Economics 2A (ECON2001)– 2021-2021
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
Economics 2A (ECON2001)– 2021-2021
December 2020 Degree Exam
QUESTION 1
A firm delivers groceries to people’s homes during COVID restrictions and operates as a price-taker. The firm’s total costs are given by () = 4 + 2 + , where A>0 is a positive constant and q is the number of deliveries.
• Find and draw on a chart (quantity q on the horizontal axis and costs on the vertical axis) the marginal cost, the average variable cost and the average total cost functions of this firm. Include a short explanation on how you derived them.
• Set up the profit maximization problem that the firm faces. What is the profit-maximizing level of output for the firm if the market price is equal to 40? What is/are the variable(s) that the firm can control here? What is the producer surplus for this firm at the profit- maximizing level of output? Derive it and show the producer surplus on a chart.
• If fixed costs were to decrease, how would the firm change its profit-maximizing level of output? Please explain briefly. How large can the fixed costs be for the firm to have non- negative profits? What would the firm do if fixed costs were higher than that?
Solution of first bullet point: The marginal cost can be obtained taking the first derivative of the total cost function: = 2 + 4. The average variable cost is obtained dividing the total variable cost by q: = + 4. Students should show this on a graph as well.
Solution of second bullet point: Profit maximization with respect to q: − () =
40 − 4 − 2 − . Setting the first order condition with respect to q to zero we get ∗ = 18. Since the firms is a price taker, the only control variable for the firm is its output q. The producer surplus is obtained by calculating the area above the marginal cost curve (see above) and below the equilibrium price. This is simply the area of a triangle, because the marginal cost curve is linear:
= = 324. Students should show this on a chart as well.
QUESTION 2
Fatima owns a computer, which is worth 512 pounds. Because she is using it so much for online learning, she feels that it is possible that the computer will crash. If it happens, she can only sell the
parts for 64 pounds. The utility that Fatima gets from using the computer is () = 3, with c being the value of the computer. There is a 10% chance that the computer crashes.
• Plot Fatima’s utility function. On the graph, indicate the utility she would obtain if the computer crashed, and if it didn’t crash. What is the expected value of the computer? What is Fatima’s expected utility? Show both on the same graph. What can we say about Fatima’s attitude towards risk?
• Imagine that Fatima can purchase an insurance that pays her 400 pounds if her computer crashes. The insurance cost is y pounds, which needs to be paid whether the computer crashes or not. Derive the monetary value (i.e. in pounds) that Fatima owns in both states of the world when the insurance is available, and its expected value. Is Fatima facing any risk if she buys this insurance policy?
• How can you find the highest level of y for which Fatima still gets this insurance policy? [you need to set up the problem, no need to solve it]. In your own words, what is one possible reason why in reality we rarely see insurance that fully compensates people for their losses?
Solution of first bullet point: The graph should show the utility function or at least the two points
mentioned above. The expected value (EV) of the computer is = 0.9 × 512 + 0. 1 × 64 = 467.2.
Fatima’s expected utility, on the other hand, is () = 0.9512 + 0. 164 = 0.9 × 8 + 0. 1 × 4 =
7.6. The utility function is concave (can motivate using the chart, or discuss second derivative),
implying that Fatima is risk averse.
Solution of second bullet point: Let us see what Fatima’s consumption bundle can be if she buys this
insurance. If the computer does not crash, Fatima has 512 (value of the computer) minus y (paid to
the insurance company). If it crashes, she has 64 (value of the computer) plus 400 (from the
insurance company) minus y (paid to the insurance company). Because 400+64<512, this is less than
full insurance: Fatima still faces some risk and will have higher ex-post utility if the computer does
not crash.
Solution of third bullet point: The critical value will be the one where utility without insurance will
be precisely as high as utility with insurance (which is a function of y). If y is any higher than that
critical value, Fatima will find it optimal not get insurance:
0.93512 − + 0. 1 3464 − = 7.6
Because Fatima is risk averse, she is willing to lower the expected value to lower the risk.
Full insurance is relatively rare in reality. One of the possible reasons is that the insurance companies may be worried that in the case of full insurance the individual who has taken up the insurance may behave recklessly, thus, increasing the risk of negative outcomes. This is referred to as moral hazard [The students do not need to mention this and may mention something slightly different from this].
