Solutions to Sample Questions 3

1. Please state the Second Welfare Theorem, making clear what conditions

are required for it to hold.

The Second Welfare Theorem states that every Pareto efficient allocation can be sustained as a competitive equilibrium, given appropriate lump sum redistribution and sufficient convexity. [Please make sure you know the formal definition of a “competitive equilibrium.”]

1a. How does the Second Welfare Theorem relate to the First Welfare

Theorem?

The First Welfare Theorem states that every competitive equilibrium is Pareto-efficient. Under the right conditions, the Second Welfare Theorem shows that the reverse is also true.

2. Explain clearly how the Second Welfare Theorem is relevant to the issue

of decentralization.

The Second Welfare Theorem gives conditions under which a given Pareto-efficient allocation can be decentralized – that is, provided using the price mechanism by the private market, featuring optimizing individuals who take prices as given.

3. Explain clearly why lump-sum transfers may be needed to decentralize a

given allocation in the economy. Here, you should refer to a two-person

Edgeworth Box diagram, and explain precisely how a given allocation can be decentralized, under the right conditions.

[We saw the relevant diagram in class.] Start at a point on the contract curve – the point we wish to decentralize. If indifference curves are convex, then we can draw a line through the point that is tangent to these indifference curves, whose slope depends on the price ratio of the two goods. As long as the initial endowments of the two agents do not lie on this line but are instead given by another point in the box, then the government must adjust endowments to move individuals onto this line.

4. Why might being able to decentralize an allocation be important? Here,

you should think about the relative merits of central planning versus

decentralization.

If an allocation can be decentralized, this shows that the market mechanism could be used to provide that allocation of resources. A number of economists have argued that the market mechanism will generally be superior to the (extreme) alternative of central planning, as markets are able to make better use of local information, in particular, information about individual preferences.

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.