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C 29 AG Advanced Economics 2

1) Firms 1 and 2 are participating in a quantity setting oligopoly, in which they face the common (linear) inverse demand function, p = a – b ( x 1 + x 2) where firms produce outputs x 1 and x 2 , selling them at the market price p . The parameters a and b are constant, but undefined. Their cost functions are also linear, with total cost C i = F i + c i x i , so that each firm faces fixed costs F i and constant marginal costs x i .

You may assume that F 1 < F 2 and that c 1 > c 2 .

Your task is to write a report in which you set out the outcome of competition between the firms in these three situations:

a) They make their production decisions simultaneously and without cooperating;

b) They make their production decisions simultaneously while seeking to maximise joint profit; and

c) They make their production decisions sequentially, with firm 1 setting output first.

In each case, you should state expressions for firm outputs, the market price, and firm profits. You should compare firm output and profits. In doing so, comment on the circumstances in which you consider that it would be reasonable to expect firm 2 to agree to cooperate with firm 1, whether by joint output setting, or by conceding market leadership.

(100 marks)

2) Consider the situation where F has an endowment, K , of capital, and G has an endowment, L , of labour. To create value from these endowments, they produce quantities of goods R and S , generating utility from consumption.

The factors of production are perfect complements in production. The production functions are:

R ( K R , L R ) = min(2 K R , L R ) and S ( K S , L S ) = min( K S , 2 L S ). The requirements of market clearing must be fulfilled in equilibrium.

The utility functions are U F = 2ln R F + ln S F , and U G = 2ln R G + ln S G .

Your task is to write a report in which you set out how F and G arrange production and consumption so that there is efficient use of factors in production, and the distribution of goods for consumption completes the Walrasian equilibrium.

You should be aware that since factor inputs are perfect complements, we do not find offer curves when analysing production. You should instead derive a market

clearing condition, from which you should be able to obtain the relative price of the factor inputs, and hence the optimal distribution of goods, and then the utility achievable by both consumers, noting how this varies with the ratio of capital: labour endowments.

(100 marks)

3) Begin by sketching a diagram showing in state space the endowment for a consumer ( W 0 , W 1), where W S is wealth in state S , with W 0 > W 1 . The probability

of state 0 is . The consumer derives utility from wealth W S , U ( W S ) = W S , where 0 < < 1.

Your task is to write a report in which you demonstrate that this consumer is risk averse. Then show that where the consumer is able to trade wealth in state 0 for increased wealth in state 1, expected wealth remains constant along a line whose slope is the odds ratio.

Then, by forming a Lagrangean function, or otherwise, show that if the consumer is offered actuarially fair insurance, she will purchase it. Hence, explain how it is possible for a population whose members face identically, independently distributed risks of incurring loss in state 1 to form a mutual insurance society, which offers full, fair insurance against loss. Explain why the society might be vulnerable to moral hazard among its members.

(100 marks)

Questi ons with answers in red

1) Firms 1 and 2 are participating in a quantity setting oligopoly, in which they face the common (linear) inverse demand function, p = a – b ( x 1 + x 2) where firms produce outputs x 1 and x 2 , selling them at the market price p . The parameters a and b are constant, but undefined. Their cost functions are also linear, with total cost C i = F i + c i x i , so that each firm faces fixed costs F i and constant marginal costs x i .

You may assume that F 1 < F 2 and that c 1 > c 2 .

Your task is to write a report in which you set out the outcome of competition between the firms in these three situations:

a. They make their production decisions simultaneously and without cooperating;

b. They make their production decisions simultaneously while seeking to maximise joint profit; and

c. They make their production decisions sequentially, with firm 1 setting output first.

In each case, you should state expressions for firm outputs, the market price, and firm profits. You should compare firm output and profits. In doing so, comment on the circumstances in which you consider that it would be reasonable to expect firm 2 to agree to cooperate with firm 1, whether by joint output setting, or by conceding market leadership.

(100 marks)

Students should realise that they are being asked to explore the model of competition in quantities , in which firms have a degree of monopoly power. The novel element of this question is the introduction of fixed costs, and good answers will note that with firm 2 having higher fixed costs, but lower marginal costs , when firm output is high enough, firm 2 has lower average costs.

Answers should demonstrate that :

- in Cournot competition, firm 2’s lower marginal cost means that it produces more than firm 1, but that it may make a smaller profit than firm 1 because of its higher fixed costs .

