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MODEL ANSWERS
QUESTION 1 [20 marks]
A consumer has the utility function where x and y are the two goods.
(a) Find the marginal rate of substitution between the two goods when x = 25 and y=200. [5 marks]
Solution:
See Besanko and Braeutigam p.83
Along an indifference curve with given U, the slope is given by
When x=25 and y=200, the MRSxy = -(200 ÷ 25) = -8.
(b) Suppose the consumer has a budget of $2,700 to spend, and that the price of good x is $90 and the price of good y is $15. What is the optimal consumption bundle of x and y? [5 marks]
Solution:
See Besanko and Braeutigam pp.132-133.
The Lagrangian is:
First-order conditions:
Whence
(c) What is the demand curve equation for x if the price of y is fixed at $1 and the budget remains $2,700? [5 marks]
Solution:
Optimality requires
Substituting into the budget constraint gives the demand curve:
(d) Holding the price of y at Py=1, use a linear approximation to the demand curve to calculate roughly the change in consumer surplus when the price of x falls from Px=10 to Px=6.
Solution:
Diagrammatically, we need the sum of areas A and B in the diagram:
A = $4 x 135 = $540
B = 0.5 x 4 x 90 = $180
A + B = $720 which is roughly the consumer surplus.
QUESTION 2 [20 marks]
(a) Using clearly-labelled diagrams, show how the labour supply curve is derived from the labour/leisure tradeoff faced by a representative worker.
[5 marks]
SOLUTION
See Besanko and Braeutigam pp.172-175. The student is required to draw some variant of Figure 5.24 or Figure 5.26. A backward-bending segment is not necessary but credit goes to students who show it and relate it to the income and substitution effects.
x
Px
10
6True demand curve, x=1350 ÷ Px
Linear approximation
135 225
A B
(b) Suppose a tax on wage income is imposed, and that as a result labour supply falls. Does this mean that the substitution effect is more powerful than the income effect, or the other way round? Explain your answer carefully.
[5 marks]
From the worker’s point of view the tax is a reduction in the wage rate. This relates directly to slide 67 in my Lecture 6, which showed the case where the tax increased labour supply (see also Besanko and Braeutigam p.175 Figure 5.26):
To get the opposite result asked for here– a fall in labour supply – the substitution effect must dominate over the income effect (J is the “decomposition bundle”):
Careful and thorough labelling of the diagram is to be rewarded along with quality of explanation.
B
U1
U2
24
BLwith tax on wages
BLwithout tax
A
Dollars of wagesW
Hours of leisureL
A tax on wages flattens the slope of BL, inducing both income and substitution effects
In this case, the net effect is an increase in labour supply A => B:
income effect outweighs substitution
B
U1
U2
24
BLwith tax on wages
BLwithout tax
A
Dollars of wagesW
Hours of leisureL
A tax on wages flattens the slope of BL, inducing both income and substitution effects
In this case, the net effect is a fall in labour supply (increase in
leisure) A => B: substitution effect JB outweighs income effect AJ
Substitution effect
Income effect
J
(c) Returning to the initial situation, suppose all workers are charged a lump-sum tax payable in cash. The amount of the tax is the same regardless of hours worked. Could this tax have the effect of reducing labour supply? Explain your answer, using an appropriate diagram. [5 marks]
SOLUTION:
There are two possible diagrams the student might use. One is the case of a tax on non-wage income, as set out in my Lecture 6 slide 65:
The point here is that no change in the relative price of labour and leisure has occurred, so there is no substitution effect. The effect of falling income is always negative for leisure and positive for labour, given that leisure is a normal good. So labour supply rises from (24-L1) to (24-L3) (ignore L2, which is labour supply if non-wage income is zero).An alternative diagram from Lecture 6 slide 61 showing the effect of a poll tax imposed on a peasant population with no access to cash income except from wages:
A
Hours of leisureL
U2
U3
24
BLwages
Dollars
U1
BLwith the tax
}Non-wage income
L1L2 L3
B
C
BLwith non-wage income
Taxing non-wage income brings the consumer back down to
BLwith the tax
Optimum is now at B
Hours in village activities
Money wageincomeper day
U1 U2
24
BL1
U3
U4
A
Solution: The colonial Government imposed a poll tax that had to be paid in cash, and prohibited villagers from selling their agricultural products for cash
Poll taxB
VB
This produced a new corner solution at B with villagers obliged to work enough time to earn sufficient cash to pay the tax and with village activity cut back to VB
The optimal solution at B is a corner, not a tangency. In the absence of the tax no labour would be supplied and 24 hours of leisure would be taken.
