a 𝑐 the point that shows how much of each of the two inputs
that the firm needs in toto if it is to produce 30 juggernauts in the Flint plant and 10
juggernauts in the Inkster plant. Draw the firm’s isoquant for producing 40 units of output if
it can split production in any manner between the two plants. Is the technology available to
this firm convex?
Question 2.
A firm has the production function 𝑓(𝑥, 𝑦) = min{2𝑥, 𝑥 + 𝑦}. Sketch a couple of production
isoquants for this firm. A second firm has the production function 𝑓(𝑥, 𝑦) = 𝑥 + min{𝑥, 𝑦}. Do either
or both of the firms have constant returns to scale? Draw a couple of isoquants for the second firm.
Question 3
Suppose the production function has the form 𝑓(𝑥1 , 𝑥2 , 𝑥3 ) = 𝐴𝑥1𝑎 𝑥2𝑏 𝑥3𝑐 , where 𝑎 + 𝑏 + 𝑐 > 1.
Prove that there are increasing returns to scale.
Question 4.
Suppose that the production function is 𝑓(𝑥1 , 𝑥2 ) = (𝑥1𝑎 + 𝑥2𝑎 )𝑏 , where 𝑎 and 𝑏 are positive
constants. For what positive values of a and b are there decreasing returns to scale? Constant returns
to scale? Increasing returns to scale?
Question 5.
Suppose that a firm has the production function 𝑓(𝑥1 , 𝑥2 ) = √𝑥1 + 𝑥22 .
a. The marginal product of factor 1 (increases, decreases, stays constant) ……?….. as the
amount of factor 1 increases. The marginal product of factor 2 (increases, decreases, stays
constant) ……..?....... as the amount of factor 2 increases.
b. This production function does not satisfy the definition of increasing returns to scale,
constant returns to scale, or decreasing returns to scale. How can this be? Find a
combination of inputs such that doubling the amount of both inputs will more than double
the amount of output. Find a combination of inputs such that doubling the amount of both
inputs will less than double output.
Question 6.
Brother Jed takes heathens and reforms them into righteous individuals. There are two inputs
needed in this process: heathens (who are widely available) and preaching. The production function
has the following form: 𝑟𝑝 = min{ℎ, 𝑝}, where 𝑟𝑝 is the number of righteous persons produced, ℎ is
the number of heathens who attend Jed’s sermons, and 𝑝 is the number of hours of preaching. For
every person converted, Jed receives a payment of 𝑠 from the grateful convert. Sad to say, heathens
do not flock to Jed’s sermons of their own accord. Jed must offer heathens a payment of 𝑤 to attract
them to his sermons. Suppose the amount of preaching is fixed at 𝑝̅ and that Jed is a profitmaximizing prophet.
a. If h < 𝑝̅ , what is the marginal product of heathens? What is the value of the marginal
product of an additional heathen?
b. If h > 𝑝̅ , what is the marginal product of heathens? What is the value of the marginal product
of an additional heathen in this case?
c. Sketch the shape of this production function in the graph below. Label the axes, and indicate
the amount of the input where h = 𝑝̅ .
d. If w < s, how many heathens will be converted? If w > s, how many heathens will be
converted?
Question 7.
1
Irma’s handicrafts have the production function 𝑓(𝑥1 , 𝑥2 ) = (min{𝑥1 , 2𝑥2 })2 , where 𝑥1 is the
amount of plastic used, 𝑥2 is the amount of labor used, and 𝑓(𝑥1 , 𝑥2 ) is the number of lawn
ornaments produced. Let 𝑤1 be the price per unit of plastic and 𝑤2 be the wage per unit of labor.
a. Irma’s cost function is 𝑐(𝑤1 , 𝑤2 , 𝑦)?
b. If 𝑤1 = 𝑤2 = 1, then Irma’s marginal cost of producing 𝑦 units of output is 𝑀𝐶(𝑦) = ? The
number of units of output that she would supply at price 𝑝 is 𝑆(𝑝) =? At these factor prices,
her average cost per unit of output would be 𝐴𝐶(𝑦) = ?
c. If the competitive price of the lawn ornaments, she sells is 𝑝 = 48, and 𝑤1 = 𝑤2 = 1, how
many will she produce? How much profit will she make?
d. More generally, at factor prices w1 and w2, her marginal cost is a function 𝑀𝐶(𝑤1 , 𝑤2 , 𝑦) =?.
