QUESTION
SCHOOL OF ECONOMICS
ECONOMICS 2101
MICROECONOMICS 2
S1 2012
ASSIGNMENT 4
1 2 3 Total
This assignment is to be handed in no later than Friday June 1
st
at 16:00 at the School of
Economics Assignment Box #2 (West Lobby, ASB ground floor)
Important: Please add (and sign) the cover sheet to your assignment. Not doing so will
result in a penalty of 5 points. The cover sheet can be downloaded from:
http://www.asb.unsw.edu.au/schools/economics/Documents/Economics%20Assignment%20Cover%20Sheet.pdf
Exercise 1 [12 points]: Consider a competitive industry with a large number of firms, all of
which have identical cost functions
(
)
for and
(
)
(That is, a firm’s
cost of producing zero units of output is zero). Suppose the demand curve for this industry is
(
)
.
a) What is the supply curve of an individual firm in the short run? If there are firms in the
industry, what will be the industry supply curve in the short run? (3 points)
b) What is the smallest price at which the product can be sold? (3 points)
c) What will be the equilibrium number of firms in the industry in the long run? (2 points)
d) What will be the equilibrium output of the industry in the long run? (1 point)
Suppose that the demand curve for the industry becomes
(
)
e) What will be the new equilibrium number of firms in the industry in the long run, and how
much profit will each firm make? (3 points)
1
Exercise 2 [8 points]: A monopolists has an inverse demand curve given by
(
)
and a cost curve given by
(
)
.
a) What is its profit-maximising level of output?
b) Suppose the government decides to put a tax on this monopolist so that for each unit it sells
it has to pay the government $2. Determine its level output under this form of taxation?
c) Suppose now that instead of a quantity tax, the government puts a lump sum tax of $10 on
the profits of the monopolist. Determine its level of output?
d) Would your answer to c) change if the government increases its profits tax to $20? Explain.
Exercise 3 [10 points]: Assume an industry in which there are only two firms 1 and 2. The
marginal cost of production of each firm is $6. The inverse demand function is given by
(
)
, where
with
and
standing for the quantities produced by
each firm.
a) You are told that the firms are “Cournot competitors”. What does it mean?
b) Write down the profit maximisation problem of firm 1 and firm 2.
c) Determine the best response function of each firm.
d) Compute the Cournot equilibrium quantities and profits for each firm.
Now assume that firm 2 is a “Stackelberg leader”.
e) Explain what is a “Stackelberg leader”.
f) Determine the equilibrium quantities and profits for each firm and compare these with your
answers in d).
Now assume that firm 1 and firm 2 collude by maximising the sum of their profits. (Hint: Read
Section 27.10 of the text).
g) Determine the equilibrium quantities of output of each firm, the sum of their profits and
compare these with your answers in d).
ECON2101 Assignment 2 S1 2012 2
SOLUTION
Exercise 1
a) The cost function of the firm is given by c(y) = y2/4 + 16 for all values of y>0. Thus it can be said that in the short run the variable cost is y2/4 and 16 is the fixed cost.
In the short run the supply will be based on the value by equating marginal revenue with the marginal cost and the profit is greater than the average variable cost.
In this case the average variable cost will be AVC/y or y2/4y. Thus the average variable cost is y/4.
The marginal cost is given by differentiating the cost curve thus we get marginal Cost (MC) as y/2. Thus the average variable cost is minimized at y=0 thus the marginal cost curve is the short run supply curve. This is the equation of straight line which is sloping upwards. Since the cost equation of each firm is same the supply curve of each firm will also be same which will be y/2. If there are n firms in the market the industry supply curve will be ny/2.
b) As given in the cost function, there is one part which is the fixed cost and the other part is the variable cost. The average variable cost as derived above is y/4. The average fixed cost thus is 16/y. Thus the long run average cost is 16/y + y/4. This is minimized by deriving it to the first order which will be -16/y2 +1/4. Equating it to zero we get y = 8. The price thus will be by putting y=8 in the long run average cost equation. Thus we get equilibrium price as
16/64 + 8/4 = 9/4
c) The minimum output for each firm as calculated above is 8. The aggregate demand is given by the function 867 – 2p. Putting the value of p as 9/4 as calculated above we get the aggregate demand as 867 – 9/2 = 862.5. Since the supply by each firm is 8 the total number of firms is given by 862.5/8 = 107.8 =108 approx.
d) The equilibrium output of the industry has been calculated in part c) as shown above as 862.5
e) Since the new demand curve is given by 850 –2p. The new aggregate demand of the industry will be 850 – 9/2 = 845.5. Since there is no change in the cost function the supply from each firm will remain same and thus supply from each firm will remain same. Thus the new number of firms will be 845.5/8 = 105.7 = 106 approx. Thus it can be said that as the demand decrease the firms tend to exit from the market.
Exercise 2
a) The demand curve gives the price in the monopoly market as 12 – y. Thus if the y units is demanded the total revenue will be y(12 – y).
Thus the marginal revenue will be the first differential of the total revenue which will be
12 – y – y = 12 – 2y.
The marginal cost will be the first differential of the cost curve which is y2. Thus the marginal cost will be 2y.