- when forming a cartel , firm 2’ s lower marginal cost effectively means that it pays firm 1 to cease production, while meeting its fixed costs

- in Stackelberg competition, firm 1 chooses its output so that firm 2 chooses the ou tput on its reaction function where firm 1 maximises its profits . Firm 1 can now produce more than firm 2, and make greater profits.

Good answers will consider the advantages which firm 2 possesses, arguing that while firm 1 might benefit initially from incumbency, that as market demand increases , firm 2 has the capacity to serve the market at lower average cost than firm 1. A cartel arrangement seems unlikely to survive given firm 2’s cost

adva ntages ( and good answers may relate that to the challenge of firm 1 imposing punishment on firm 2) , and if we think of fixed costs as existing in the short run only , we expect to see firm 1 having to adapt to the new competition, or leave the market .

.

2) Consider the situation where F has an endowment, K , of capital, and G has an endowment, L , of labour. To create value from these endowments, they produce quantities of goods R and S , generating utility from consumption.

The factors of production are perfect complements in production. The production functions are:

R ( K R , L R ) = min(2 K R , L R ) and S ( K S , L S ) = min( K S , 2 L S ). The requirements of market clearing must be fulfilled in equilibrium.

The utility functions are U F = 2ln R F + ln S F , and U G = 2ln R G + ln S G .

Your task is to write a report in which you set out how F and G arrange production and consumption so that there is efficient use of factors in production, and the distribution of goods for consumption completes the Walrasian equilibrium.

You should be aware that since factor inputs are perfect complements, we do not find offer curves when analysing production. You should instead derive a market clearing condition, from which you should be able to obtain the relative price of the factor inputs, and hence the optimal distribution of goods, and then the utility achievable by both consumers, noting how this varies with the ratio of capital: labour endowments.

(100 marks)

Students should recognise the Leontief production functions , and that for market clearing in the factor markets, the output expansion paths for goods R and S intersect , giving the relative factor prices and the marginal costs of both goods. Good answers will explain that with these considerations, the production possibility frontier will consist of two linear segments, with a corner where there is full employment of resources .

Taking the ratio of marg inal costs to be the marginal rate of transformation , answers should note that this is not well defined at the corner, and so will concentrate on obtaining the constrained maximum in exchange for F and G, given outputs and factor incomes . Given the form of the utility functions , 2/3 of expenditure will be on good R and 1/3 on good S

Good answers will obtain expressions for relative factor prices from the full empl oyment condition , and hence will obtain expressions individual income , obtaining the Walrasian income as the intersection of offer curves.

3) Begin by sketching a diagram showing in state space the endowment for a consumer ( W 0 , W 1), where W S is wealth in state S , with W 0 > W 1 . The probability

of state 0 is . The consumer derives utility from wealth W S , U ( W S ) = W S , where 0 < < 1.

Your task is to write a report in which you demonstrate that this consumer is risk averse. Then show that where the consumer is able to trade wealth in state 0 for increased wealth in state 1, expected wealth remains constant along a line whose slope is the odds ratio.

Then, by forming a Lagrangean function, or otherwise, show that if the consumer is offered actuarially fair insurance, she will purchase it and be fully insured. Hence, explain how it is possible for a population whose members face identically, independently distributed risks of incurring loss in state 1 to form a mutual insurance society, which offers full, fair insurance against loss. Explain why the society might be vulnerable to moral hazard among its members.

(100 marks)

Students should be able to draw a diagram in which each point shows state conting ent wealth . It is sufficient to differentiate the wealth function twice to co nfirm concavity , and then infer risk aversion . The constant expected wealth line can b e obtained by taking the differential of expected wealth , and setting the value of the differential to zero.

The Lagrangean is f ormed by adding the expected utility function to the product of the multiplier and the constant expected wealth line, written so that the realised expected wealth is no greater than the endowment . Standard first order conditions confirm that the optimum is found where wealth is equal across states, so that the consumer bears no risk, and is fully ensured, with the insurer making zero expected profit .

Risk a verse consumers can form a mutual insurance society assuming that risks are iid because within a population, there is no risk, but a proportion of people who suffer loss ( a fuller explanation might note that variance tends to decrease inversely with the number of members of the society, but that requires a capacity t o meet losses through borrowing .)

Moral hazard can be explained as the result of people making less effort to avoid a bad state of the world when insured, so not facing the full costs of loss. With insurance , we show this as leading to an increase in the probability of loss, so that full, fair insurance, based on the ex ante probability of loss, will lead to certain income which is greater than expected wealth based on ex post probabilities , and so losses for the society .