INDUSTRIAL ORGANIZATION AND FIRM STRATEGY
ECON - 3742, section A01-Winter 2022
Assignment #1: due February 11
Individual or in groups of up to 2 students
*Submission in the dropbox in UM learn under Assignments: submit only under the name of one student but
be sure to include the name and ID student number of the other student. If you forget to include it then you
cannot include it after the due date.
True/False questions (20 points)
1. (4 points) In a principal-agent relationship between owner and manager with hidden e¤ort, the owner can
design a wage scheme that insures the optimal …rst best e¤ort by the manager regardless of the risk aversion
of the manager. Justify your answer.
2. (4 points) Consider a monopoly that faces an inverse demand curve and has a linear cost function.
The monopoly would be indi¤erent when maximizing pro…ts between either choosing quantities or choosing
prices.
3. (4 points) A multiproduct …rm that as monopoly power over several products sets lower prices than
separate …rms (each controlling a single product) when the products are substitutes or when there are
economies of scope.
4. (4 points) In the dominant …rm model (à la Hotelling) an increase in the marginal cost of the dominant
…rm (with constant marginal costs) implies that pro…ts necessarily decrease.
5. (4 points) Suppose that an industry has 10 …rms where the market shares are ordered from the most
to the least dominant …rm f0:5; 0:37; 0:05; 0:03; 0:02; 0:01; 0:01; 0:005; 0:004; 0:001g. The Her…ndahl index is
IH = 0:3.
Problems (80 points)
6. (20 points) Consider a monopolist facing a linear inverse demand curve p (q) = a bq, where q denotes
units of output and b > 0 represents the slope of the inverse demand curve. This …rm faces cost function
C (q) = F + cq, where F denotes its …xed costs (can contain sunk costs), c represents the monopolist’s
(constant) marginal cost of production and assume a > c 0.
a. (4 points) Find the monopolist’s pro…t maximizing output and label it q m . Verify if it is positive.
b. (3 points) What is the market price pm and the pro…t level m ? Is the pro…t level always positive? If not
what is the condition for the pro…t level to be positive? Explain
0
c. (2 points) Find the absolute value of the price elasticity of demand = q q(p) p evaluated at q m as de…ned
in the textbook. Is the elasticity greater than one? Find the markup on price or Lerner index, de…ned as
C 0 (q)
L (q) = p(q)p(q)
, at q m . What happens with L when increases? Is it true that "A pro…t maximizing
monopolist decreases its markup as demand becomes more price elastic? Explain.
d. (3 points) Find the socially optimal output level q which is the value that maximizes consumer surplus
(price equals marginal cost). Is it larger or smaller than the pro…t maximizing output, q m , that you found
in part (a)? Is it true that q = 2q m ? Explain.
e. (3 points) Illustrate q m and q in a graph where price is the vertical axis and q is the horizontal axis. Be
sure to include the MR and MC curves as well as the inverse demand curve.
1
f. (2 points) Let us say that a regulator wants to induce the monopolist to produce q instead of q m . Would
the monopolist have positive pro…ts at q ? Suppose F represents only sunk costs, would the monopolist
produce q in this case? Explain.
g. (3 points) Suppose F are only sunk costs and that the regulator of the monopoly establishes the price
the monopoly takes as given and then decides to produce the level that maximizes pro…ts. What is the price
level the regulator would need to establish for the monopoly to be induced to produce the social optimum
level q ? How would your conclusion change if F are …xed costs? Explain.
7. (15 points) Consider the same setup as in the previous exercise but now cost function is convex i.e.
C (q) = cq 2 .
m
m
a. (3 points) Find qconvex
, pm
convex and convex which denote the output produced, price and pro…ts for the
monopolist with convex cost function. Explain.
b. (2 points) Represent what you …nd in a graph like that of 1.d).
c. (5 point) Under what conditions does the monopolist with convex cost function produce less than the
monopolist with a linear cost function as in 1)? Justify your answer.
d. (5 points) Find the socially optimal output level qconvex . Is it larger or smaller than the social optimal
output when the monopolist faces a linear cost function? Explain.
8. (20 points) Two …rms competing à la Hotelling with …xed locations and symmetric constant
marginal costs. Suppose there is a line between 0 and 1 in which two …rms locate themselves, …rm 0 being
at point x0 = 0 and …rm 1 being at point x1 = 1. Each …rm has a linear cost function C (qi ) = cqi where
qi represents output level of …rm/product i = 0; 1, parameter is such that c > 0. Consumers are located
between 0 and 1 and are distributed uniformly according the the graph below.
