ECO336Y

Solutions to Sample Questions 4

This note provides sample solutions to the fourth set of sample questions, posted on the course website.

Please note that an answer earning full marks need not be very long, but it should indicate that you have a clear understanding of the relevant issue. (Needless-to-say, you should also write clearly, and in complete sentences.) On the term test, for questions where written answers are required, I will indicate how long an answer earning full marks would need to be. Unnecessarily long answers will be penalized.

1. Person A will trade two loaves of bread for five apples and remain equally happy. At the same time, Person B will gladly exchange three loaves of bread for ten apples. Is the allocation of loaves and apples between Persons A and B Pareto-efficient? Please justify your answer. [Two sentences.] (2 points)

It is clear that the two people have different tradeoff rates at the existing allocation of apples and bread. This means that gains can be made by reallocating goods, implying that the existing allocation cannot be Pareto-efficient.

[In more detail, it is straightforward to demonstrate that mutually beneficial trades can make both people better off. At the existing allocation, Person B values loaves more highly. That person would need 3.333 apples if she sacrificed 1 loaf in order to remain as well off, while Person A would need only 2.5 apples if he sacrificed one loaf. Thus mutually beneficial trade could work as follows: Person A gives one loaf to Person B and receives 2.5 apples in return, remaining as well off as before. But Person B would have been willing to give up 3.333 apples in order to get the extra loaf. Thus Person B gains 0.833 apples, and this gain can then be split between the two people, leading to mutual gain.

Essentially, the person who values loaves more highly would acquire more loaves, and that would lead to a Pareto-improvement.]

2. Explain why Monte-Carlo experiments are useful in determining the properties of estimators. [Aside: an estimator is a statistical 'recipe' that is applied to data in order to obtain estimates. Least squares is perhaps the most widely used estimator.] In answering this question, you should focus on the case of the least squares estimator, and specifically whether it gives unbiased estimates or not, depending on whether the error term is uncorrelated with the explanatory variables.

Monte-Carlo experiments are useful because they allow the researcher to determine whether the estimator (that is, the statistical recipe applied to the data at hand in order to calculate estimates) provides a reliable means of learning about the underlying parameter estimates. The reason such experiments are so useful is that the researcher can fix the 'true' parameter values before-hand. Random errors can then be generated that are uncorrelated with the explanatory variables, and the parameter values, explanatory variables and errors can be used to calculate the associated dependent variable, based on the statistical model being considered. (In class, the model we considered related house of work to a constant, and individual wage, and an error term.)

With the artificial data set in hand, least squares can then be applied to the data to obtain parameter estimates. The exercise can then be repeated, generating a new set of individual error terms and calculating a new set of implied dependent variables (eg. hours of work), and using the resulting data to calculate new parameter estimates, and so on. By averaging over all the resulting sets of parameter estimates, the researcher can then check to see whether the parameter estimates are, on average, equal to the true values - that is, the estimator is unbiased.

Knowing that one's estimator is unbiased is important when analyzing real, as opposed to artificial, data. One wants to know that the recipe, on average, will give the right answer. [Aside: In the case of least squares, it is possible to prove directly that the estimates are unbiased, under the right conditions. In other cases, a direct proof is not available, in which case Monte-Carlo simulations are especially useful.]

Question 3 (17 points)

Suppose a firm faces a horizontal demand curve, with price P = $4. The firm’s total cost curve is given by

C = Q²,

where Q is the firm’s output.

3a) Does the firm have any market power? Please explain. [One sentence.] (2 points)

As the firm is a price taker, it has no market power.

3b) Write down a general expression (in terms of Q) for this firm’s profit level. (3 points)

Profit = TR – TC = P Q – C(Q) = 4Q – Q²

3c) Solve for the firm’s optimal output choice, showing your workings. [Hint: your answer should give Q* equal to some number.] (6 points)

The firm’s optimal output Q* solves

d (Profit)/d Q = 0 where

d (Profit)/d Q = 4 – 2Q

Q* = 2

Aside: What are we doing here? We are finding the output level at which profits are maximized, which is equivalently the output level at which the profit function has a slope of zero.

A second aside: strictly, we should also check that the profit function is maximized, rather than minimized, at Q* = 2. We would do this by checking that the second derivative of the profit function is negative, evaluated at Q* = 2, which it is, being equal to –2 throughout.

3d) What is the firm’s marginal cost at the optimal output level? (2 points)

MC = d C(Q)/ d Q = 2Q.

If Q* = 2, then MC = 4.

