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Finance 30210
Practice Midterm #1 Solutions
1) Suppose that you have the opportunity to invest $50,000 in a new restaurant in
South Bend. (FYI: Dr. HG Parsa of Ohio State University has done a study that
shows that 59% of restaurants fail within the first three years!).
a) Given the following data, what is your opportunity cost here? Explain.
Asset Annual Return
5 year Government Bond 1.25%
DJIA (Stocks) 7%
“Junk” Bonds (CCC or below) 13%
Note: CCC bonds have an average default rate of 27%
The opportunity cost of the $50,000 investment would be the
returns that you could have earned elsewhere. However, the
“elsewhere” has to be an opportunity that is easily risky. In the
example, government bonds would not be an equally risky
alternative (government bonds are much safer). The Stock market
is also probably much safer given the 59% failure rate for
restaurants. So we should use the 15% per year junk bond rate.
Opportunity Cost = .013 * 50,000 = $650
b) Now, suppose that as a part owner, you are allowed to eat for free as
often as you like. How does this change your calculation from (a)?
Given that you can eat for free, we would need to deduct your savings
in food costs from the above number.
2) Suppose that Amtrak builds a new train line from Chicago to Los Angeles.
Unfortunately, the train line passes through thousands of acres of cornfields in
Iowa. When the train passes through the cornfields, it throws off sparks that
destroy the corn. The corn farmers take Amtrak to court in an attempt to get the
train line shut down.
a) What would be the “right” outcome in this case? Explain.
The “right” outcome is a little bit ambiguous. It depends on what we
mean by right. Let’s assume that the right outcome is the efficient
outcome. In that case, we need to figure out the profits of Amtrak and
costs to farmers. Let’s assume that Amtrak earns $50,000 per year in
profits from the train line. If the cost to the farmers is $20,000, then
the efficient thing to do would be to rule against the farmers and
award the property rights to Amtrak.
b) The Coase theorem states that as long as negotiation between the two
parties involved is relative costless, the “right” outcome will result
regardless of how the judge might rule. Explain.
Suppose that the judge rules for Amtrak. We have the efficient
outcome and no side payments are needed.
Amtrak Gains: $50,000
Farmer’s Loss: $20,000
Suppose that the judge rules for the Farmers. Then Amtrak will be
willing to make a payment to the Farmer’s (i.e. buy them off) to get the
land rights. Suppose that Amtrak pays the farmers $25,000 (we know
the payment will be somewhere between $50,000 and $20,000).
Amtrak’ Gain: $25,000
Farmer’s Gain: $5,000
3) Consider the following productivities:
United States England
Services 6 Units/hr. 3 Units/hr.
Manufacturing 2 Units/hr. 6 Units/hr.
a) Calculate the opportunity cost of services in the US and England
US: 33.6
2 Units of manufacturing per unit of services
England: 23
6 Units of manufacturing per unit of services
b) Calculate the opportunity cost of manufacturing in the US and England. Who
has the comparative advantage in services?
US: 32
6 Units of Services per unit of manufacturing
England: 5.6
3 Units of Services per unit of manufacturing
The US has a comparative advantage in services while England has a
comparative advantage in manufacturing.
c) Suppose that the average price of Services is $20 per unit and the average
price of manufacturing is $20. What trade pattern will emerge? What will
wages be in England and the US?
With a relative price of $20/$20 = 1 units of services per unit of
manufacturing, the US will produce and export services while England
produces and exports manufacturing.
In England, wages will be 6*$20 = $120/hr.
In US, wages will be 6*$20 = $120/hr.
d) Suppose that the inflation rate in England is 3% while the inflation rate in the
US is 5%. How is your answer in (c) affected
In England, wages and prices will rise by 3% per year while in the US, wages
and prices will rise by 5% per year, but relative prices are unaffected so
production and trade patterns do not change.
4) Suppose that you have the following demand and supply curve for sneakers:
PQ
PQ
s
d
2200
3400
a) Solve for the equilibrium price and quantity.
So, we need to set supply equal to demand and solve for an equilibrium
price.
