Demand function for tickets for a rock concert has been estimated

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1. (i) Demand function for tickets for a rock
concert has been estimated to be

ln
Q = 3.737 – 1.372 ln P +1.823 ln I

where Q denotes number of tickets
(in thousands), P the (average) ticket price and I the average income of the concert goers.

Determine the values of the price
elasticity of demand and the income elasticity of demand.

(ii) In a recent study it has been estimated that
the own price elasticity of demand for a special type of U.S. manufactured
automobile tires is – .75, while the income elasticity of demand is 1.1 and the
cross price elasticity of demand with respect to foreign imports is 1.4. The current sales volume for the U.S.
manufactured tires is 5 million unites per year. It is anticipated that the price of the
foreign imports will rise by 5%.
(a) Assuming that average income of
the target group of customers will not change, calculate the number of tires
that the tire manufacturers will be able to sell if they plan to increase their
own price by 3%.

(b)
According to newly released economic data It is now expected that over the next
year the average income of the target group of consumers in the U.S. will grow
by 4%. Calculate the amount (%) by which the U.S. tire manufacturers can adjust
their price if they wish to increase their sales volume by 8.4% over the
initial 5 million units. Assume the same increase in foreign tire prices as
above.

(iii) Bright Future, Ltd (BF) is a nonprofit
foundation providing medical treatment to emotionally distressed children. BF
has hired you as a business consultant to design an employment policy that
would be consistent with its goal of providing the maximum possible service
given its limited financial resources.
You have determined that the service (Z) provided by BF is a function of
its medical staff input (M) and social worker staff input (S) which is given
by:

Z
= M + .5S + .5 MS – S2

BF’s staff budget for the coming year is
$1,200,000. Annual employment costs are
$30,000 for each social worker staff member (S) and $60,000 for each medical
staff member (M).

(a) Using
the Lagrangean multiplier approach calculate the optimal (i.e., service maximizing) combination of medical
and social worker staff. Determine the
optimal amount of service provided by BF.

(b) Calculate BF’s marginal cost. Explain your answer.

(c) Using Excel-Solver verify your answer to (a).
(Showyour work. Show the spreadsheets in
detail. Show the Solver window embedded on the relevant worksheet so that the
commands in the Solver window become directly visible and are linked to the
cells of the worksheet. To show the solver window, use print screen command on
your key board and then create a MS Word document using paste. See The Solver
supplement for help.)

(iv) In year 2014 price of a (500 ml.) bottle of
wine made by Richard Mondavi was $14.00.
Fifty (thousand) bottles were sold. It was estimated that the own price
elasticity of demand for wine of that type was -1.4. Assuming a straight line demand curve, determine the
demand equation for Mondavi’s wine.
Show and explain all your calculations.

2 (i) The production function for a firm is given
by

q = L.7K.4

where q
denotes output; Land K
labor and capital inputs.

(a) Determine
marginal product of labor. Show whether or not the above production function exhibits diminishing marginal
productivity of labor.

(b) Calculate the output (or production)
elasticity with respect to labor.

(c) Determine the
nature of the Return to Scale as exhibited by the above production function.
Show and explain all calculations.

(ii) A
firm’s production technology is given by the production function

.0/msohtmlclip1/01/clip_image002.gif”>
where
L represents labor hours, K machine hours and q the amount of output.
The market wage and rental rates are, w= $64 and r = $128.
The
firm is operating in the long run where it can adjust both inputs.

(a) Suppose
that the firm currently is using twice as many units of labor as capital.
Is it minimizing its long run
total cost? If so why so and if not why
not? Explain. If it is not minimizing its long run cost, how
should it adjust its input usage? Explain. Provide appropriate calculations.

(b) Suppose that the firm wants to produce 128
units of output. Determine the cost minimizing combination of L and
K. Calculate the resulting long run
total cost. Show and explain all calculations.

(c) Calculate
the short run total cost if q =128 and w=
$64 and r = $128, but capital, K
is fixed at 1.

(d) Without assuming a specific numerical
production target, but using w= $64 and r = $128 calculate the equation for
the long run total cost function (in terms of q).
(Hint: Assume that the level
of output is q. Using the above w, r
values first determine the least cost combinations of L
and K)

(e) Using Excel- Solver verify your
answers to (b) above.
(Show
your work. Show the spreadsheets in detail. Show the Solver window embedded on
the relevant worksheet so that the commands in the Solver window become
directly visible and are linked to the cells of the worksheet. To show the
solver window, use print screen command on your key board and then create a MS
Word document using paste.)