QUESTION 3
The market for search engines has only two firms, which we call Goo (firm 1 for the notation) and Gle (firm 2). Goo has constant average cost (1) = 2. Gle has average cost equal to (2) = 2 . Demand for the good they produce as a function of the price is () = 5 − .
• First assume firms compete in quantity. Derive the reaction functions and how much each firm produces in a Cournot equilibrium. Draw the reaction functions on a graph, carefully labelling axis and curves. Calculate profits of each firm. For each step, briefly comment on the logic behind them.
• Now, calculate quantities produced, price and profits if Gle is the Stackelberg leader. In your own words, discuss the differences between profits of each firm in the Stackelberg equilibrium and in the Cournot-Nash equilibrium.
• If both firms agreed to form a Cartel to maximize joint profits, how much would each firm produce? Assuming that the firms are sure that the cartel is stable, would the firms agree to let another firm enter the market (and the cartel)? Why or why not?
Solution of first bullet point: The reaction function of Goo (firm 1) is found by setting the first order conditions with respect to 1 from profit maximisation to be equal to zero, and solving the resulting equation for 1 . After having substituted for the price p using the inverse demand function, the maximisation is the following: 1 (5 − 1 − 2)1 − 21, after having noted that constant average costs equal to 2 imply total costs equal to 21 . Multiplying through, taking the FOCs with respect to 1 and solving for 1 as a function of 2 we get 1 = . Doing the very same steps for firm Gle and noting that total costs of firm 2 are , we obtain the reaction function of firm 2, which is 2 = . Graphically, the Cournot equilibrium quantities are at the intersection point of the two reaction functions above. Arithmetically, to derive such quantities, we solve the system of equations given by the two reaction functions above. Substituting the second equation into the first we get
1 = , which we can then solve for 1 to get = 1 (c is for Cournot here). Substituting this back into the reaction function for 2 we then get = 1. Using the inverse demand function, we get the price, which is = 5 − 1 − 1 = 3. Profits of firm 1 are equal to 1, profits of firm 2 are equal to 2. Students should include comments for each step.
Solution of second bullet point: Being the Stackelberg leader means that firm 2 can decide how much to produce before firm 1 gets to make that decision. Firm 2 can take account of the way firm 1 will react to its decision. This means that firm 2 maximises its profits subject to firm 1’s reaction function. In practice, we solve this by substituting firm 1’s reaction function into firm 2’s profit maximisation problem, to get 2 5 − 2 − 2 − . Setting the first order conditions with respect to 2 to be equal to zero and solving for 2, we get that = ≈ 1. 17 (s for Stackelberg). Using firm 1’s reaction function, we then obtain = ≈ 0.92. Equilibrium price can be derived from the inverse demand function: = 5 − − ≈ 2.92. It is then straightforward to calculate the profits of firm 1 (which are equal to 0.85, or 0.84 depending on simplification, or 121/144 if reported as fraction) and firm 2 (equal to 49/42 or 2.04). Firm 2 benefits from being the Stackelberg leader in terms of profits, while firm 1 has lower profits. By having a first-mover advantage firm 2 is able to set a quantity that is higher than in Cournot equilibrium, knowing the best response of firm 1 will be to set a lower quality to prevent prices from falling too much.
Solution of third bullet point: Maximisation of joint profits: 1, 2 (5 − 1 − 2)(1 + 2) −
21 − . We can take first-order conditions with respect to quantity of each firm and then solve the
2-equation system. Result (students should include steps): 1 = ; 2 = 1. Possible to note that
quantities are lower, and price is higher than in the Cournot equilibrium. This is typical of cartels: agree to lower quantities/limit competition to keep prices high.
Perhaps counterintuitively, provided that the cartel is stable it is always going to be weakly optimal
to let firms join. The intuition is that a stable cartel behaves like a monopolist, so having access to an additional technology cannot make the monopolist worse off.
QUESTION 4
A market is characterised by a monopoly. The firm’s total cost function for producing X units of the
good is () = 1 2 . Assume that the monopolist faces the demand function A monopolist faces
market demand = 15 −
• On a graph, show the inverse demand, marginal revenue, marginal cost curves. Set up the profit maximization problem that the firm faces. Derive the profit maximizing price, quantity and the resulting profit of the monopolist, including all the intermediate steps. Show the equilibrium price and quantity on the graph. What is the socially optimal price? Derive and explain.