400 3 200 2
200 5
40
P P
P
P
Note: as a check, we can plug 40 into both supply and demand to make
sure the quantities are equal.
400 3 400 3 40 280
200 2 200 2 40 280
d
s
Q P
Q P
b) Calculate consumer expenditures on sneakers
Consumer expenditures would be price times quantity…
$40*280 = $11,200
c) Calculate the elasticity of demand at the equilibrium found in (a)
% 403 .43
% 280
Q Q P
P P Q
So, at this price, a 1% rise in price will lower expenditures by
.43%
d) Would a 5% increase in price cause consumer expenditures to rise or fall?
Explain.
A 5% rise in price would be from $40 to 40(1.05) = $42. This will
cause a 5(.43) = 2.15% drop in quantity to 280(1-.0215) = 274.
$42*(274) = $11,508. So, spending increases.
In general, if the elasticity is smaller than 1 in absolute value, a
rise in price cases an increase in expenditures.
Price
Quantity
40
280
11,200
S
D
5) Suppose that you have the following demand curve:
IPQ 001.4120
You know that the current market price is $10 and average income is $40,000.
a) Calculate the markets consumer surplus.
First, at an income of $40,000, solve for the price where quantity equals
zero.
120 4 .001 40,000
160 4
0 160 4
40
Q P
Q P
P
P
Now, solve for the quantity at a price of $10
120 4 .001 40,000
160 4
160 4 10 120
Q P
Q P
Q
So, Consumer surplus is (1/2)(40-10)(120) = $1,800
b) Calculate the market’s total willingness to pay.
Actual consumer expenditures are $10*120 = $1200, so total willingness
to pay is $1200 + $1800 = $3000
Price
Quantity
40
120
D
CS
10
6) Suppose that you have the following demand curve.
IPQ 005.6400
Q Represents quantity demanded, P represents price and I represents average
income.
You know that the current market price is $20 and average income is $20,000
a) Calculate current demand.
At a price of $20, we have Q = 400 – 6($20) +.005($20,000) = 380
b) Calculate the price elasticity of demand.
206 .31
380
Q P
P Q
c) Calculate the income elasticity of demand
20,000.005 .26
380
Q I
I Q
7) Suppose that you are concerned about drug use in the US. You are interested in
what the impact would be if authorities could be more effective at getting drugs
off the streets. The DEA has estimated the following data:
Elasticity of Demand for Cocaine: -.55
Elasticity of Supply: 1
Current Market Price Cocaine: $80 per gram
Current Cocaine Sales (annual): 950M grams
a) We are using a simply supply/demand framework:
d
s
Q a bP
Q c dP
Use the data above to find the parameters a,b,c, and d.
d
s
Q a bP
Q c dP
Use the data above to find the parameters a,b,c, and d.
950.55 6.53
80
950 6.53 80 1472.4
9501 11.88
80
950 11.88 80 0
d
s
Qb
P
a Q bP
Qd
P
c Q dP
b) As a check of the estimated model, solve for the equilibrium price and
quantity.
1472.4 6.53
11.88
1472.4 6.53 11.88
$80
11.88*80 950
d
s
Q P
Q P
P P
P
Q
c) Suppose that the DEA is able to seize 100M grams of cocaine and take
it off the market. What will happen to the equilibrium price and
quantity?
So, we need to subtract 100 from the market supply and resolve for
price and quantity. Intuitively, what should happen is that the seizure
will force the price up, which creates incentives for more supply.
1472.4 6.53
11.88 100
1472.4 6.53 11.88 100
$85.4
11.88*85.4 100 914
d
s
Q P
Q P
P P
P
Q
d) How will cocaine revenues for drug dealers be affected?
Revenues prior to the DEA seizure were $80*(950M) = $76,000M =
$76B. After the seizure we have $85.40(914M) = $78,055M =
$78.055B
Looks like the drug dealers win!
e) What happens to consumer surplus?
First, find the price where demand drops to zero.
