ECF5040 Industry Economics Assignment 2 solution / answers
1 Productivity as a Cost Shifter
In class we saw how productivity can both be thought of as a shifter of the production
function and as a shifter of the cost curve. This exercise will walk you through how to
show the latter under a Cobb-Douglas production function. The firm’s problem is that
of cost minimization subject to the constraint the firm faces, which is the production
function itself.
minL,KTC = wL + rK
subject to Q = ALαK1−α
You are to derive that the total cost function, after optimization, looks like:
TC =
α−α
(1 − α)1−α
Q
A
wαr1−α
The point of the exercise is to show how TC depends on Q and A, total output produced
and productivity. We start with two preliminary questions followed by the steps of the
derivation.
1. What is the definition of α in the production function? Please give a formal definition.
2. What is the definition of A in the production function?
3. Start by taking the derivative of the Lagrangean (see below) with respect to the first
choice variable, labor, and setting the resulting first order condition to 0. Rearrange
the equation to have the factor price, w, on one side, everything else on the other.
Repeat the same thing for capital (capital’s factor price is r). Your starting equation
is
minL,KwL + rK − λ(Q − ALαK1−α)
4. Divide the first equation you got by the second equation to obtain a relationship
between the ratio of factor prices and the ratio of factor quantities. Rearrange this
equation to have L on one side, and everything else on the other.
5. Rearrange the production function such that K is on one side, and everything else
on the other.
6. Substitute the final equation you obtained in 4) for L into the expression for K you
obtained in 5). Rearrange such that you have an equation where K is on one side
and α,w,r,Q, and A on the other side.
7. Substitute the expression for K in 6) into the expression for L you obtained in 4).
8. The last step is to substitute the expressions from 7) for L and from 6) for K into
the TC function. You should now have TC = α−α
(1−α)1−α
Q
Awαr1−α.
These steps are not the only way to obtain the final expression for TC, it does not
matter which route you follow as long as you explain what steps you are taking and
show the algebra.
9. How does the total cost value in optimum depend on productivity? And on output?
2 Market Structure and Productivity: A Concrete Example
Read the Syverson (2004) Market Structure and Productivity: A Concrete Example paper
uploaded among the week6 materials on Moodle. Note, to answer the questions you will
need to read the whole paper, not just the introduction distributed in class. Please answer
the following questions
1. What is the stylized fact the paper starts out from?
2. What does the paper hypothesize and test?
3. Why is the concrete market particularly suited for testing the general idea outlined
in 2)?
4. What is demand density?
5. What is the role of substitutability across producers within the local market?
6. What is the role of substitutability across producers that are in different markets?
7. The model described in the paper refers to one market. The goal of the model is
to derive implications on the relationship between demand density in a market and
various characteristics of the firms active in a given market in equilibrium. These
implications can then be tested in the empirical part of the paper by using the
fact that demand density is different across different markets in the dataset. In
the next few questions we will focus on understanding how one such implication is
derived, namely the relationship between demand density and the productivity (or
cost) distribution of firms operating in the market in equilibrium.
Describe the demand side of the market. What is the market in the model? What
is the mass or density of consumers on the market? Does every consumer on the
market face the same price? Why or why not? What is the price a given consumer
faces? Explain the components to this price.
8. The supply side of the model consists of two stages. In the first stage, a large
number of firms consider whether to pay sunk cost s in order to be told what
their marginal cost ci would be if they started production. Before learning about
their ci they do not know anything about how productive they are relative to other
potential entrants. This ci is a random number whose distribution is summarized
by the probability density function g(c) with support [0, cu] (this just means for no
producer can the marginal cost be below 0 or above cu).
In the second stage, those that have decided to pay s to learn their ci think about
whether to start producing or not. What do they take account of when making this
decision?
9. If a firm decides to start production, they pay fixed cost f to start production,
they get a location assigned to them on the unit circle and they set their price pi,
called factory door price. (Factory door price just means the price is not inclusive
of transport costs.)
Because we know all consumers purchase on the market (this is an assumption made
in the paper), and because of some restrictions on parameters of the model, it is
the case that between any two neighboring firms there will be a marginal consumer
that is indifferent between buying from the firm to their right and the firm to their
left (firm i and j). Let us denote the distance of this indifferent consumer from firm
i towards the direction of firm j by xij . Write down the condition that determines
xij . Explain both the right hand side and the left hand side of this condition in
detail.
10. The expected profit of a firm i is given by equation (3) in the paper. Firm i
maximizes this expected profit with respect to its choice variable, its factory door
price pi. Show how to get the optimal pi starting from equation (3).
11. Equations (6a), (6b) and (6c) give firm i’s markup (pi − ci), expected maximized
market share (2E(xi)) and expected maximized profits (E(πi)). Explain how you
can see that the expression for markup and for expected market share are both
decreasing in firm i’s own cost ci.
12. (6c) shows the expected maximized profits as a function of firm i’s own cost among
others. Recall that in the second stage of the firm’s problem, once it knows its
own cost, but does not know where it will be located yet or the costs of other firms
potentially on the market, firm i makes its decision whether to enter the market and
start production based on whether it expects to make profits from entry. At this
stage, all the firm is able to calculate is its expected profit (6c). The firm will thus
enter the market if its cost realization ci is such that its expected profit is positive.
There will be some firms that got a cost draw ch which is too high for them to make
a profit in expectation. Setting (6c) equal to zero tells us what is the threshold
value of c, call it c∗ above which firms would make a loss in expectation and below
which firms would make a profit. Derive this cost level c∗ using equation (6c).