At these factor prices and an output price of p, the number of units she will choose to supply
is 𝑆(𝑝, 𝑤1 , 𝑤2 ) =?
Question 8.
A firm uses labor and machines to produce output according to the production function 𝑓(𝐿, 𝑀) =
1
1
4 𝐿2 𝑀2 , where 𝐿 is the number of units of labor used and 𝑀 is the number of machines. The cost of
labor is $40 per unit and the cost of using a machine is $10.
a. Draw an isocost line for this firm, showing combinations of machines and labor that cost
$400 and another isocost line showing combinations that cost $200. What is the slope of
these isocost lines?
b. Suppose that the firm wants to produce its output in the cheapest possible way. Find the
number of machines it would use per worker. (Hint: The firm will produce at a point where
the slope of the production isoquant equals the slope of the isocost line.)
c. Sketch the production isoquant corresponding to an output of 40. Calculate the amount of
labor units and the number of machines that are used to produce 40 units of output in the
cheapest possible way, given the above factor prices. Calculate the cost of producing 40 units
at these factor prices: 𝑐(40, 10, 40) = ?
d. How many units of labor and how many machines would the firm use to produce 𝑦 units in
the cheapest possible way? How much would this cost?
Question 9.
The prices of inputs (𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) are (4, 1, 3, 2)
a. If the production function is given by 𝑓(𝑥1 , 𝑥2 ) = min{𝑥1 , 𝑥2 }, what is the minimum cost of
producing one unit of output?
b. If the production function is given by 𝑓(𝑥3 , 𝑥4 ) = 𝑥3 + 𝑥4 , what is the minimum cost of
producing one unit of output?
c. If the production function is given by 𝑓(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) = min{𝑥1 + 𝑥2 , 𝑥3 + 𝑥4 }, what is the
minimum cost of producing one unit of output?
d. If the production function is given by 𝑓(𝑥1 , 𝑥2 ) = min{𝑥1 , 𝑥2 } + min{𝑥3 , 𝑥4 }, what is the
minimum cost of producing one unit of output?
Question 10.
After carefully studying Shill Oil, T-bone Pickens decides that it has probably been maximizing its
profits. But he still is very interested in buying Shill Oil. He wants to use the gasoline they produce to
fuel his delivery fleet for his chicken farms, Capon Truckin’. In order to do this Shill Oil would have to
be able to produce 5 million barrels of gasoline from 8 million barrels of oil. Mark this point on your
graph. Assuming that Shill always maximizes profits, would it be technologically feasible for it to
produce this input-output combination? Why or why not?
Question 11.
Farmer Hoglund has discovered that on his farm, he can get 30 bushels of corn per acre if he applies
no fertilizer. When he applies 𝑁 pounds of fertilizer to an acre of land, the marginal product of
fertilizer is 1 − 𝑁/200 bushels of corn per pound of fertilizer.
a. If the price of corn is $3 a bushel and the price of fertilizer is $𝑝 per pound (where 𝑝 < 3),
how many pounds of fertilizer should he use per acre in order to maximize profits?
b. Write down a function that states Farmer Hoglund’s yield per acre as a function of the
amount of fertilizer he uses.
c. Hoglund’s neighbour, Skoglund, has better land than Hoglund. In fact, for any amount of
fertilizer that he applies, he gets exactly twice as much corn per acre as Hoglund would get
with the same amount of fertilizer. How much fertilizer will Skoglund use per acre when the
price of corn is $3 a bushel and the price of fertilizer is $𝑝 a pound? (Hint: Start by writing
down Skoglund’s marginal product of fertilizer as a function of 𝑁.)
d. When Hoglund and Skoglund are both maximizing profits, will Skoglund’s output be more
than twice as much, less than twice as much or exactly twice as much as Hoglund’s? Explain.
e. Explain how someone who looked at Hoglund’s and Skoglund’s corn yields and their fertilizer
inputs but couldn’t observe the quality of their land, would get a misleading idea of the
productivity of fertilizer.
