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ECO 204, 2011 Summer, Test 3 Solutions This test is copyright material and may not be used for commercial purposes without prior permission

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S. Ajaz Hussain, Dept. of Economics, University of Toronto

University of Toronto, Department of Economics, ECO 204 2011 Summer Ajaz Hussain

TEST 3 SOLUTIONS

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QUESTION 1 [40 POINTS]

Consider an economy in which all agents live for two periods: (agents are “young” in and

“old” in ). There is a single good in the economy (say “corn”), the base period is , and the

real interest rate is .

Consider an individual who is endowed with income in respectively (income is in

units of corn) and her preferences over consumption (in units of corn) in are represented by

the utility function:

Here and are consumption (in units of corn) in respectively, and are

parameters. The consumption set is *( ) +.

(1.1) [5 POINTS] Does this individual have monotone preferences over consumption in and ?

Show all calculations and state any assumptions.

Answer:

To check if she has monotone preferences let’s see if the marginal utilities of and (recall that

)

For all bundles ( ) > 0

For all bundles ( )> 0


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(1.2) [5 POINTS] Suggest a positive monotonic transformation of this individual’s utility function and use it

throughout this question. Show all calculations and state any assumptions.

Answer:

The agent has the utility function:

Adding to both sides yields another utility function representing the same preferences:


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(1.3) [5 POINTS] Write down the inter-temporal budget constraint and interpret the and intercepts

(let be the -axis variable). Show all calculations and state any assumptions.

Answer:

The lifetime inter temporal budget constraint is:

Lifetime Value of Consumption = Lifetime Value of Income

We can value over time in terms of present or future values. Let’s do the latter:

( ) ( )

If is on the y-axis then:

( ) ⏟

( )

The y-axis intercept is the Future Value Income . Dividing both sides by ( ) shows that the x axis

intercept is

.

The inter-temporal budget constraint is shown below:

E

Observe that the inter-temporal budget constraint always passes through the endowment income

bundle.


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(1.4) [10 POINTS] Solve this individual’s inter-temporal Utility Maximization Problem and express all

solutions to the UMP in terms of both Present Value Income and Future Value Income (for example,

express in terms of PV income, as well as FV income). You are expected to use the appropriate

inequality/equality constrained optimization method. Show all calculations and state any assumptions.

Answer:

The utility function is a log form of the Cobb-Douglas production function and from consumer theory we

know that the optimal choice must be in the interior of the consumption set, i.e. and we can

therefore dispense with the non-negativity constraints .

The UMP is:

( ) ( )

( )

Form the Lagrangian:

, ( ) -

The FOCs are (the circled numbers indicate the order in which the FOCs are solved):

( )

❶

( )

❷

, ( ) -

❸ ( )

Setting ❶ = ❷ shows that at the optimal solution the indifference curves are tangent to the budget

line:


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( )

( )

The left side is the | | and the ride side is the absolute value of the budget line slope. From this

isolate any variable (say) :

( )

Substitute in the budget constrain and solve for

( )

( )

( )

(

)

( )

That is, consumption tomorrow is a constant fraction of future value income. To express in terms of

present value income, note that:

so that:

( )

Now:

( )


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( )

( )

That is, consumption today is a constant fraction of present value income. To express in terms of future

value income, note that:

so that:

Finally, we can solve for . From above, take any FOC, for example:

❷

This can be expressed in terms of by noting that:

so that:

( )

In summary, the solutions in present and future value terms are:


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Present Value Terms Future Value Terms

( )

( )


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(1.5) [5 POINTS] True or false: regardless of the real interest rate , the marginal utility of Future Value

Income is always positive. Show all calculations and state any assumptions.

Answer:

By the envelope theorem the marginal utility of a small change in is:

, ( ) -

This will be always positive (i.e. an increase in the real interest rates always makes the agent happier) so

long as :

( )

( )

This condition is always true whenever .


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(1.6) [5 POINTS] Under what conditions will higher real interest rates make this individual “happier”?

Show all calculations and state any assumptions.

Answer:

By the envelope theorem the marginal utility of a small change in is:

, ( ) -

, ( ) ( ) -

, -

,

-

Assuming we know that in which higher interest rates make the agent happier whenever

– that is, whenever the agent is a saver when she’s young. When does that happen?

[

]

( )

That is, when the | | through the endowment point is less than the budget line slope: when the

indifference curve at the endowment point is flatter than the budget line.


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(1.7) [5 POINTS] Suppose that ; i.e. the individual has the same endowment income in

each period. Given that the individual has the same income in each period, does this mean she’ll

consume the same amount of corn in each period? Show all calculations and state any assumptions.

