ECO336Y

Solutions to Sample Questions 5

 

Here are my solutions.  Let me know if anything is unclear.

 

 

Total Points = 47

 

1)  If property rights are not defined over a resource such as a river, explain why this may give rise to externality problems. (2 points)

 

Ownership of a resource helps ensure that the owner takes account of the opportunity costs of using the resource in different ways.  If property rights are not defined over a resource, then individuals may use the resource in ways that have undesirable consequences for others, and there is no clear mechanism for ensuring that the external costs and benefits felt by others are take into account adequately.

 

2)  If markets fail, is government intervention always justified?  Please set out your reasoning.  [Two sentences.]  (4 points)

 

If markets fail (because of externalities or market power), this provides a reason for considering government intervention as a means of correcting the failure.  However, government intervention is often costly and ineffective, and in some cases, the costs outweigh the benefits of the government’s actions, so that government intervention is not always justified.

 

3)         [Again] Please explain why an indifference curve drawn in leisure-income space (that is, with leisure on the horizontal and income on the vertical axis) slopes down. [Two sentences.] (4 points)

 

An indifference curve in the leisure-income space shows all points that give an individual the same utility level. We usually think of more work (and thus less leisure) as bad, and more income as good; so if we start from a point on a given indifference curve where an individual has a lot of leisure and then move to a new point where the individual has less leisure, he/she must be compensated through higher income, so the slope will be negative.

 

Question 4 (8 points)

 

Consider a firm that pollutes the environment in the process of producing a good.  Let Q be the output of the good.  The firm’s and society’s marginal benefit schedule is given by

 

MB = 7 – Q.

 

Suppose the firm’s private marginal cost schedule is given by

 

MPC = 3 + Q/2,

 

and the marginal social damage schedule is given by

 

MSD = 4.5.

 

4a)  What level of output will the firm choose to produce if it ignores the pollution problem?  (2 points)

 

The firm will produce at the output level where

 

MPC = MB, or

 

3 + Q/2 = 7 - Q.

 

Solving for Q, we get that

 

Q* = 8/3.

 

4b)  What is the socially optimal level of output?  (3 points)

 

The socially optimal level is found where MSC = MB.  Here,

 

MSC = MPC + MSD.

 

Therefore,

 

MSC = 7.5 + Q/2,

 

which is an increasing function of Q.  Note that the MSC curve is everywhere about the MB curve.  Thus the optimal output level is zero.

 

4c)  Please give clear intuition for your answer to part b).  (3 points)

 

In this case, the external damage associated with producing the good is so high that once these damages are added into the private cost, the full social costs outweigh the benefits at all levels of output.  So the socially optimal output level is zero.

 

 

Question 5 (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.

 

5a)       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.

 

5b)       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

 

5c)       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.

 

5d)       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.

 

5e)       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 6 (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 (Ci), the level of police expenditures (Ei), and the level of unemployment (Ui).  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:

 

Ci = a0 + a1 Ui + a2 Ei + epsiloni

 

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

 

6a)       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 + a1Ui, 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.

 

6b)       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(Ci|Ei = 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.