Econ 3010: Intermediate Price Theory (microeconomics)
Professor David L. Dickinson
PROBLEM SET #2
INPUTS AND PRODUCTION
1)
Consider the production function with three variable inputs
f ( x1 , x 2 , x3 )  x12 x 22 x32 . Prove that this production function exhibits increasing returns to
scale.
2)
Write out a production function specific to the following technology: Output can
be produced with either human workers or robots. To produce each unit of output, we
must use either 50 humans or 10 robots.
3)
Prove that the technical rate of substitution (the TRS) is equal to the ratio of
marginal product of input x2 to input x1 (Hint: recall that as we move from one point to
another along a given isoquant, the total amount of output remains constant and does not
change).
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4)
Assume a firm with production function f ( x1 , x 2 )  x12 x 2 2 . What is the firm’s
TRS?
5)
Suppose the Jackie makes furniture using her labor=x1 and wood=x2. Her
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production function is f ( x1 , x 2 )  x1 3 x 2 3 . What is the general equation for one of
Jackie’s isoquants? Draw her isoquant for the level of output Q=4 (include at least 4-5
different points (x1,x2) that yield output level Q=4). Now suppose that her wood supply
is fixed in the short-run at 8 units. Draw Jackie’s production function (I’d suggest
varying the input level of x1 from 0 to 8 to 24 to 40 to 80 in order to get a good view of
this production function). Finally, how would the production function differ if her wood
supply were fixed at 64 instead?
PROFIT MAXIMIZATION AND COST MINIMIZATION
1)
Jiffy Lawn is a local lawn mowing company. The company has only two inputs,
both variable: people hours=x1 and riding lawn mower hours=x2. Suppose that it takes
exactly two hours of someone using the riding mower in order to produce one unit of
output, which in this case is an acre of mowed lawn. Write down the production function
for the company.
Now, suppose that workers are paid the market wage of $10 per hour, and it costs
$20 per hour for Jiffy Lawn to rent a riding lawn mower. Clients pay Jiffy Lawn $70 for
each acre of lawn that is mowed. What is the profit maximizing choice of inputs and
output, and the resultant profits, for Jiffy Lawn?
Suppose rather that Jiffy lawn does not want to mow more than 10 acres of lawn.
Graph the company’s profit maximizing choice in this case, and what are the firm’s
profits.
2)
Let a firm’s production function be described by f(x1.x2)=3x1+x2. What would be
the general form of the conditional factor demands for x1 and x2 (recall that the
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conditional factor demands describe the optimal level of the input, in terms of input
prices and the level of output)?
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3)
Suppose a firm has a Cobb-Douglas technology of the form f ( x1 , x 2 )  x1 3 x 2 3 .
What are the marginal products of each input? What is the technical rate of substitution,
the TRS? Use the condition that the TRS=w1/w2 at the profit maximizing choice in order
to find the conditional factor demand functions for x1 and x2 (hint: set the TRS=w1/w2,
solve this relationship for x1 and substitute this into the production function to solve for
the conditional factor demand x2*=x2*(w1,w2,y). Do the same to solve for x1*).
4)
Assume that a firm has a perfect complement technology:
f ( x1 , x2 )  min 1  x1 , 1  x2 . What would the firm’s cost function be (i.e., costs
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based on optimally chosen inputs)?
5)
I’ve often looked at the bag of, say, potato chips and been intrigued by the
ingredients list. It is not uncommon to see something like: Potatoes, salt, and one or
more of the following (canola oil, corn oil, safflower oil).
Consider the part that reads “one or more of the following (canola oil, corn oil,
safflower oil). Does this imply perfect complement technology for these “oil” inputs, or
perfect substitute technology? What is your reasoning? Are there different implications
for the firm if the price of, say canola oil rises, but other input prices remain fixed, and
we assume perfect complement versus substitute technology?
    
COSTS CURVES AND FIRM SUPPLY
1)
Suppose a firm has cost function c(y)=3y+10.
Find the firm’s ATC, AVC, AFC, and MC curves, and graph them all together.
What is the firm’s optimal level of output in the short-run if P=$5 for output?
What if P=$2 for output.
2)
Prove that a firm in the short-run will want to shut down for all P<AVC.
3)
Mr. Goodwrench fixes cars. Specifically, in order to fix s cars, it costs Mr.
Goodwrench c(s)=4s2+20s+200. Find Mr. Goodwrench’s ATC, AVC, and MC and graph
them together. Highlight Mr. Goodwrench’s short-run supply curve.
Suppose the price he can get for each fixed car is P=$92. Highlight Mr.
Goodwrench’s profit maximizing choice of cars fixed. Also, highlight on the graph his
total revenue, his total costs, and his profits.
Finally, what price does Mr. Goodwrench require in order to remain in the carfixing industry in the long-run?
4)
Suppose that a firm has total cost function c(y)=20y2+4500. Find all the firm’s
cost curves and graph the firm’s optimal choice point, assuming P=$1000 for output, on a
graph that includes the cost curves ATC, AVC, MC.
5)
Consider the problem you just did in questions (4). Find the firm’s producer
surplus, as well as the profits it makes at its optimum. Highlight these on the graph of the
firm’s optimal choice point (either a new graph or the graph you made for question (4)).
What is the relationship between PS and profits?