13. Notice that the above derived c∗ is still a function of endogenous objects, such as
E(c). E(c) is the expected cost of those firms that will be operating in the market in
equilibrium. So, in order to be able to determine c∗ only as a function of exogenous
objects (like demand density) and parameters, we need to go one step back, to the
first stage of the firm’s problem.
Equation (7) tells us the expected profit conditional on deciding to start manufacturing
after seeing one’s own cost draw. (The |ci < c∗ means conditional on the cost
draw of firm i being below the threshold cost c∗ above which we know firms will
decide not to start production.)
Determining the value of c∗ as a function of exogenous objects and parameters only
thus requires us to think through when firms would decide to pay the sunk cost s in
the first stage of the firm’s problem. They would decide to pay it if their expected
profit from operating E(πi|ci < c∗) covers at least the fixed cost f they need to pay
to start manufacturing and the sunk entry cost s to be able to obtain a cost draw ci.
Equation (10) shows the expected value of entry, as a function of c∗. The value of c
which equates V e to 0 tells us the threshold c∗ now only as a function of parameters
and exogenous objects.
Now with the expression for c∗ in hand, we can determine how according to this
model c∗ varies with demand density D, in other words we can determine whether
dc∗
dD is positive or negative. Section II/B of the paper shows that this expression is
negative. This means that if D is larger, the equilibrium cutoff c∗ is smaller. This
means that the truncation point c∗ is going to be a lower number, i.e. only firms
with lower costs will be entering the market. In other words only more productive
firms will be entering the market when demand density is higher. (Notice how useful
it is to think of productivity as a shifter of the cost curve (something we showed in
problem 1 of this homework) to grasp the intuition of how demand density changes
the productivity distribution of firms in the market in equilibrium.)
What does this dc∗
dD < 0 result imply about the minimum productivity levels of
producers in denser markets?
14. What does this dc∗
dD < 0 result imply about the average productivity levels of producers
in denser markets?
15. What does this dc∗
dD < 0 result imply about the productivity dispersion in denser
markets?
16. The above are the major implications of the model which the author tests in the
data. How is the concept of demand density used in testing the hypothesis of the
author?
17. What can you read off of Fig 1 and Fig 2? The evidence from these graphs is just
suggestive.
18. In the data it is not possible to observe costs ci of firms. However, it is possible
to measure productivity in the data which is inversely related to costs. The author
uses a method called TFP index method to recover elasticities αlt, αkt, αmt, αet.
With these in hand he can recover ait as a residual. See equation (15). Explain
how this method works. How does this method solve the endogeneity problem we
discussed in class?
19. How does the author test their hypothesis? Explain the regression outlined in (16).
20. Look at Table 3. What does the coefficient estimate on demand density -0.014 mean
(Model 1, first row).
3 Strategic Bundling
On week 4 we discussed entry barriers. One strategic entry barrier is bundling: when
a firm offers two of its products as a bundle at a lower price relative to the prices for
the same products bought separately. This strategy is often used when the firm is (close
to) a monopolist in one market, but its market share on the other product’s market is
threatened. In this exercise we learn through an example how this strategy indeed can
deter entry.
Imagine there is a manufacturer that sells plain paper and colored paper by the box. Let’s
call this firm “firm P” (P for paper). The plain paper is $10/box and the colored one is
$30/box. Its customers are retailers that buy both plain and colored paper. Currently P
is a monopolist in both markets, but faces potential entrants in the plain paper market.
The current prices are very profitable for the manufacturer, the margin it makes on each
box is $5 for plain and $15 for colored paper. Firm P sells 1M boxes of each type, so total
profits are $20M. It faces the threat of entry on the plain paper market.
1. Which price should the monopolist change if it wants to deter entry? How should
it change the price? Why?
2. The monopolist isn’t quite sure yet what to do (it has not yet hired any economists),
so it decides to wait and see what happens if new entrants enter the market. After
entry the price of the plain paper drops to $7.5/box. Firm P hires an economist.
The economist does some research and determines that the $7.5/box price is just
enough to cover the costs of the typical entrant in the long run (all entrants have
the same cost structure). It also calculates that firm P’s new profit is $15.5M. P
lost part of its original revenue/box on the plain paper, and also it now sells fewer
boxes as some retailers now buy from the new entrants.
P now comes up with the following strategy: it offers a bundle of a box of plain and
a box of colored paper for $37. Customers can continue to buy the two products
separately from P, at unchanged prices if they wish ($10 for the plain, $30 for the
colored).
Suppose now you are a retailer (all retailers in the market are identical, so they will
make the same decision). Remember, retailers resell both products, so you need to
have them both in your store. You have options to buy colored from P at $30, plain
from a new entrant at $7.5, plain from P at $10, or a box of plain+a box of colored
(a bundle) from P at $37. What do you do?
3. What do the entrants do now?
4. What’s the profit of the typical entrant now? (All entrants have identical cost
structure.)
5. How much is the profit of P now? Compare this to the profit before entry and also
to the profit that materalized when entry happened (recall the economist calculated
profits of P were $15.5M then). (Assume for simplicity that the total number of
boxes demanded from both the plain and the colored paper has been the same
during this whole time period, irrespective of the price changes, so 1M box of plain
and 1M box of colored.)
6. Under this setup and calculations it seems that to deter entry, the monopolist is
better off introducing the bundle at $37 even before entry happens, in fact to deter
entry.