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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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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
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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S. Ajaz Hussain, Dept. of Economics, University of Toronto
(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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S. Ajaz Hussain, Dept. of Economics, University of Toronto
(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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Page 5 of 29
S. Ajaz Hussain, Dept. of Economics, University of Toronto
(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:
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
( )
( )
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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S. Ajaz Hussain, Dept. of Economics, University of Toronto
( )
( )
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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S. Ajaz Hussain, Dept. of Economics, University of Toronto
Present Value Terms Future Value Terms
( )
( )
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S. Ajaz Hussain, Dept. of Economics, University of Toronto
(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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Page 10 of 29
S. Ajaz Hussain, Dept. of Economics, University of Toronto
(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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S. Ajaz Hussain, Dept. of Economics, University of Toronto
(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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Page 12 of 29
S. Ajaz Hussain, Dept. of Economics, University of Toronto
QUESTION 2 [25 POINTS]
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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Page 13 of 29
S. Ajaz Hussain, Dept. of Economics, University of Toronto
(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?
IPO $100/share Nature
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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Page 14 of 29
S. Ajaz Hussain, Dept. of Economics, University of Toronto
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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S. Ajaz Hussain, Dept. of Economics, University of Toronto
(2.3) [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 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?
IPO $100/share Nature
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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S. Ajaz Hussain, Dept. of Economics, University of Toronto
. /√ .
/√
√
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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S. Ajaz Hussain, Dept. of Economics, University of Toronto
(2.4) [5 POINTS] Suppose
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?
IPO $100/share Nature
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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S. Ajaz Hussain, Dept. of Economics, University of Toronto
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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Page 19 of 29
S. Ajaz Hussain, Dept. of Economics, University of Toronto
(2.5) [5 POINTS] Suppose
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?
IPO $100/share Nature
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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S. Ajaz Hussain, Dept. of Economics, University of Toronto
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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Page 21 of 29
S. Ajaz Hussain, Dept. of Economics, University of Toronto
QUESTION 3 [35 POINTS]
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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S. Ajaz Hussain, Dept. of Economics, University of Toronto
(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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S. Ajaz Hussain, Dept. of Economics, University of Toronto
(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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S. Ajaz Hussain, Dept. of Economics, University of Toronto
(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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S. Ajaz Hussain, Dept. of Economics, University of Toronto
(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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S. Ajaz Hussain, Dept. of Economics, University of Toronto
(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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S. Ajaz Hussain, Dept. of Economics, University of Toronto
(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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S. Ajaz Hussain, Dept. of Economics, University of Toronto
√ 0.983666
0.967599
Thus, the most you’d pay for an actuarially fair full insurance policy is