Answer:

Suppose . In that case:

[

]

[

]

[

]

( )

[

] ( )

[

] ( )

Does this mean that ? Let’s check:

[

]

[

] ( )

( )

( )

If this condition holds for then the agent will engage in consumption smoothing.

The interpretation of this condition is that if there is a bundle on the 45 degree line (where:

) such that the indifference curve is tangent to the budget line, then the agent will engage in

consumption smoothing (in fact can check that this will be true even if ).


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Ajax Inc. wants to raise capital by issuing stocks through an IPO (“initial public offering”). The board of

directors must decide whether to issue stock either at a price of $100 per share or at $50 per share.

There is uncertainty about the number of shares that will be sold at each price. Suppose the board of

directors believes that:

If the stock is issued at $100 per share then the market will buy either 2m shares with

probability or 1m shares with probability ( )

If the stock is issued at $50 per share then the market will buy either 3m shares with probability

or 2.5m shares with probability ( )

(2.1) [5 POINTS] Graph the decision tree for this decision making under uncertainty problem.

Answer:

Decision?

IPO $100/share Nature

2m shares

$200m

1m shars

$100m

IPO

$50/shareNature

3m shares

$150m

2.5m shares

$125m


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(2.2) [5 POINTS] Suppose

. If Ajax is to issue shares at a price of $100 a share, what must the

probability of selling 2m shares be? Assume Ajax is risk neutral. Show all calculations and state any

assumptions.

Answer:

With

the decision tree becomes:

Decision?


2m shares

$200m

1m shars

$100m

IPO

$50/shareNature

3m shares

$150m

2.5m shares

$125m

If Ajax is risk neutral and issued shares at $100/shares, then it must be that:

, - , -

( ) ( )

(

) (

)


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That is, as long the chance of selling 2m shares at a price of $100/share is at least 30%, Ajax will shares

at $100/share.


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. If Ajax is to issue shares at a price of $100 a share, what must the

probability of selling 2m shares be? Assume Ajax is risk averse and has the utility function √

(where is millions of dollars). Show all calculations and state any assumptions.

Answer:

With

the decision tree becomes:

Decision?


2m shares

U($200m)

1m shares

U($100m)

IPO

$50/shareNature

3m shares

U($150m)

2.5m shares

U($125m)

If Ajax is risk averse and issued shares at $100/shares, then it must be that:

, - , -

( ) ( ) ( ) ( ) ( ) ( )

√ ( )√ (

)√ (

)√

√ ( ) (

)√ (

)√

√ (

)√ (

)√


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. /√ .

/√

√

Now that Ajax is risk averse, for him to issue shares at $100/share, the probability of selling 2m shares

at a price of $100/share must be at least 33.6484% (higher that the 30% probability when Ajax was risk

neutral).


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and = # calculated in part (2.3). Assuming Ajax is risk averse and has the

utility function √ (where is millions of dollars) what is the certainty equivalence to the

uncertainty of issuing stocks at a price of $100 per share? Show all calculations and state any

assumptions.

Answer:

Assume . The decision tree becomes:

Decision?


2m shares

U($200m)

1m shares

U($100m)

IPO

$50/shareNature

3m shares

U($150m)

2.5m shares

U($125m)

By definition, the certainty equivalence to any uncertainty is:

( )

Thus, for the $100/share IPO:

( ) ( ) ( ) ( )

√ √ ( )√

√


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How does one interpret this? Ajax gets the same utility from issuing shares at a guaranteed price of

$130/share versus the uncertainty of issuing stocks at $100/share (the uncertainty stems from

uncertainty about how many shares will be sold; at the CE price of $130/share, Ajax will surely sell 33%

of 2m shares plus 66% of 1m shares).


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and = # calculated in part (2.3). Assuming Ajax is risk averse and has the

utility function √ (where is millions of dollars) what is the certainty equivalence to the

uncertainty of issuing stocks at a price of $50 per share? Show all calculations and state any

assumptions.

Answer:

Assume . The decision tree becomes:

Decision?


2m shares

U($200m)

1m shares

U($100m)

IPO

$50/shareNature

3m shares

U($150m)

2.5m shares

U($125m)

By definition, the certainty equivalence to any uncertainty is:

( )

Thus, for the $50/share IPO:

( )

( )

( )

√

√

√

√


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Notice the for $50/share IPO is the same as the for $100/share because we calculated the

fore $100/share at the probabilities for which Ajax was indifferent between the $50/share

and $100/share IPO.

How does one interpret this? Ajax gets the same utility from issuing shares at a guaranteed price of

$130/share versus the uncertainty of issuing stocks at $50/share (the uncertainty stems from

uncertainty about how many shares will be sold; at the CE price of $130/share, Ajax will surely sell 20%

of 3m shares plus 80% of 2.5m shares).