1472.4 6.53
0 1472.4 6.53
225.48
dQ P
P
P
So, prior to the seizure, CS = (1/2)(225.48 – 80)(950) = $69,103M =
$69.103B
After the seizure, CS = (1/2)(225.48 – 85.40)(914) = $64,016M =
$64.016B
The drug user lose!
8) Suppose that you observed the following set of data:
Average Business School tuition: $30,000
Average Salary for non-MBA’s: $50,000 per year
Average MBA salary: $90,000 per year.
The length of an MBA program is 2 years and is assumed that and MBA will have
a working career of 20 years after graduation. Further, suppose that, instead of
going to get an MBA, you could keep your current non-MBA job and invest what
you could have used to pay for tuition, risk free, at 4% per year.
a) Is this set of data consistent with market equilibrium? Explain.
We need to figure out the present value of $40,000 per year, starting two
years from now, at an interest rate of 4%.
(Note: I did this calculation in excel…I wouldn’t expect you to calculate
it!)
2 3 22
40,000 40,000 40,000... $539,583
1.04 1.04 1.04PV
The opportunity cost of the MBA is the tuition plus the lost salary for two
years which is 2(50,000 + 30,000) = $160,000.
So, the benefits outweigh the costs at an interest rate of 4% (this says that
the return to an MBA is higher than 4%). With these numbers, the interest
rate would need to be over 20% for the costs to outweigh the benefits!
b) If your answer to (a) is no, how will markets adjust?
If an MBA was strictly preferred to working without an MBA, demand for
MBA degrees should rise, pushing up tuition. Further, as the number of
MBAs increases, their salaries should drop.
9) Suppose that a busy restaurant charges $9 for its octopus appetizer. At this price,
an average of 48 people order the dish each night. When it raises the price to $12,
the number ordered per night falls to 42.
a) Assuming that demand is linear, find the demand curve the restaurant
faces.
So, I know that the coefficient in from of price is 2 (a $3 increase in price
lowers sales by 6)
2Q A P
I also know that at a price of $9, sales are 48.
48 2 9
66
66 2
A
A
Q P
b) What price should the restaurant charge to maximize revenues?
So, revenues are price times quantity….
266 2 66 2R PQ P P P P
So, to maximize revenues, take the derivative with respect to price and set
it equal to zero.
66 4 0
16.50
P
P
66 2 16.50 33
16.50 33 $544.50
Q
R
10) Suppose that you are a cattle rancher. You are deciding when to take your cattle to
market to sell. You currently have a herd of 100 cattle. Each cow currently
weighs 650 pounds and is gaining 50 pounds per month. Your feed costs are $40
per month per cow. Cattle prices are currently $8 per pound, but have been falling
at the rate of $0.10 per month. If you are maximizing profits, how many month
from now should you sell your cows?
So, we have profits equal total revenues minus total costs where (t is time in
months)
8 .1P t
100 650 50Q t
100 40TC t
2
2
8 .1 65,000 5000 4000
520,000 40,000 6500 500 4,000
520,000 29,500 500
Profits t t t
Profits t t t t
Profits t t
So, take the derivative with respect to t and solve
29,500 1,000 0
29.5
t
t
8 .1*29.5 $5.05
100 650 50*29.5 212,500
4,000 29.5 $118,000
$5.05 212,500 $118,000 $955,125
P
Q
TC
Profits
11) Suppose that you are a pizza shop. Your profits depend on your sales of pizza
and beer. Specifically, your profits as a function of Pizza sales (P) and beer sales
(B) is given by:
2 280 120 140 8 12 4Profits P B P B PB
Solve for the profit maximizing choices for gasoline and heating oil.
We need to take the derivatives with respect to P and B
120 16 4 0
140 24 4 0
P B
B P
Now, we need to solve the above system for P and B. First, solve the first
equation for B.
120 16 4 0
30 4
P P B
B P
Now, plug into the second to get P
140 24 30 4 4 0
140 720 96 4 0
580 92 0
6.3
30 4 6.3 4.8
P P
P P
P
P
B
12) Suppose that your sales are a function of both price (P) and advertising expenses
(A) given by
2 23,000 8 25 2 .5 3Q p A pA p A
Solve for the combination of price and advertising that maximizes sales.