1
Buy from 0
Buy from 1
x*
0
1
x
Consumers value the product …rm i = 0; 1 in ri > 0 and incur a disutility from travelling to the location of
the product which is assume to be a linear function of distance. The mass of consumers is normalized to 1 since
that is the area in the graph. Indirect utility function for consumers is then V (ri ; pi ; x) = ri
jxi xj pi
where pi denotes the price of product/…rm i = 0; 1 and x is the consumer (point) that is located in the line
between 0 and 1. Each consumer wants at most one unit of one of the products, the one that is closest
in distance. Note that the term jxi xj represents the distance between consumer x and the location xi
of …rm i = 0; 1. For example the consumer located at x = 0:5 has a distance to either …rm of 0.5 since
jx0 xj = j 0:5j = 0:5 and jx1 xj = j1 0:5j = 0:5. Assume > 31 jr1 r0 j > 0 where is the location
di¤erentiation of the products among the …rms.
The demand for each …rm depends on the existence of an "indi¤erent" consumer denoted x who is the
one point between 0 and 1 that achieves the same utility regardless of the product purchased. This means
that this indi¤erent x consumer satis…es
r0
jx0
x j
p0 = r1
jx1
x j
p1
or replacing x0 = 0 and x1 = 1 the indi¤erent consumer is determined to be after rearranging
x =
r0
r1 + p1
2
p0
+
1
2
Hence the demand for …rm 0 is the mass of consumers to the left of x (base of x times the height of one):
Q0 (p0 ; p1 ) = r0 r12+p1 p0 + 12 while the demand for …rm 1 is the mass of consumers to the right of x (base
of 1 x times the height of one): Q1 (p0 ; p1 ) = 12 r0 r12+p1 p0 . Note that if r0 = r1 and p0 = p1 then the
demand for each …rm is 0:5 which is equal to x as represented in the graph above. For di¤erent values of
r0 6= r1 and/or p0 6= p1 the indi¤erent consumer is not located at 0:5 but will be a point between 0 and 1.
2
a. (5 points) Find the best response price pi (pj ) …rm i = 0; 1 for any pj that …rm j = 0; 1 chooses (Hint:
maximize pro…ts choosing pi for …rm i = 0; 1 taking as given pj ).
b. (5 points) Graph the two best response functions in a space where the vertical axis is p1 and the horizontal
axis is p0 . Find the Nash equilibrium (p1 ; p2 ). Are the equilibrium prices above marginal cost when r1 = r0 ?
Are they increasing in ? If so, what does that mean? Explain.
c. (5 points) What happens to the equilibrium prices if r1 increases while r0 remains constant? Interpret
what you …nd.
d. (5 points) Find the pro…ts at equilibrium. Are they positive? What happens to pro…ts when increases?
Interpret what you …nd.
9. (10 points) Each of two …rms has one job opening. Suppose that (for reasons not discussed here but
relating to the value of …lling each opening) the …rms o¤er di¤erent wages: …rm i = 1; 2 o¤ers the wage wi .
Imagine two workers, each of whom can apply to only one …rm. The workers simultaneously decide whether
to apply to …rm 1 or to …rm 2. If only one worker applies to a given …rm, that worker gets the job; if both
workers apply to one …rm, the …rm hires one worker at random (probability 0.5 for hiring worker 1 and
probability 0.5 of hiring worker 2) and the other worker is unemployed (which has a payo¤ of zero). Suppose
workers are risk neutral i.e. u (w) = w.
a. (2 points) Construct the normal form game. (Hint: payo¤s in the normal form game are expected utility
payo¤s)
b. (2 point) Solve for the pure strategy set of Nash equilibria of the workers’ normal-form game when
0:5w1 < w2 < 2w1 . At a pure strategy Nash equilibrium would a worker secure a job? Explain
c. (3 points) Find the mixed strategy Nash equilibrium (p1 ; p2 ) and verify that pi 2 (0; 1) for i = 1; 2.
(Hint: …nd the expected value for worker i = 1; 2 for each A1 and A2 assuming the worker j chooses A1 with
probability pi and A2 with probability 1 pi and then equate these two expected values to …nd pi )
d. (3 point) Solve for the set of Nash equilibria of the workers’ normal-form game when 0:5w1 > w2 . At
equilibrium would a worker secure a job? Explain
10. (15 points) Consider a N consumers that have linear-quadratic utility function
ui (q0 ; q1 ; q2 ) = aq1 + aq2
1
bq12 + 2dq1 q2 + bq22 + q0
2
where q0 is the Hicksian composite commodity with price normalized to one. Assume b > jdj > 0 which
implies that products are di¤erentiated. Consumer i has a budget constraint yi = p1 q1 + p2 q2 + q0 where yi
is the real income of i = 1; ::; N .
a. (3 points) From the budget constrain isolate q0 as a function of the other variables and then plug it in
the utility function. Verify that the Hessian matrix associated with this utility function is negative de…nite
(Hint: …nd second order partial derivatives with respect to q1 and q2 and verify the conditions for the Hessian
to be negative de…nite)
b. (3 points) Find the …rst order conditions of utility maximization and the demand functions (q0 (p1 ; p2 ) ; q1 (p1 ; p2 ) ; q2 (p1 ; p2 ))
for consumer i.