In monopoly the marginal revenue is equal to marginal cost thus we get
12 – 2y = 2y or 4y = 12 or y = 3. Thus at equilibrium price will be 12 – 3 = 9. Thus the profit is
9 x 3 – 32 = 24
b) If the $2 per unit tax is introduced this will change the cost function to y2 + 2y. Thus we get the marginal cost as 2y + 2. Thus the profit maximization will be given by
12 – 2y = 2y + 2 or 4y = 10 or y = 2.5. Thus the level of output will decrease.
c) In case lump sum tax of $10 is introduced, the cost function will be y2 – 10. The marginal cost will be 2y and thus the value of y will be 3. This level is same as that is calculated in part a).
d) In case lump sum tax of $20 is introduced, the cost function will be y2 – 20. The marginal cost will be 2y and thus the value of y will be 3. This level is same as that is calculated in part a). Thus it can be said that the level of output will remain same as that in part c) but the profit will decrease as the tax will increase (Stanford University). Thus in other words the profit will be maximum in case the output remains same however the amount of profit that is made in both cases will be different.
Exercise 3
a) Cournot Competitors: It is used to describe the industry structure where there is competition in the market on the output. Thus it can be said that there is no price war but the competition is based on the volume that can be fulfilled in the market and thus the output produced by the firms. Another important feature is that there in product differentiation (Gaudet & Salan, 1990).. This implies that the products that are produced in the market are homogenous (Lipsey & Chrystal, 2007). In this type of market structure the number of firms is fixed and contrary to perfect competition is bound to remain fixed and not increase.
b) The inverse demand function is 36 – 4Q = 36 – 4 (q1 + q2). Thus the total revenue of firm 1 will be (36 – 4(q1 + q2))q1 and the total cost is TC1(q1) where TC1(q1) is the total cost function of the firm 1.
Similarly, the total revenue of firm 2 will be (36 – 4(q1 + q2))q2 and the total cost is TC2(q2) where TC2(q2) is the total cost function of the firm 1.
The profit maximization for the two firms will be the first differentiation of the profit equation of the two firms that has been shown above.
Thus the profit maximization of firm 1 will be
36- q2 -8q1 – MC(q1). The marginal cost is $6. Thus the profit maximization for firm1 will be
36- q2 -8q1 – 6 = 30 – 8q1 – q2. Similarly the profit maximization of firm 2 will be 30 – 8q2 – q1.
c) The best response function is obtained from the profit maximization of the two firms. The best response function is given by equating the profit maximization to zero.
Thus for firm 1 we get q1 = (30-q2)/8 or b1(q1) = (30-q2)/8 and similarly the best response function of firm 2 is b2(q2) = (30-q1)/8
d) The equilibrium is by substituting the value of q1 in the other equation. This has been done as shown below.
q1 = (30-q2)/8 and q2 = (30-q1)/8
Solving the two equations we get
q1 = (210 + q1)/64
Or q1 = 210/63
Thus q1 = 3.33 and after substituting q1 we get q2 = 3.33
e) Stackelberg leader refers to the situation in the market where the followers in the market moves behind the leaders in the market and thus make strategic move in the market as made by the market leader (Demiguel & Xu, 2009). Generally the followers provide competition to the stackelberg leader or the market leader in case they identify the advantage they have over the leader. Stackelberg leader is the subset of Cournot Equilibrium.
f) In this case considering that firm 1 is the market leader and thus considering the demand function of firm 2 we get P = 36 – 4 (q1 + q2). Thus the total revenue is [36 – 4 (q1 + q2)]q2
Thus the marginal revenue is 36 –q1 -8q2. Equating marginal revenue to which is $6 we get
q2 = (30 – q1)/8. Similarly the demand for the firm 1 will be P= 36 – 4 (q1 + q2). Substituting the value of q2 we get P = 36 -4q1 – 4q2 Thus we get P = 42 – 7q1.
Thus the marginal revenue is 42 – 8q1.
Equating this with the marginal cost which is $6 we get 8q1 = 36 or q1 = 4.5.
Thus q2 will be 3.1875.
g) In case of colluding the total output will be 4.5 + 3.1875 = 7.6875. Thus this will be the total output in the above situation and thus the profit will be maximized even if shared which is better than the cournot profit. This can be said as the for the firm 1 which will be the market leader will have the demand function of firm 2 we get P = 36 – 4 (q1 + q2). Thus the total revenue is [36 – 4 (q1 + q2)]q2
Thus the marginal revenue is 36 –q1 -8q2. Equating marginal revenue to which is $6 we get
q2 = (30 – q1)/8. Similarly the demand for the firm 1 will be P= 36 – 4 (q1 + q2). Substituting the value of q2 we get P = 36 -4q1 – 4q2 Thus we get P = 42 – 7q1.
Thus the marginal revenue is 42 – 8q1.
Equating this with the marginal cost which is $6 we get 8q1 = 36 or q1 = 4.5.
Thus q2 will be 3.1875
This is same as that of part f) the difference being that the profit will be shared in proportion of the quantity sold. Since the profit of both the firms will be different it will mean that the profit will be adjusted and both the firms although will not be competing will have a leverage in the market and thus will be beneficial for both. This can be said as the mix of both Cornout and Sackelberg.
References:
Gaudet G. & Salant S.W. (1990). Uniqueness of Cornout Equillibrium. Available at: http://pareto.uab.cat/xmg/Docencia/IO-en/IOReadings/CournotEq/%20GaudetSalant.pdf
Lipsey R.G. & Chrystal K.A. (2007). Cournot equilibrium (Oxford University Press). Available at
http://www.oup.com/uk/orc/bin/9780199286416/01student/interactive/lipsey_extra_ch09/page_03.htm
Stanford University. (n.d.).The Basics of Monopolies. Available at: http://www.stanford.edu/~sandersn/1A/Monopoly_Handout.pdf
Demiguel V. & Xu H. (2009). A Stochastic Multiple-Leader Stackelberg Model: Analysis, Computation, and Application
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