INDUSTRY SUPPLY AND EQUILIBRIUM
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1)
Assume the market for #2 pencils (the old ones, not the mechanical ones) is
perfectly competitive and currently in LR equilibrium. Now suppose that all schools
across the country decide to celebrate the history of the #2 pencil and want to use them in
all their classrooms (o.k., I’m trying to come up with something creative to say that
market demand goes up).
Do a detailed graphical analysis of the SR effects and LR adjustment process in this
competitive market.
2)
Suppose that every firm and potential firm in a market faces the same marginal
costs, MC=2y (in other words, this is the supply curve for all firms in the market), and
minimum ATC is at $2. Graph the market supply curves for a situation where there are
1, 2, 3, 4, or 5 firms in the market.
If we now assume that market demand is given by Qd=25/4-(5/4)p, how many
firms will enter the market and what will be the LR equilibrium price? (hint: write the
supply curve as a function of N, the number of firms in the market, and the find N
necessary to produce p=$2 (because this is the long-run minimum price needed for
profits=0). This should help)
3)
Suppose that the market for corn is perfectly competitive, and the market supply
curve is given by S(p)= 5P-5, whereas market demand is given by D(p)=40-10P
(quantity, Q, is measure in bushels). Graph equilibrium supply and demand in this
market.
Suppose now that the government decides that wheat farmers should get at least
$3.50 for each bushel of wheat. What are the consequences of this policy? What if the
government promises to buy up all the wheat that farmers are willing to supply at a price
of $3.50 per bushel, even if market consumers do not demand those bushels?
4)
Use graphical analysis to show that a per-unit subsidy on a good is economically
inefficient (i.e., it lowers total surplus relative to the free market outcome).
Does the inefficiency result depend on whether the subsidy is explicitly offered to
buyers or sellers?
MONOPOLY, PRICE DISCRIMINATION, AND MONOPOLISTIC
COMPETITION
1)
Consider the following two statements: a) Demand for heroin in highly
inelastic. b) The supply of heroin in monopolized by the mafia.
Can you identify anything inconsistent in these statements? (Hint: think about
elasticity of demand along a linear demand curve and the relationship of d and marginal
revenue).
2)
Consider a “natural” monopolist. Suppose that the government wants to regulate
the price a natural monopolist can charge to be no more than its marginal costs. What is
the problem with this type of regulatory policy?
3)
Show that the monopolist’s profit maximizing decision will lead to a price markup that is a function of the elasticity of demand.
4)
Suppose that Placebo Medicine is a monopoly seller of a fabulous new drug
“Relief-trex”. Placebo has total costs given by TC=y2. Suppose that market demand for
Placebo’s medicine’s is Qd=10-p.
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Placebo medicine is deciding whether to charge one single price for its drug, or
whether to use its amazing negotiating skills to charge a different price to each hospital in
its market. Assume that Placebo is so skilled that it could negotiate a price equal to each
hospital’s willingness-to-pay for the drug. Graph and show profits under each of the two
different scenarios: charging a single price, and charging a different price=WTP for each
customer. In which situation are profits highest? Is each outcome efficient or inefficient,
and why?
5)
Assume the market for hotel rooms is monopolistically competitive. Use
graphical analysis to show how short-run profits by the typical firm, like Holiday Inn,
will be reduced to zero in LR equilibrium.
Also, if hotels empirically operate at about a 20% vacancy rate, how is this
consistent with profit maximization under our assumption of a monopolistically
competitive market?
OLIGOPOLY
1)
Pete’s Petro (PP) and Gina’s Gas-World (GG) are the only two supplier of
gasoline in the town of Podunk. Their gasoline they sell is identical. Market Demand for
gasoline in Podunk is given by D(p)=60-P. PP and GG have identical marginal cost of
gasoline production at MC=30.
Use the Cournot model to predict the outcome in the gasoline market in Podunk.
Graph the predicted Cournot equilibrium.
2)
Let’s continue the example of the duopoly gasoline market in Podunk. You have
a Cournot equilibrium prediction for Gina and Pete. Suppose that Gina and Pete both
decide to produce some random quantity, say qpp=4, and qgg=25. Use the reaction
functions to show how Gina and Pete will converge to the Cournot equilibrium (hint:
start with one of them reacting to the quantity of the other, and then the other reacts to the
new quantity of the competitor (after the competitor has already reacted first), etc.).
3)
One more time with Gina and Pete. Suppose now that Gina stumbles upon a great
new technology that lowers her marginal cost of gasoline supply to MCgg=20. Calculate
the new Cournot equilibrium, and show on a graph of reaction curves how this new
technology changes the Cournot equilibrium picture relative to before. What happens to
Gina’s market share and to the market price as a result of Gina’s lower MC?
4)
Assume that Shilo Shoes is a dominant firm in the shoe market. It is much bigger
than all other firms, and can produce a lot more shoes relative to all the smaller firms that
make up the rest of the market supply for shoes (these smaller firms behave competitively
also). Assume that market demand for shoes is given by D(p)=100-P (quantity is
measure in terms of pairs of shoes). Shilo Shoes has marginal costs=$10 per pair of
shoes. The supply curve for the competitive fringe firms is given by S(Yf)=(1/3)P.
5)
Let’s continue the example of Shilo Shoes. What is Shilo’s market share from the
price leadership equilibrium in question (4)? Now suppose that some new firms enter the
fringe supply of the market such that now, the fringe supply is given by S(Yf)=(1/2)P.
Find the new price leadership outcome and graph the outcome relative to the
previous outcome (you can do this on your previous graph if you like). What has
happened to the market price and Shilo’s market share as a result of the additional fringe
supply?
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