Intertemporal choice:
Question 1:
Decide whether each of the following statements is true or false. Then explain why your answer is
correct, based on the Slutsky decomposition into income and substitution effects.
(a) “If both current and future consumption are normal goods, an increase in the interest rate will
necessarily make a saver save more.”
(b) “If both current and future consumption are normal goods, an increase in the interest rate will
necessarily make a saver choose more consumption in the second period.”
Question 2:
Laertes has an endowment of $20 each period. He can borrow money at an interest rate of 200%,
and he can lend money at a rate of 0%. (Note: If the interest rate is 0%, for every dollar that you save,
you get back $1 in the next period. If the interest rate is 200%, then for every dollar you borrow, you
must pay back $3 in the next period.)
a. Illustrate his budget set in the graph.
(b) Laertes could invest in a project that would leave him with m1 = 30 and m2 = 15. Besides
investing in the project, he can still borrow at 200% interest or lend at 0% interest. Draw the new
budget set in the graph above. Would Laertes be better off or worse off by investing in this
project given his possibilities for borrowing or lending? Or cannot one tell without knowing
something about his preferences? Explain.
(c) Consider an alternative project that would leave Laertes with the endowment m1 = 15, m2 =
30. Again, suppose he can borrow and lend as above. But if he chooses this project, he cannot do
the first project. Draw the budget set available to Laertes if he chooses this project. Is Laertes
better off or worse off by choosing this project than if he did not choose either project? Or
cannot one tell without knowing more about his preferences? Explain
UNCERTAINTY
Question 1:
It is a slow day at Bunsen Motors, so since he has his calculator warmed up, Clarence Bunsen (whose
preferences toward risk were described in the last problem) decides to study his expected utility
function more closely.
(a) Clarence first thinks about big gambles. What if he bet his entire $10,000 on the toss of a
coin, where he loses if heads and wins if tails? Then if the coin came up heads, he would
have 0 dollars and if it came up tails, he would have $20,000. His expected utility if he took
the bet would be __________, while his expected utility if he didn’t take the bet would be
___________. Therefore, he concludes that he would not take such a bet.
(b) Clarence then thinks, “Well, of course, I wouldn’t want to take a chance on losing all of my
money on just an ordinary bet. But what if somebody offered me a really good deal?
Suppose I had a chance to bet where if a fair coin came up heads, I lost my $10,000, but if it
came up tails, I would win $50,000. Would I take the bet? If I took the bet, my expected
utility would be __________. If I didn’t take the bet, my expected utility would be
___________. Therefore, I should take the bet.”
Question 2:
Socrates owns just one ship. The ship is worth $200 million dollars. If the ship sinks, Socrates loses
$200 million. The probability that it will sink is .02. Socrates’ total wealth including the value of the
ship is $225 million. He is an expected utility maximiser with von Neuman- Morgenstern utility U(W)
equal to the square root of W. What is the maximum amount that Socrates would be willing to pay
order to be fully insured against the risk of losing his ship?
a. $4 million
b. $2 million
c. $3.84 million
d. $4.82 million
e. $5.96 million
Labour Supply
1. Aloo Tikki and Pav Bhaji work in fast food restaurants. Aloo Tikki gets Rs 320 an hour for the
first 40 hours that she works and Rs 480 an hour for every hour beyond 40 hours a week. Pav
Bhaji gets Rs 400 an hour no matter how many hour he works. Each has 80 hours a week to
allocate between work and leisure and neither has any income from sources other than
labour. Each has a utility function, U= cr, where c is consumption and r is leisure. Each can
choose the number of hours to work.
