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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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