3e) In the absence of externalities, does this firm produce at the socially optimal output level? Please explain. [Two sentences.] (4 points)

In the absence of externalities, the product price captures the full social benefits of an additional unit of output, and the marginal cost captures the full cost to society of an extra unit of output. Here, P=4 and MC=4 at Q*, so the firm’s optimum output level is also society’s optimum output level.

Question 4 (worth 12 points)

A number of social scientists have studied the impact of police expenditures on the level of crime to see whether higher expenditures lower the crime rate significantly. Here, we repeat the standard type of analysis, collecting data on a random sample of communities. Our data set includes, for each community i, the crime rate (C_i), the level of police expenditures (E_i), and the level of unemployment (U_i). Suppose that the unemployment rate takes on just two values: 0.1 in low-unemployment communities, and 0.2 in high-unemployment communities. We estimate the following model using ordinary least squares:

C_i = a₀ + a₁ U_i + a₂ E_i + epsilon_i

where a₀, a₁, and a₂ are our coefficient estimates, and epsilon_i is a random error term.

4a) Consider the least squares line that corresponds to these estimates, drawn with crime expenditures on the horizontal axis and the crime rate on the vertical axis. Is the intercept of this line in the low-unemployment community higher or lower than in the high-unemployment community? And does this make sense? Please explain. [Three sentences.] (6 points)

The intercept term is given by a₀ + a₁U_i, assuming that the error can be ignored – on average it is zero. As long as a₁>0, as seems plausible, so an increase in unemployment (in moving from low- to high-unemployment communities) leads to higher crime, as we would expect.

4b) Suppose our actual estimates are: a₀ = 5, a₁ = 2, and a₂ = -1. Predict the average crime rate in high-unemployment areas if police expenditures = 2 per year. Please set out your working. (6 points)

Given these estimates, the average crime rate is given by:

E(C_i|E_i = 2) = 5 + 2 X 0.2 – 2 = 3.4,

where the expression on the left reads “the expected value of the crime rate in community i, given that police expenditures in that community are equal to 2.” Here, we assume that the expected value of the error term equals zero, which is standard.

5a. Suppose a firm takes prices as given, and the price it faces for its product is $6. Let the firm's total cost function be given by C(Q) = Q^2, where Q is output (and ^2 denotes 'raising to the power 2').

i. Write down the firm's profit function.

Profit = TR – TC = 6Q – Q^2

ii. Solve for the firm's profit maximizing output level. What is it?

Q* solves d(profit)/dQ = 0 or 6 – 2Q = 0. This implies Q* = 3.

iii. At this output level, what is the firm's average revenue, marginal revenue, and marginal cost?

AR = P = 6, MR = 6, MC = dC/dQ = 2Q, and if Q = 3, then MC = 6 also.

5b. Now suppose that the firm has market power. By selling more output, the firm is forced to lower its price. Let the firm's demand curve be given by

P = 6 - Q.

The firm still faces the same cost function as in 5a.

i. Write down a general expression for the firm's total revenue.

TR = PQ = (6 – Q)Q

ii. Write down a general expression for the firm's profit.

Profit = TR – TC = (6 – Q)Q – Q^2 = 6Q – 2Q^2

iii. Solve for the firm's profit maximizing output level. What is it?

Q* solves d(profit)/dQ = 0, implying that 6 – 4Q = 0, so Q* = 3/2.

iv. At this output level, what is the firm's marginal cost and the firm's average revenue?

AR = P = 6 – 3/2 = 9/2.

MC = 2Q = 3 [and note that MR = 6 – 2Q = 3 also]

Clearly, MC < AR.

v. Is this level of output efficient (in the sense that the extra cost to society equals the extra benefit)? If not, please calculate the efficient level of output, explaining your reasoning carefully.

The efficient output level occurs where MC = AR, or 6 – Q = 2Q, implying Q* = 2. Thus the monopolist is under-producing.

vi. Intuitively, what is the problem with monopoly from society's point-of-view? [two sentences]

Monopoly leads to less production than is socially optimal. At the monopoly output level, the marginal benefit to society exceeds the marginal cost of an additional unit of output, so by increasing output, the total net benefit to society would increase.

vii. Why does monopoly create a potential role for government? [two sentences]

Monopoly is an instance of market failure that creates a potential role for government, either to regulate the monopoly directly, or to increase competition by promoting entry to the industry. Government intervention usually has costs associated with it, to it is important to consider whether government intervention will in fact improve matters.