(a) How many hours will Pav Bhaji choose to work?
(b) At what levels of c and r will Aloo Tikki have kinks in its budget “line”. Draw the budget
line indicating the regions where she would be if she does not/does work overtime.
(c) If Aloo Tikki was paid Rs 320 an hour no matter how many hours she worked, for how
many hours she would work and what would be the total income she would have made a
week?
(d) Will Aloo Tikki choose to work overtime? What is the best choice for Aloo Tikki if she
works overtime. How many hours a week she would work?
(e) Suppose that the jobs are equally aggregable in all other respects. Since Aloo Tikki and
Pav Bhaji have the same preferences, they will be able to agree about who has the better
job. Who has the better job?
2. Sam’s utility function is 𝑈(𝐶, 𝑅) = 𝐶 − (15 − 𝑅)2 , where 𝑅 is the amount of leisure he has
per day. He has 16 hours a day to divide between work and leisure. He has an income of Rs
1600 a day from non-labour sources. The price of consumption goods is Rs 80 per unit.
(a) If Sam can work as many hours a day as he likes but gets zero wages for his labour, how
many hours of leisure will he choose?
(b) If Sam can work as many hours a day as he wishes for a wage rate of Rs 800 an hour, how
many hours will he choose to work?
(c) If Sam’s non-labour income decreased to Rs 400 a day, how many hours would he
choose to work?
(d) Suppose that Sam has to pay an income tax of 20 percent on all of his income, and
suppose that his before-tax wage remained at Rs 800 an hour and his before-tax nonlabour income was Rs 1600 per day; how many hours would he choose to work?
Tax Incidence
1. Each firm in a perfectly competitive constant cost industry has the long-run constant cost
function 𝐶(𝑥) = 𝑥 3 − 50𝑥 2 + 750𝑥. The market demand curve for the product is 𝑋 =
2000 − 4𝑝.
(a) What is the long-run supply curve of the industry?
(b) How many firms are there in the long-run equilibrium?
(c) A sales tax of 20% of the market price is imposed on the product. How many firms will
there be in the new long run equilibrium?
(d) Suppose instead of a sales tax, an excise tax of Rs 50 per unit is imposed. How many
firms will there be in the new long run equilibrium?
(e) Suppose, the government instead of imposing any tax, pays some subsidy on that
number of firms in the industry increases by 3 from the situation in (b). How much is the
per unit subsidy?
2. Suppose the market for widgets can be described by the following equations:
Demand : 𝑃 = 10 − 𝑄
Supply: 𝑃 = 𝑄 − 4
Where, P is the price per unit and Q is the quantity in thousands of units. Then,
(a) What is the equilibrium price and quantity?
(b) Suppose the government imposes a tax of Rs 80 per unit to reduce widget consumption
and raise government revenues. What will be the new equilibrium quantity? What price
will the buyer pay? What amount will the seller receive per unit?
(c) Suppose the government has a change of heart about the importance of widgets to the
happiness of people. The tax is removed and a subsidy of Rs 80 per unit is granted to the
widget producers. What will be the equilibrium quantity? What price will the buyer pay?
What amount per unit (including the subsidy) will the seller receive? What will be the total
cost to the government?
3. Assume India at present imports all of its coffee. The annual demand for coffee by Indian
consumers is given by the demand curve Q = 250 – 10P, where Q and P stands for the
quantity and the market price per unit respectively. World producers can harvest and ship
coffee to Indian distributors at a constant marginal (=average) cost of Rs 640 per unit. Indian
distributors can in turn distribute coffee for a constant Rs 160 per unit. The Indian coffee
market is competitive. Parliament is considering a tariff on coffee imports of Rs 160 per unit.