N
X
c. (4 points) Find the aggregate demands Qj (p1 ; p2 ) =
qji (p1 ; p2 ) for j = 0; 1; 2. Which of these demand
i=1
functions do not have income e¤ects? Justify your answers.
d. (3 points) The consumer surplus for consumer i is de…ned as CS i (q0 ; q1 ; q2 ) = ui (q0 ; q1 ; q2 ) yi . Using
the de…nition of utility function, real income and the …rst order conditions of part b) determine explicitly
N
X
CS i (q0 ; q1 ; q2 ) for consumer i = 1; ::; N . Find the aggregate consumer surplus CS
CS i (q0 ; q1 ; q2 ). Is
i=1
the consumer surplus a¤ected by income e¤ects? Justify your answer.
e. (2 points) Determien if the following statement is true. "Given that the aggregate consumer surplus is
just a rescale of consumer surplus of an individual then it is without loss of generality as a modelling choice
to only consider a representative consumer in the analysis". Justify your answer.
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Bonus questions (10 points, not a mandatory exercise to submit) Consider a principal-agent model of
hidden e¤ort or moral hazard in which owner of a …rm does not observe the action by manager and has to
design a wage contract or compensation scheme that can only be based on observed pro…ts. Problem is that
expected pro…ts not only depend on the unobservable action of top management, but also on some unobservable random component. Assume the owner is risk neutral and only cares about maximizing expected pro…ts
(e) minus the payment w to the manager. Expected pro…ts are de…ned as (e) = p (e) + (1 p (e))
where > > 0 are constants and p (e) 2 (0; 1) denotes the probability of achieving high pro…t which
depends on the e¤ort e 2 [0; emax ] that the manager exerts such that p (0) = 0, p (emax ) = 1 and p0 (e) > 0,
p0 (0) > a , p0 (emax ) < a , p00 (e) < 0. Pro…t function captures the idea that the owner does not observe
if pro…ts come from high e¤ort from manager or not. Assume that the manager has the following utility
function
U (w; e) = u (w) ae
where u (w) is the utility of wage income w such that u0 (w) > 0 and u00 (w) 0 (if u00 (w) = 0 the manager
is risk neutral while if u00 (w) < 0 the manager is risk averse), a > 0 is the weight on the disutility of
e¤ort. Assume that the manager can obtain unemployment insurance that gives her 1 < w0 < w while being
unemployed in which case the e¤ort is zero.
a. (2 points) Find the …rst best optimal e¤ort e (as if the e¤ort would be observable) that the owner
would want the manager to take assuming that the manager is risk neutral i.e. u (w) = w. (Hint: maximize
B (e) = (e) w ( (e)) subject to u (w ( (e))) ae = u (w0 ))
b. (2 points) Represent the optimal e¤ort in a graph where the horizontal axis is e and the vertical axis is
increases? When a increases? Explain.
p0 (e ). What happens with the optimal e¤ort level when
(Hint: use the …rst order condition of optimal e¤ort found in part a.)
c. (2 points) Suppose the owner o¤ers a wage contract that takes the form w ( ) =
R where R is the rent
the manager would pay the owner out of gross pro…ts, independent of e¤ort provided. Find the optimal e¤ort
level that the manager would choose given any value of R. (Hint: maximize U (w ( (e)) ; e)
(e) R ae
). Would it be the same e¤ort level found above? Explain. Determine the value of R the owner would
o¤er the manager at the optimal e¤ort level under the assumption the manager is risk neutral (Hint: use
the participation constraint of the manager evaluated at the optimal e¤ort level e ).
d. (4 points) Consider again the owner wants to set up a wage scheme that takes the form w ( (e)) R
where R > 0 is independent of e. Find the optimal e¤ort e level that the manager would choose under
the assumption that the manager is risk averse i.e. u (w) = ln w. (Hint: maximize expected utility of the
manager i.e. EU (w ( (e)) ; e) = p (e) ln + (1 p (e)) ln
R ae) Represent the optimal e¤ort e in a
graph similar to the one used in part b above. Determine the value of R the owner would o¤er the manager
just so that manager accepts the job (Hint: use the participation constraint of the manager to …nd R given
that it binds at e ). Is it the case that e < e ? (Hint: use the …rst order conditions found for each of the
e¤ort levels to conclude) Explain. If = w0 would it be the case that R > R ? Explain
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