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You must answer ONE question from this section, using a separate document for

each question (the weighting is noted against the question/sub question).

Section A

Question 1

There are two types of workers, H (high skilled) types and L (low skilled) types. Workers’

utility function is:

??

?? ??

?? ?? for L types

??

?? ??

?? ?? for H types

where ?? ?? is the compensation schedule posted by firms, ?? ?? is the cost of acquiring an

education of e years for an H type, ?? ?? is the cost of acquiring an education of e years for

an L type. Suppose that ?? ??

0.5?? 2 and ?? ??

0.75?? 2 . Suppose that the value of the

marginal product of an H type is 6?? and the value of the marginal product of an L type is 3??.

Suppose that 1/3 of the workers are H types, but an employer cannot directly distinguish an H

type from an L type.

Find a separating equilibrium and derive the equilibrium compensation schedule posted by

firms at this equilibrium. Compare the equilibrium investments in education made by both

types with respect to the levels of investment that they would make in the case of full

information.

(50%)

Continued Overleaf/

3

Question 2

A firm has two partners, and each has the utility function ?? ??, ??

????, where ?? is the amount

of leisure consumed and ?? is the per capita amount of income. The income generated by the

firm depends on the amount of effort ?? supplied by each partner. Each partner s time

endowment is ?? 24 (which is the maximum amount of leisure consumption) and thus ??

?? ??. Let ?? 2 be the income generated per unit of effort when the two partners cooperate

in production. The partners share equally the income generated by their joint effort.

Derive the optimal values of ?? ?? , ?? ?? and ?? ?? when the workers cooperate to maximise

the income generated by their partnership and the effort supplied is observable. Let

????

?? ?? , ?? ?? denote the optimal bundle.

Derive the partnership equilibrium values of ?? , ?? and ?? when effort is not observable,

and each worker maximizes their own utility function. Let ??

?? ,??

be the

individual s choice at equilibrium under the partnership arrangement. Compare ??

?? , ?? with ????

?? ?? , ?? ?? by providing a diagrammatic discussion.

Consider the following grim trigger strategy:

Choose ?? ?? in the first period and in every subsequent period as long as the other

partner supplied ?? ?? in each previous period.

Choose ?? in period t and in every subsequent period if the other partner did not supply

?? in period t-1.

Suppose that the common discount factor is 0 ?? 1. Show that there is a Nash equilibrium

strategy of the infinitely repeated partnership game in which each partner adheres to the above

grim trigger strategy.

(50%)

Continued Overleaf/

4

You must answer ONE question from this section, using a separate document for

each question (the weighting is noted against the question/sub question).

Section A

Question 1

There are two types of workers, H (high skilled) types and L (low skilled) types. Workers’

utility function is:

??

?? ??

?? ?? for L types

??

?? ??

?? ?? for H types

where ?? ?? is the compensation schedule posted by firms, ?? ?? is the cost of acquiring an

education of e years for an H type, ?? ?? is the cost of acquiring an education of e years for

an L type. Suppose that ?? ??

0.5?? 2 and ?? ??

0.75?? 2 . Suppose that the value of the

marginal product of an H type is 6?? and the value of the marginal product of an L type is 3??.

Suppose that 1/3 of the workers are H types, but an employer cannot directly distinguish an H

type from an L type.

Find a separating equilibrium and derive the equilibrium compensation schedule posted by

firms at this equilibrium. Compare the equilibrium investments in education made by both

types with respect to the levels of investment that they would make in the case of full

information.

(50%)

Continued Overleaf/

3

Question 2

A firm has two partners, and each has the utility function ?? ??, ??

????, where ?? is the amount

of leisure consumed and ?? is the per capita amount of income. The income generated by the

firm depends on the amount of effort ?? supplied by each partner. Each partner s time

endowment is ?? 24 (which is the maximum amount of leisure consumption) and thus ??

?? ??. Let ?? 2 be the income generated per unit of effort when the two partners cooperate

in production. The partners share equally the income generated by their joint effort.

Derive the optimal values of ?? ?? , ?? ?? and ?? ?? when the workers cooperate to maximise

the income generated by their partnership and the effort supplied is observable. Let

????

?? ?? , ?? ?? denote the optimal bundle.

Derive the partnership equilibrium values of ?? , ?? and ?? when effort is not observable,

and each worker maximizes their own utility function. Let ??

?? ,??

be the

individual s choice at equilibrium under the partnership arrangement. Compare ??

?? , ?? with ????

?? ?? , ?? ?? by providing a diagrammatic discussion.

Consider the following grim trigger strategy:

Choose ?? ?? in the first period and in every subsequent period as long as the other

partner supplied ?? ?? in each previous period.

Choose ?? in period t and in every subsequent period if the other partner did not supply

?? in period t-1.

Suppose that the common discount factor is 0 ?? 1. Show that there is a Nash equilibrium

strategy of the infinitely repeated partnership game in which each partner adheres to the above

grim trigger strategy.

(50%)

Continued Overleaf/

4

x2

2

A risk neutral manager has utility function u(x, y) = 50x + y, where x is the amount of

leisure consumed and y is labour income. The manager is endowed with T = 24 of x and zero

units of y. The manager’s best alternative opportunity provides a level of utility of uy = 624. If

the manager supplies e units of efforts then the firm profit R will be 30e + }, where & is a

random variable with expected value zero. R is profit before deducting the manager’s pay.

Suppose that the owner of the firm offers the manager the compensation contract y = OR + F,

where 0 Sos 1 and F is a constant. Derive the manager’s effort supply function. Show that

effort increases when 0 increases.

Derive the optimal contract that maximizes the owner’s expected profit by employing

the manager’s effort supply function

What is the owner’s expected profit, the manager’s expected utility and the effort

supplied by the manager under the contract that maximises the owner’s expected

profit?

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Explanation & Answer:

300 Words

Tags:

utility function

Economic of Business

condition satisfied

Utilizing maximization

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