(iii) Summit Farms hires unskilled daily workers to pick strawberries in their
fields. The actual production depends on the
availability of workers and weather related factors. Summit Farms has the following
data on the number of workers used, L and the
amount of production, q, measured in pounds of strawberries for 15 days.

Daily Production Data

Day

L

q

1

5

250

2

7

385

3

8

442

4

6

331

5

6

324

6

8

442

7

10

500

8

11

478

9

12

432

10

9

490

11

10

480

12

10

494

13

8

317

14

7

399

15

8

448

(a) Using
Excel’s Regression package estimate a production function for Summit Farms which is of the form: q =a
L3 + bL2. Provide the detailed regression output generated by Excel. Identify the estimates of
the coefficients a and b. Write the equation for the estimated production
function. Identify the value of R2
.

If you are using a Mac you can use the linest function key for running
the regression.

(b) Using
the estimated function and Excel’s charting tool plot the Total Product of Labor (TPL), the Average Product of Labor
(APL) and the Marginal Product of Labor Curves. Are these
graphs and thus the estimated cubic production function consistent with standard TPL, APL and MPL curves?

3. (i) The Write Easy Company manufactures a variety
of pens selling for $2.98 each. Sales have averaged 10,000 units per
month during the last year. Recently Write Easy’s closest competitor, Joy Write
Company cut its prices on similar pens from $3.49 to $2.59. As a result Write
Easy’s sales declined to 8,000 units per month.
(a) Calculate the arc cross price elasticity of
demand between Write Easy’s and Joy Write’s pens.

(b) If Write Easy knows its own arc price
elasticity of demand for its pens to be -2.2, what price would they have to
charge in order to restore their monthly sales
back to 10,000 units? (Assume that Joy Write maintains its price at $2.59.)

Show and explain all calculations.

(ii) Dirt
Diggers (DD) is an excavating firm that excavates roadside ditches for laying drainpipe. Its output follows
the following production function:

Q = 10L – .1L2

where L
denotes labor hours and Q the length of the ditch in meters. DD hires labor
at the going wage rate of $12 per hours.

(a) DD has received an offer to excavate 250
meters for a lump sum price of $500. Should it accept the offer? Explain with appropriate calculations.

(b) Suppose
instead of the previous offer DD is offered as much or as little excavation work at a price of $2.00 per meter dug. Should it accept the offer?
If it does, calculate its profit and the optimal (profit maximizing) output
(meters dug) and labor usage.

(iii) As
the manager of an 80-unit motel you know that all units are occupied when you charge $60 a day per unit. Each occupied room costs $30 for service and
maintenance a day. You have also
observed that for every x dollars increase in the daily rate above $60, there
are 2x units vacant. Determine the daily
price that you should charge in order to maximize profit. Calculate the number of occupied units. Assuming that fixed cost is $550 calculate
optimal profit.

(Hint:
You may like to determine the demand function first.)

(iv) A firm is selling a product with a constant
marginal cost of $60. The own price elasticity of demand for the product has
been estimated to be -1.5. Calculate the profit-maximizing price that the firm
should charge.

(v) The
domestic demand and supply function for oil for a small country is given by: Qd= 210 – 1.5p
and Qs = – 140 + 2p, where p
is the price per barrel and Qd
Qs
are the quantities in million barrels.

(a) Use
Excel to calculate quantity demanded and quantity supplied for p =
$70, 75,80….140 (in $5 increments).
Determine the equilibrium price and quantity
in absence of any oil import.

(b) Assume
that OPEC can sell unlimited quantity of oil at $80 per barrel. Using your
calculations in (a), determine the equilibrium price, amount of domestic consumption,
quantity supplied by domestic producers and the amount of oil import. (Assume that at a
given price, the amount of import is
the gap between domestic demand and domestic production.)

(c) Now,
suppose the country’s government imposes a limit on the amount of oil that can be imported
from OPEC at their price of $80. Given that the limit
is set at 35 million barrel,
use Excel and the spreadsheet created in part
(a) above to calculate the aggregate supply of oil that includes the domestic
supply and the import from OPEC. Calculate the domestic price for oil,
domestic consumption and domestic production.

 

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