Thus whichever analysis is used, the lump-sum tax cannot have the effect of reducing labour supply.
QUESTION 3 [20 marks]
(a) If the price of labour increases by 35% while all other input prices remain the same, would the long-run total cost of producing a given output rise by more than 35%, less than 35%, or 35% exactly? Explain your answer carefully, using a clearly labelled diagram or diagrams. Assume that the underlying production function is Cobb-Douglas.
SOLUTION:
The production function has smooth convex isoquants which ensures that substitution of other inputs for labour is possible in the long run. As the relative price of labour has increased, such substitution will reduce the share of labour in the total input mix, and this will bring the total cost increase down below 35%. The question is looking for students’ understanding of the intuition of factor substitution. But there is also another step in the argument that cuts in the same direction.
Only one isoquant is needed for quantity of output , and the argument can be made with just two inputs, K and L. The optimal input bundle with the first set of factor prices is at point A where isocost line C0C0 is tangent to the isoquant. Raising the price of labour causes the isocost lines to steepen and the optimal input bundle for changes to at point B, where the isoquant is tangent to isocost line C1C1.
.
C1
C1
C0
C0
K0
K1
L0L1
A
L
K
B
If no substitution had taken place when the price of labour rise, production would have taken place at A but with isocost line JJ. By inspection, B is a lower-cost combination at the new price ratio. Therefore if the cost of the original bundle at A rises by less than 35%, then so too must the new bundle at B cost less than 35 more than the original total cost.
.
Before the increase the total cost at A was
After the increase, total cost at A is
The cost of the bundle at A has increased as the isocost line rotates from C0C0 to JJ by
So total cost of producing at A has risen less than 35%, from which it follows that the input bundle at B costs, at the new prices, less than 135% the cost of the bundle at A at the original prices.
There will be lots of ways for students to demonstrate this result – marks for elegance, conciseness, completeness…
B
C1
C1
C0
C0
K0
K1
L0L1
A
L
K
J
J
(b)Would a perfectly competitive firm continue to produce if price fell below its minimum level of short-run average variable cost? Clearly explain your answer.
[5 marks]
SOLUTION:
This is the definite shut-down price even for a firm with all its fixed costs sunk; see Besanko and Braeutigam p. 308 Figure 9.2. There is no way the firm can do better continuing to produce than by shutting down in this situation.
The answer will need to show good understanding of the principles behind the shut-down price. If the firm cannot recover all its avoidable costs (variable costs plus non-sunk fixed costs) it is better-off to shut down and avoid them.
With all fixed costs sunk, the minimum of AVC sets the shutdown price.
(c) Would a perfectly competitive firm continue to produce if price fell below the minimum of its short-run average cost? Clearly explain your answer.
[5 marks]
SOLUTION:
See Besanko and Braeutigam p. 31 Figure 9.3. The answer to this question depends on the extent to which fixed costs are non-sunk and hence avoidable. If all fixed costs are non-sunk, then the firm will shut down at any price below SAC. If all fixed costs are non-sunk, then the shutdown point is reached only when price falls to the minimum of AVC as in (b) above. With some fixed costs sunk and some non-sunk, the shutdown price will lie somewhere between AVC and SAC – students should reproduce the diagram from Figure 9.3 and the accompanying analysis.