(a) If there is no tariff, how much do consumers pay for a unit of coffee? What is the
quantity demanded?
(b) If the tariff is imposed, how much will consumers pay for a unit of coffee? What is the
quantity demanded?
(c) Calculate the lost consumer surplus.
(d) Calculate the tax revenue collected by the government.
(e) Does the tariff result in a net gain or a net loss to the society as a whole?
Portfolio Diversification
1. Farmer Ravi has a pasture located on a sandy hill. The return to him from this pasture is a
random variable depending on how much rain there is. In rainy years the yield is good; in dry
years the yield is poor. The market value of the pasture is Rs 2,40,000. The expected return
from this pasture is Rs 40,000 with a standard deviation of Rs 8000. Every inch of rain above
average means an extra Rs 8000 in profit and every inch of rain below average means
another Rs 8000 less profit than average. Farmer Ravi has another Rs 2,40,000 that he wants
to invest in a second pasture. There are two possible pastures that he can buy.
(a) One is located on low land that never floods. This pasture yields an expected return of Rs
24,000 per year no matter what the weather is like. What is Ravi’s expected rate of
return on his total investment if he buys this pasture for his second pasture? What is the
standard deviation of his rate of return in this case?
(b) Another pasture that he could buy is located on the very edge of the river. This gives very
good yields in dry years but in wet years it floods. This pasture also costs Rs 2,40,000.
The expected return from this pasture is Rs 24,000 and the standard deviation is Rs 8000.
Every inch of rain below average means an extra Rs 8000 in profit and every inch of rain
above average means another Rs 8000 less profit than average. If Ravi buys this pasture
and keeps his original pasture on the sandy hill, what is his expected rate of return on his
total investment? What is the standard deviation of the rate of return on his total
investment in this case?
(c) If Ravi is a risk averter, which of these two pastures should he buy and why?
2. Alia has a choice of two assets : The first is a risk-free asset that offers a rate of return 𝑟𝑗 , and
the second is a risky asset (a china shop that caters to large mammals) that has an expected
rate of return of 𝑟𝑚 and a standard deviation of 𝜎𝑚 .
(a) If 𝑥 is the percent of wealth Alia invests in the risky asset, what is he equation for the
expected rate of return on the portfolio? What is the equation for the standard deviation
of the portfolio?
(b) By solving the second equation above for 𝑥 and substituting the result into the first
equation, derive an expression for the rate of return on the portfolio in terms of the
portfolio’s riskiness.
(c) Suppose that Alia can borrow money at the interest rate 𝑟𝑗 and invest it in the risky asset.
If 𝑟𝑚 = 20, 𝑟𝑗 = 10, and 𝜎𝑚 =10, what will be Alia’s expected return if she borrows an
amount equal to 100% of her initial wealth and invests it in the risky asset?
(d) Suppose that Alia can borrow or lend at the risk-free rate. If 𝑟𝑚 = 20%, 𝑟𝑗 = 10%, and
𝜎𝑚 =10%, what is the formula for the “budget line” Alia faces? Plot the budget line.
(e) Which of the following risky assets would be Alia prefer to her present risky asset,
assuming she can only invest in one risky asset at a time and that she can invest a
fraction of her wealth in whichever risky asset she chooses? State which of the assets are
“better”, “worse” or “same”.
i)
Asset A with 𝑟𝐴 =17% and 𝜎𝐴 =5%
ii)
Asset B with 𝑟𝐵 =30% and 𝜎𝐵 =25%
iii)
Asset C with 𝑟𝐶 =11% and 𝜎𝐶 =1%
iv)
Asset D with 𝑟𝐷 =25% and 𝜎𝐷 =14%
(f) Suppose Alia’s utility function has the form 𝑢(𝑟𝑥 , 𝜎𝑥 ) = 𝑟𝑥 − 2𝜎𝑥 . How much of her
portfolio will she invest in the original risky asset?
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