We need to take the derivatives with respect to p and A
8 2 0
25 2 6 0
A p
p A
Now, we need to solve the above system for p and A. First, solve the first
equation for p.
8 2 0
2 8
A p
p A
Now, plug into the second to get A
25 2 2 8 6 0
25 4 16 6 0
9 2 0
4.5
2 8 2 4.5 8 1
A A
A A
A
A
p A
13) We need to enclose a field with a fence. We have 500 feet of fencing and a
building is on one side and so won’t need any fencing. Determine the
dimensions of the field that will enclose the largest area.
So, the problem we have is:
,
maxx y
xy subject to 2 500x y
So, set up the lagrangian
500 2l xy x y
take the derivatives with respect to x and y and set them equal to zero
2 0
0
y
x
From the second equation, we have x . Plug into the first equation to get
2
yx
Building
Field x x
y
Now, use the constraint with 2
yx
2 500
2 5002
2 500
250
125
x y
yy
y
y
x
14) Suppose that Apple is selling IPads in both the US and Europe. Sales in each
country are a function of the level of advertising and given by
2
2
12 6 1.2
8 2 .2
US US US
E E E
S A A
S A A
Solve Apples’ maximization problem; maximize total sales across the two
districts subject to a total advertising budget of $4M. How would a $1M increase
in Apples’ advertising budget influence sales?
So, the problem we have is:
2 2
,max 12 6 1.2 8 2 .2US US E E
x yA A A A subject to 4US EA A
So, set up the lagrangian
2 212 6 1.2 8 2 .2 4US US E E US El A A A A A A
take the derivatives with respect to A(US) and A(E) and set them equal to zero
6 2.4 0
2 .4 0
US
E
A
A
Solve each for lambda and set equal to each other
6 2.4 2 .4
4 2.4 .4
2.4 4 .4
6 10
US E
US E
US E
US E
A A
A A
A A
A A
Now, use the constraint…
4
6 10 4
7 14
2
2
6 2.4 6 2.4 2 1.2
US E
US US
US
US
E
US
A A
A A
A
A
A
A
So, a $1M increase in the advertising budget will raise sales by approximately
1.2(1) = $1.2M.
15) In the game blackjack, face cards are worth 10 points, aces are worth 1 or 11, and
all other cards are worth their face value. You are dealt two cards with the objective
of getting more points than the dealer. A “Blackjack” is 21. Assuming a fresh
deck (i.e. no cards have been dealt), what are the odds of getting blackjack?
First, let’s figure out the odds of getting an ace and then a 10 point card.
(Remember, once the ace has been dealt, there are only 51 cards left)
Prob(Ace) = 4/52
Prob (10 point card/an ace has been dealt)) = 16/51 (4 10s, 4 jacks, 4 queens, 4
kings)
So, the probability of an ace first blackjack is (4/52)(16/51) = 64/2652
Now, the alternative would be to get a 10 point card first, and then an ace
Prob(10 point card) = 16/52
Prob(ace/a 10 point card has been dealt) = 4/51
So, the probability of an ace second blackjack is (16/52)(4/51) = 64/2652
So, the total probability of a blackjack is 128/2652 = .048 (4.8%) – about 1 in 20.
16) Assuming two decks of cards (again, assume a fresh deck), if the dealer is showing
an ace, and you have not drawn any additional cards yet, what are the odds that the
dealer has blackjack?
If the first card dealt to the dealer is an ace, then there are 103 cards remaining in
the two decks, and 32 of those cards are worth 10 points. So, the odds are 32/103
(31.07%).
Now, looking at your hand, if you have one 10 point card, there are 101 cards
remaining and 31 of those cards are worth 10 points, so the odds of a dealer
blackjack are 31/101 (30.69%)
If you have 2 ten point cards, there are 101 cards remaining and 30 of those are
worth 10 points, so, we have 30/101 (29.70%)
If you have no 10 point cards, then the odds are 32/100 (31.68%)
17) Suppose that you are playing craps. If you roll the dice 10 times, what are the odds
that 4 of your rolls come up with a total of seven?