(d) Suppose firm i’s long-run production function is and that in the short run K is fixed at 900. How much labour will be demanded in the short run at a wage of 10?
[5 marks]
SOLUTION:
The cost-minimising firm will hire labour to equate its marginal product with the wage.
The trap is that students may be tempted to look for the long-run optimal combinations of K and L, which would give the long-run demand for labour. Besanko
and Braeutigam pp.248-251 show (Figure 7.15) the short-run expansion path contrasted with the long-run one, and in their Learning-by-Doing Exercise 7.6 part (c) show how to get labour demand with only one of the inputs variable if output is known. In this case output is not known but the wage is.
Here, the short-run production function is
Marginal product of labour is
Labour demanded is found by setting marginal product equal to the wage:
QUESTION 4 [20 marks]
(a) Suppose that the supply curve for an individual firm is
There are 1000 identical firms in the industry.
There are two groups of consumers in the market. The representative consumer of type A has demand and the representative consumer of type B has demand .
There are 40,000 Type A consumers and 10,000 Type B consumers.
Find the market equilibrium price and quantity.
[15 marks]
SOLUTION:
The industry supply curve for is
The choke price for Type A consumers is and their aggregated demand curve is
The choke price for Type B consumers is P = 20 and their demand curve is
The market demand curve is therefore
[up to 8 marks for correct derivation including identification of the choke price and the “kink” at P=20, if the final answer has not been successfully found]
To see which segment of the demand curve will be relevant, check for excess demand or supply when P=20:
At this price there is excess supply, so equilibrium will be on the lower segment of the demand curve.
[3 marks for using the correct supply and demand curves for the calculation, if final answer is wrong]
Solving
The price is above the industry shutdown price of P=5, so this is the answer.
[keep 1 mark in reserve for this last point, ensuring the price is above shutdown]
(b) Explain, using a clearly-labelled diagram and accompanying text, how a negative network externality can lead to the observed demand curve having a different slope from the demand curve that would be obtained simply from aggregating individual consumers’ constrained-optimisation-based demands.
SOLUTION:
See Besanko and Braeutigam p.172 Figure 5.23. The negative-externality case gives a steeper demand curve at industry level than the curve that would result from aggregation of individual demands.
QUESTION 5 [20 marks]
Consider the simultaneous-move game below.
Player 2Left Center Right
Top -1, 3 3, -1 5, -2Player 1 Middle 3, -1 -1, 3 5, -2
Down -2, 5 0, 6 1000, 4
(a) Find the pure-strategy Nash equilibria of this game (if any). [5 marks]
(b) Apply iterated deletion of strictly dominated strategies to obtain a 2x2 payoff matrix. Explain your work. [5 marks]
(c) Consider the 2x2 payoff matrix obtained in part (b). Find the mixed-strategy Nash equilibrium of that game. [5 marks]
(d) Now we return to the payoff matrix from part (a). Suppose that player 1 moves first. Draw and label the game tree. What is the subgame-perfect equilibrium outcome? [5 marks]
Answer:
a. Let us mark the best responses in the payoff matrix.Player 2
Left Center RightTop -1, 3* *3, -1 5, -2
Player 1 Middle *3, -1 -1, 3* 5, -2
Down -2, 5 0, 6* *1000, 4There is no pure strategy profile such that each player’s strategy is a best response to the opponent’s strategy. Thus, there is no pure-strategy Nash equilibrium.