• Calculate the consumer, surplus, producer surplus and deadweight loss (DWL) from the monopoly and carefully show it on a chart. Briefly include intermediate steps, comment on them and discuss in your own words what is the reason why the monopoly produces an inefficiency.
• Imagine that the government tries to fix this inefficiency and aims to maximize the total surplus. What could the government do in this case? Please discuss things mentioning quantities from this specific exercise. In your own words, explain which challenges the government may face while implementing such policy.
Solution of first bullet point: Students should construct a chart and populate it with what is required
in the question. Setting up the monopolist’s profit maximisation problem:
1
(15 − ) − 2
We can then multiply through and derive the first-order condition, from which we get = 6 (M
for monopolist), = 15 − = 9. = − () = 9 ∗ 6 − 9 = 45.
Solution of second bullet point: We obtain the socially optimal price by setting the price equal to
the marginal cost and solve for quantity: = . After substituting in the inverse demand, we get
15 − = and thus ∗ = 10 and ∗ = 5. = (10−6)(9−3) = 12.
minimum quantity [Other ideas might be correct as well] There are probably many reasons why it is
hard to do this in reality, among which informational frictions certainly play an important role.
Resit exam – summer 2021
QUESTION 1
Mark owns a bar by the beach, which is valued at 400,000 pounds. The beach sometimes has strong winds, and Mark is worried that particularly strong winds could damage his business. The probability of that event is 30 percent. If that happens, the damage is 80,000 pounds. Mark can buy insurance to mitigate this risk: Mark can pay p pounds for every pound of insurance.
• Let denote the value of the bar in case of wind and denote the value of the bar if there is no wind. Write the equation for your state-contingent budget constraint (taking account that insurance is available), with alone on the left-hand side of the inequality. What is the slope of that budget constraint?
• Mark’s utility function is () = . What is Mark’s MRS? Briefly describe each step of your answer. Write down a condition that characterizes the optimal consumption bundle, i.e. a condition that you could use to solve for ( , ). Comment on how you will then solve for the optimal insurance policy.
• How much should Mark spend on insurance if = 0.3 (recall that p is the insurance premium for every pound of insurance)? In your own words, why are such insurance policies uncommon in reality? How much will he spend if = 0.4 ? Derive and briefly explain your findings.
Solution of first bullet point: Without insurance, = 400,000 and = 320,000. Thanks to the availability of an insurance policy, Mark and can decide to pay a premium to be given a transfer from the insurance company in case of damages. Given that the price of one pound of insurance is p, we can write = 400,000 − (where X is the amount you insure. If there is no damage, Mark enjoys the full value of the bar minus the amount paid to the insurance company) and = 320,000 + − (if there are damages, the insurance pays Mark the capital insured X, and still costs pX to buy). We can then solve for X in both equations, substitute and rearrange, to get = 400,000 + 320,000 − .
Solution of second bullet point: Given the utility function above, after some simplification, we get that the MRS is − . The solution will be such that the MRS is equal to the slope of the state-
contingent budget constraint, which means the optimal insurance policy will satisfy = 0.3
Solution of third bullet point: At p=0.3 the insurance policy is fair. Since Mark is risk averse, he will purchase full insurance. In reality, it is likely that asymmetric information and/or moral hazard are present and might limit the scope of full insurance.
QUESTION 2
The market for smartphones is characterised by the following inverse demand function () = 70 − , for firm Alpha and firm Beta, where the total quantity y is given by = ℎ + . The
technology is the same for both firms and total costs are ( ) = 10 + for = {ℎ, } . Derive step-by-step quantities produced by each firm, equilibrium price and profits under the following cases (you can approximate using two decimal places):
• Simultaneous competition in quantities (include a graph with the firms’ reaction functions).
• Firm Beta decides first. Briefly comment on how the results (quantity of each firm, total quantity, equilibrium price, profits of each firm) compare to the first case.
• Explain in your own words whether it is typically an advantage or a disadvantage to be the
first mover in this type of environments where firms have the same total cost function? Why is that the case?
2021-12-06