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You own a house currently valued at 1m. You have a habit of throwing wild parties by inviting

econ faculty who may cause damage (“loss”) to your house. You estimate the potential damage (the

“loss”) to your house at . Let the probability of your house being damaged = .

(3.1) [5 POINTS] Suppose you don’t purchase insurance. What is the value of your house in each state of

uncertainty (damage, no damage), as well as the expected value of your house? Show all calculations

and state all assumptions.

Answer:

Assuming you don’t buy insurance:

No Insurance

No Loss State Loss State Expected Value ( ) ( ) ( )( ) ( )

Notice that value of the house depends on the state.


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(3.2) [5 POINTS] What is actuarially fair insurance? Provide one “proof” for actuarially fair insurance.

Show all calculations and state all assumptions.

Answer:

Actuarially fair insurance means that the price per dollar of insurance equals the probability of loss. For

example, if the probability of loss is 5% then the insurance company will charge you $0.05 in insurance

premium for every dollar of insurance. That is, for an insurance policy of $1,000 the (total) insurance

premium will be 0.05(1,000) = $50.

There are two ways to show that at the very least insurance companies will charge actuarially fair

insurance rates. Let price per dollar of insurance, amount of the insurance policy. Then:

Proof #1: Assume insurance industry is competitive so that on this insurance policy

Expected profit = 0

( )(Insurance Premium) (Insurance Premium – Payout) =0

( ) ( )

Proof #2: Assume insurance industry is regulated and required to maintain sufficient balances to cover

expected payout

Insurance Premium Collected = Expected Payout

( )( ) ( )


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(3.3) [5 POINTS] Suppose you purchase an actuarially fair insurance policy in the amount of (i.e. if the

house is damaged, the insurance company will pay you ). What is the value of your house in each state

of uncertainty (damage, no damage), as well as the expected value of your house? Show all calculations

and state all assumptions. Do not assume that you have purchased a full insurance policy.

Answer:

Assuming you buy an insurance policy:

Insurance

No Loss State Loss State Expected Value ( )( ) ( ) ( )( ) ( )

Notice that value of the house depends on the state.


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(3.4) [5 POINTS] Suppose you purchase an actuarially fair full insurance policy for (i.e. if the house is

damaged, the insurance company will pay you ). What is the value of your house in each state of

uncertainty (damage, no damage), as well as the expected value of your house? Show all calculations

and state all assumptions.

Answer:

Assuming you buy a full insurance policy ( ):

Full Insurance:

No Loss State Loss State Expected Value

( )( ) ( )

( )

Notice that value of the house is the same in each state (so that the expected value too, regardless of

the probability of loss, equals the value of the house in either state).


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(3.5) [5 POINTS] Compare the value of your house in each state of uncertainty (damage, no damage), and

the expected value of the house from parts (3.1) and (3.4). What do you notice? Why would someone

purchase a full insurance policy? Show all calculations and state all assumptions.

Answer:

Observe:

No Insurance

No Loss State Loss State Expected Value ( ) ( ) ( ) ( )( )

Full Insurance:

No Loss State Loss State Expected Value

( ) ( ) ( )

Purchasing full insurance guarantees that no matter whether your house burns down or not, it’s value

will be always or the “No loss Value” of the house minus the insurance premium. That said, not

purchasing insurance means that your expected (not actual) value of the house will As a risk

averse person your utility from having for sure is greater than the utility of having an expected

value of You buy insurance because you want to avoid uncertainty.

Incidentally, only full insurance guarantees the same value of the house in either state. The following

table shows the value in each state and expected value for an insurance policy that doesn’t necessarily

cover full loss:

Insurance

No Loss State Loss State Expected Value ( )( ) ( ) ( )( ) ( )


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(3.6) [5 POINTS] Suppose the probability that your house is damaged by party animals econ faculty is

What is the insurance premium of a full insurance policy? Show all calculations and state all

assumptions.

Answer:

Let price per dollar of insurance. The insurance premium for a full insurance policy is:

( )

Under actuarially fair insurance so that the insurance premium for a full insurance policy is:

( ) ( )


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(3.7) [5 POINTS] Suppose the probability that your house is damaged by party animal econ faculty is

What is the maximum insurance premium you’re willing to pay to insure against damage to

your house? Assume √ where is millions of dollars.Show all calculations and state all

assumptions.

Answer:

The maximum amount you’ll pay for insurance is:

( )

S

S

F

F ( )

( )

( )

Max WTP

What is the most you’ll pay for insurance?

Premium Max Premium

The expected utility of no insurance is:

, - ( ) ( ) ( )

, - ( ) ( )

, - √ √

, - √ √

We want:

( ) 0.983666


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√ 0.983666

0.967599

Thus, the most you’d pay for an actuarially fair full insurance policy is


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