b. First, notice that for player 2 strategy Center dominates strategy Right. Thus, player 2 will never play Right and we can delete it from the payoff matrix:
Player 2Left Center
Top -1, 3* *3, -1Player 1 Middle *3, -1 -1, 3*
Down -2, 5* 0, 6*Next, consider player 1. We see that strategy Down is strictly dominated by strategy Top. So we can delete Down. We will end up with the following payoff matrix:
Player 2Left Center
Top -1, 3* *3, -1Player 1 Middle *3, -1 -1, 3*
c. Suppose that player 2 chooses Left with probability p and Center with probability 1-p. Player 1’s expected payoff from choosing Top will be -1*p + 3*(1-p). Player 1’s expected payoff from choosing Middle will be 3*p + (-1)*(1-p). Player 1 will be willing to randomize between Top and Middle only if his expected payoffs are the same. Thus, p must satisfy –p + 3(1-p) = 3p –(1-p). Solving for p yields p = ½. A symmetric argument applies to the other player. To recap, the mixed strategy equilibrium involves player 1 choosing Top with probability ½ and player 2 choosing Left with probability ½.
d. The game tree is shown below (the top numbers are player 1’s payoffs, the bottom numbers are player 2’s payoffs).
To find the subgame perfect Nash equilibrium, we use backward induction. Player 1 knows that if he chooses Top, player 2 will choose Left (as 3 is bigger than -1 and -2), so player 1 will get a payoff of -1; if player 1 chooses Middle, player 2 will choose Center (as 3 is bigger than -1 and -2), so player 1 will get -1; if player 1 chooses Down, player 2 will choose Center (as 6 is bigger than 5 and 4), so player 1 will get 0. Thus, the subgame-perfect equilibrium involves player 1 choosing Down and player 2 choosing Center. Player 1 gets 0 and player 2 gets 6.
QUESTION 6 [20 marks]
A monopolist serves a market with an aggregate demand function given by Q = 36 – 3P. The monopolist’s cost function is given by C(Q) = 2Q.
(a) What is the monopolist’s profit maximizing output? What price will he set? [5 marks]
(b) How much profit can the monopolist generate with first-degree price discrimination?
[5 marks]
(c) Suppose that the monopolist can partition his market into two separate submarkets. The demand function for submarket 1 is given by Q1 = 20 – 2P1 and the demand function for submarket 2 is given by Q2 = 16 – P2. What prices would this monopolist set if he practices third degree price discrimination? [5 marks]
(d) Now assume that the monopolist can charge different prices in the two markets. However, he is required by law to sell identical quantities in these markets, i.e. Q1 = Q2. What quantities will he choose to sell? At what prices? [5 marks]
Player 1
Player 2 Player 2 Player 2
Top Middle Down
Left Center Right Left Center Right Left Center Right
-1 3
3 -1
5 -2
3 -1
-1 3
5 -2
-2 5
0 6
1000 4
Answer:
a. The monopolist’s inverse demand is P = 12 – Q/3. Thus, the marginal revenue is given by MR = 12 – 2Q/3. The marginal cost is 2. The profit-maximizing quantity satisfies 12 – 2Q/3 = 2. Solving for Q gives us Q = 15. The profit maximizing price is P = 12 – 15/3 = 7.
b. See the graph below
Net social surplus is equal to the red area: NSS = 10*30/2 = 150. If the monopolist can implement first-degree price discrimination, he can extract the entire net social surplus as profits. So the monopolist can attain a profit of 150.
c. The inverse demands of the two markets are P1 = 10 - Q1/2 and P2 = 16 - Q2. Therefore, the marginal revenues are MR1 = 10 - Q1 and MR2 = 16 - 2Q2. Since the marginal cost is constant, in each market the monopolist will sell quantities that equalize marginal revenue and marginal cost: 2 = 10 - Q1 and 2 = 16 - 2Q2. Thus, Q1 = 8 and Q2 = 7. The corresponding profit maximizing prices are P1 = 10 - 8/2 = 6 and P2 = 16 – 7 = 9.
d. The monopolist’s joint profit is .
Since the monopolist is restricted to sell the same quantities in the two markets, we have . So the monopolist’s profit can be written as
. The optimal output choice will satisfy the
first-order condition . Thus, .
Substitute in the inverse demands to get and .