Recall, that you can roll a seven 6 ways (1+6, 6+1, 2+5, 5+2, 3+4, 4+34), so the
odds of a 7 is 6/36.
Now, the binomial distribution gives us the probability of k successes in n tries
where p is the probability of success. (Here, the probability of success is 6/36)
!1
! !
n kknP p p
k n k
So, we want 4 successes (k = 4) out of 10 rolls (n=10) where the probability of
success is 6/36
4 610! 6 30
4! 6 ! 36 36P
4 6
210 .167 .833 210 .00077 .334 .054P (5.4%)
18) Consider the following regression analysis of player performance measures and
average winnings per tournament in the PGA (Professional Golf). The data looks
at 196 players.
a) First, let’s consider driving distance (Note: The average driving distance is
287 yards with a variance of 68):
W D
Where W is average winnings and D is driving distance in yards.
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.20
R Square 0.04
Adjusted R Square 0.03
Standard Error 54041.64
Observations 196.00
ANOVA
df SS MS
Regression 1.00 23093588860.13 23093588860.13
Residual 194.00 566576795050.79 2920498943.56
Total 195.00 589670383910.92
Coefficients Standard Error t Stat
Intercept -331133.39 134365.65 -2.46
Average Drive 1315.17 467.70 2.81
a) What would be the impact on a player’s average winnings of a 20 yard
increase in his average driving distance? What would be a 95%
confidence interval for the impact of a 20 yard increase in a player’s
average drive?
The coefficient on average drive is $1,315.17 for ever yard of driving
distance with a standard error of $467.70. Therefore, a 95% confidence
interval for the coefficient would be 2 standard deviations in either
direction.
$1,315.17 +/- 2*$467.70 = [$2,250.57, $379.77]
So, a 20 yard increase in driving distance would add 20*1315.17 =
$26,303.40 to the players average salary. A 95% confidence interval
would be 20*[$2,250.57, $379.77] = [$45,011.40, $7,595.40]
b) Calculate a forecast with a 95% confidence interval for a player with a
300 yard drive.
The forecast would be
331,133.39 1315.17 300 0 $63,416.84W
To calculate a 95% confidence, we need a standard deviation for the
forecast
12 2
1ˆ/ 1
1
D DSD W D
N N Var D
2300 2871
/ 300 54,041.64 1 $54,528.57196 195*68
SD W D
So, our 95% confidence interval is $63,416.84 +/- 2*$54,528.57
(Note, we can’t have negative earnings!)
[$0, $172,473.98]
c) How far must a player be able to drive the ball on average to expect to
have positive earnings?
We have the regression equation
331,133.39 1315.17 300 0 $63,416.84W
Set winnings equal to zero and solve for D. We get D = 251.8 yards
Now, suppose that I altered the regression by taking the natural log of
winnings.
lnW D
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.108
R Square 0.012
Adjusted R Square 0.007
Standard Error 0.984
Observations 196.000
ANOVA
df SS MS
Regression 1.000 2.237 2.237
Residual 194.000 188.027 0.969
Total 195.000 190.264
Coefficients Standard
Error t Stat
Intercept 6.567 2.448 2.683
Average Drive 0.013 0.009 1.519
a) Now, what impact would a 20 yard increase in driving distance have on
average winnings?
The coefficient on average drive is .013 (1.3%) increase in average
winnings for ever yard of driving distance with a standard error of
$.9%. Therefore, a 95% confidence interval for the coefficient would
be 2 standard deviations in either direction.
1.3% +/- 2*.9% = [3.1%, -.5%]
So, a 20 yard increase in driving distance would add 20*1.3 = 26% to
the players average salary. A 95% confidence interval would be
20*[3.1%, -.5%] = [62%, -10%]
b) Calculate forecast for average winnings for a player with an average
drive of 300 yards.
The forecast would be
10.467
6.567 .013 300 0 10.467
$35,136.65
W
e
The Standard Deviation would be
12 2
1ˆ/ 1
1
D DSD W D
N N Var D
12 2
300 2871/ 300 .984 1 1.002
196 195*68SD W D
Note, that’s a standard error of over 100%!
So, if W = 10.467 +/- 2*1.002, we have W = [8.463, 12.471]
8.463
12.471
4,736.25
260,667.30
e
e
W = [$4,736.25, $260,667.30]
19) Consider the following time series regression:
P t
Where P is total non-farm payrolls in the US and t is time in months. The data used
is monthly data from Jan 1939 until August 2016 (t = 0 is Jan 1939). We have 931
observations (so, the average for time is 466 and the variance is 72,463)
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.990
R Square 0.981
Adjusted R Square 0.981
Standard Error 4867.781
Observations 932.000
ANOVA
df SS MS
Regression 1 1120288062756.780 1120288062756.780
Residual 930 22036623597.343 23695294.191
Total 931 1142324686354.120
Coefficients Standard Error t Stat
Intercept 26509.512 318.642 83.195
Time 128.864 0.593 217.437
a) On average, how many jobs do we create per year in the US?
The coefficient on time is 128,864 per month (employment is in thousands, so
this is 128,864 jobs per month).
So in a year, we create 128,864*12 = 1,546,368 jobs
b) Calculate a forecast for Non-farm payrolls for December 2016 ( t = 935) with a
95% confidence interval.
26509.512 128.864 935 0 146,997.421P
So, the forecast for payrolls in Dec. 2016 is 146,997,421.
12 2
1ˆ/ 1
1
t tSD P t
N N Var t
12 2
935 4661/ 935 4867.781 1 4878.172
931 930*72463SD P t
So, our forecast is 146,997,421 +/- 2*(4,878,172).
Now, suppose that I added seasonal dummies for the first three quarters
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.990
R Square 0.981
Adjusted R Square 0.981
Standard Error 4832.911
Observations 932
ANOVA
df SS MS
Regression 4 1120672723961.980 280168180990.495
Residual 927 21651962392.140 23357025.234
Total 931 1142324686354.120
Coefficients Standard Error t Stat
Intercept 27245.376 419.879 64.889
Time 128.853 0.588 218.984
D1 -1757.508 448.255 -3.921
D2 -488.961 448.251 -1.091
D3 -667.040 448.729 -1.487
a) Is there evidence for seasonality in employment in the US?
By the coefficients, it says that
Quarter 1 jobs created are 1,757,508 lower than the 4th quarter
Quarter 2 jobs are 488,961 lower than the 4th quarter
Quarter 3 jobs are 667,040 lower than the 4th quarter
However, only the 1st quarter dummy is statistically significant. So we have
the 1st quarter is a slow quarter for job creation!
b) Calculate a new forecast for Dec. 2016 (don’t worry about the Standard Dev.)
27245.376 128.853 935 0 147,723.088P
So, 147,723,088 jobs!
20) Suppose that I repeated the above analysis, but I converted payrolls to logs….
1 2 1 3 2 4 3ln P t D D D
Regression Statistics
Multiple R 0.9874
R Square 0.9749
Adjusted R Square 0.9748
Standard Error 0.0700
Observations 932
ANOVA
df SS MS
Regression 4 176.3536 44.0884
Residual 927 4.5464 0.0049
Total 931 180.9001
Coefficients Standard Error t Stat
Intercept 10.5350 0.0061 1731.5119
Time 0.0016 0.0000 189.5660
D1 -0.0247 0.0065 -3.7960
D2 -0.0100 0.0065 -1.5329
D3 -0.0090 0.0065 -1.3868
How does this change the analysis above?
So, now we have that payrolls increase by .0016 (.16%) per month or 12*.16
= 1.92% per year.
Seasonality is still present with payrolls in the first quarter being 2.47 percent
lower than the 4th quarter. The other quarters are not significantly different.
If I were to forecast Dec 2016 (t = 935)
12.031
ln 10.535 .0016 935 0 12.031
167,879.208
P
e
So, 167,879,208 jobs!