12
36
2
p
q
NSS
30
QUESTION 7 [20 marks]
An incumbent firm, Firm 1, faces a potential entrant, Firm 2, with a lower marginal cost. The market demand curve is . Firm 1 has a constant marginal cost of $20 and no fixed costs, while Firm 2 has a marginal cost of $10 and a fixed cost of $400.
(a) Suppose that if Firm 2 enters, the two firms will compete by simultaneously choosing quantities. What are the Cournot equilibrium quantities, price and profits if there is no government intervention? [5 marks]
(b) Assume that Firm 2 can credibly commit to produce a half of whatever output Firm 1 chooses to produce. What are the equilibrium output levels of the two firms? [5 marks]
(c) Suppose that no firm can commit. To block entry, the incumbent appeals to the government to require that the entrant incur extra costs. What happens to the Cournot equilibrium if the government causes the marginal cost of the second firm to rise to that of the first firm, i.e. $20? [5 marks]
(d) What is the lowest value of Firm 2’s marginal cost that will prevent the entry of that firm? Hint: this is the marginal cost which will set Firm 2’s Cournot equilibrium profit to zero. [5 marks]
Answer:
a. The profit of Firm 1 is . Given , Firm 1’s optimal choice of output satisfies the first-order condition
. The profit of Firm 2 is
. Its first-order condition is
. Solve this for to get . Substitute in
Firm 1’s first-order condition: . Solving for yields . Therefore, . To find the equilibrium price substitute these quantities in the inverse demand: . Firm 1’s equilibrium profit is . Firm 2’s equilibrium profit is
.
b. If Firm 2 can commit to produce , then Firm 1’s profit will be given by . Firm 1’s optimal output choice satisfies
the first-order condition . Solving for yields
. Since Firm 2 is committed to producing half of that, we have
.
c. If Firm 2’s marginal cost is $20, then the reaction functions of the two firms will be symmetric. So in equilibrium it must be true that . Substitute in Firm 1’s first-order condition from part (a) to get . The equilibrium price is . Firm will make a profit of 53.33*33.33 – 20*33.33 = 1100.889. Firm 2’s profit will be 1100.889 – 400 = 700.889.
d. Suppose that Firm 2’s marginal cost is c. Then Firm 1’s first-order condition will be , while Firm 2’s first-order condition will be
. Solving for will give us the Cournot equilibrium
output levels: . Thus, the equilibrium profit of Firm 2 is
. This expression simplifies to
. The lowest c which will prevent entry sets the equilibrium
profit of Firm 2 to zero: . Solving this equation for c yields
c = 40.
QUESTION 8 [20 marks]
Jane is a farmer who can plant either potatoes or corn on her land. Both choices are risky, and her profits will depend on the weather according to the following table:
Weather Potatoes CornCold 4000 1000
Normal 8000 8000Warm 6000 11000
There is a ¼ probability that the weather will be cold, a ½ probability that the weather will be normal, and a ¼ probability that the weather will be warm.
(a) Calculate Jane’s expected profits for each crop. [6 marks]
(b) If Jane’s utility over income is and she has $2000 from other sources (so her total income is crop profits plus $2000), which crop would she rather plant?
[7 marks]
(c) Now suppose that Jane’s utility is . She can plant a fraction α of her land with potatoes and a fraction 1- α with corn. That is, if α = 0.6, then Jane would get 60% of the potato profits and 40% of the corn profits for
whatever type of weather occurred. She has no additional income. What α will
Jane choose? Note: .
[7 marks]
Answer:
a. Jane’s expected profit from planting potatoes is 0.25*4000 + 0.5*8000 + 0.25*6000 = 6500. Jane’s expected profit from planting corn is 0.25*1000 + 0.5*8000 + 0.25*11000 = 7000.
b. If Jane plants potatoes, her expected utility will be . If Jane plants corn, her
expected utility will be . Thus, Jane will prefer to plant corn.
c. Jane’s expected utility will be
The optimal α satisfies the first-order condition
. Solving for α yields
.
