VOLUNTARY EXERCISES (LECTURE I – IV)
1) PRESENT VALUE OF EXPECTED FUTURE PROFITS
If the interest rate is 8%, what is the present value of the profits in the next 10 years?
Number of years in the
future
Profit
1
7
2
8
3
9
4
12
5
13
6
10
7
9
8
15
9
17
10
17
7/1,08 + 8/1,08² + 9/1,08³ + 12/1,084 + 13/1,085 + 10/1,086 + 9/1,087 + 15/1,088 + 17/1,089 +
17/1,0810 = 74,19
2) ELASTICITY
The Ford Company, a maker of formula 1 engines, determines that 2008 the demand curve for
its product is
P = 2000 – 50 Q
Where P is the price (in thousand euros) of an engine and Q is the number of engines sold per
month.
a) To sell 20 engines per month, what price would Ford have to charge?
b) If it sets a price of 500, how many engines will Ford sell per month?
c) What is the price elasticity of demand if price equals 500?
d) At what price, if any, will demand for Ford engines be of unitary elasticity?
a) With Q = 20, we have P = 2,000 - 50*20 = 1,000.
b) Solving the demand curve equation for Q, we have 50Q = 2,000 – P
Q = 40 - (1 / 50)*P
P = 500, Q = 40 - (1 / 50)*500 = 30.
c) ε = (dQ / dP) * (P / Q)
dQ / dP = - (1 / 50)
d) P = $500, Q = 30
ε = (-1 / 50)*(500 / 30) = -(1 / 3)
3) ELASTICITY
Given the demand function QD = 1000 – 0.5 P, find the price elasticity of demand at an output
level of 200.
η = dQ / dP * P / Q
η = - 0,5 * 1600 / 200 = - 4
4) ELASTICITY
Find the advertising elasticity for the given demand function and parameters.
QD = 1000 – 0.5 P + 0.2 A
P = 1800
A = 1000
The corresponding output level is 300.
η = dQ / dA * A / Q
η = 0.2 * 1 / 300 = 0,66
5) ELASTICITY
After the “Guinness brewery” did some statistical analysis with their data, they found out that
the demand function for their product is the following:
Q = 1200 – 30P
a) Calculate the point elasticity at P = 15.
b) Calculate the arc elasticity at P = 15. (Use P = 16 as your second reference point.)
c) Why is there a difference between the two numbers?
a) The formula to calculate the point elasticity is:
η = dQ / dP * P/Q
η = (-30) * 15 / (1200 – 30*15)
η = - 0.6
The point elasticity at a price of 15 is -0.6
b) The formula to calculate the arc elasticity is:
delta Q / delta P * ((P1 + P2)/2) / ((Q1 + Q2)/2)
Quantity demanded at a price of 15 is: 1200 – 30*15 = 750
Quantity demanded at a price of 16 is: 1200 – 30*16 = 720
η = (-30) / 1 * ((15 + 16) / 2) / ((750 + 720) / 2) = - 0.633
The arc elasticity at a price of 15 (with a reference price of 16) is -0.633
c) The different values for η are a result of the fact that you actually measure the elasticity at
different positions of the demand curve. With the point elasticity you measure the elasticity at
the point P = 15. With the arc elasticity you measure the elasticity between P = 15 and P = 16.
6) ELASTICITY
Use the given demand function to compute the following numbers.
Qx = 100 – 8Px + 0.5Py + 2I
Qx
… Quantity demanded of good x
Px
… Price of good x
Py
… Price of good y
I
… Income
Px = 7, Py = 12, I = 4.5
a) Calculate the cross price elasticity for the given values.
b) Is good y a substitute or a complementary product.
a) The formula for the cross price elasticity is:
ηxy = dQx / dPy * Py / Qx
Qx for the given numbers is: Qx = 100 – 8*7 + 0.5*12+2*4.5 = 59
ηxy = 0,5 * 12 / 59 = 0,102
The cross price elasticity is 0.102
b) Good y is a substitute product to good x. Since the quantity demanded of good x increases
as the price of good y increases (positive sign of the cross price elasticity), while quantity
demanded for good y decreases, we find an inverse relationship between the quantities
demanded of those two goods.
7) ESTIMATING DEMAND FUNCTIONS
The Nestle Company hired a consultant to estimate the demand function in the milk market.
After the consultant did some research about the market during the last year, she came up with
the following equation:
Log Q = 3500 – log 1.8 P + log 0.6 A – log 1.5 PW
Q…quantity demanded
P…price of milk
A…advertisement expenditures
PW…price of cocoa (complementary good)
a) How is the economic concept called that is described by the log-numbers?
b) What is the effect on demand if advertisement expenditures increase by 10%?
c) What is the effect on demand if the price of cocoa decreases by 1%?
d) What is the effect on demand if the price of milk decreases by 50%?
a) The log-numbers represent the elasticity’s. P, A and PW represent the price elasticity, the
advertisement elasticity, and the cross price elasticity.
b) (Price change in advertisement expenditures) * (elasticity) = change in quantity demanded.
10 * 0,6 = 6 %
c) 1 * (-) 1,5 = -1,5 %
d) 50 * (-) 1,8 = - 90 %
8) ESTIMATING DEMAND FUNCTIONS
The marketing department did some analysis of the demand of a good and came to the
following result:
Demand function: (T-values in parenthesis)
Q = 70 – 0.75 P + 0.2 I + 0.3 A + 0.05 Px
(4.5) (2.8)
(0.5)
(2.2)
(1.1)
2
R : 0.88
Q…quantity demanded
P…price of the good 1
I…income
A…advertisement expenditures
Px …price of good 2
a) What does the R2 value state and is it a good value?
b) What do the T-values represent and what do the given values suggest?
a) The R² value shows you the percentage of total variations in Q that can be explained with
these variables.
88 % is a pretty good value and shows a strong relationship between the variables and Q.
b) The t-value shows you the probability if a variable is just random or really reliable. If the
value is higher than 2 or so, it is very likely that the relationship really exists and you can
work with that number. With the given t-values you should disregard the relationship between
Q and I and between Q and PX.
9) PERFECT COMPETITION
Given a demand function QD = 1000 – 0.5 P and a supply function QS = 200 + 0.1 P. Find the
equilibrium quantity and price and the corresponding total revenue.
Equate demand and supply to find equilibrium quantity and price:
1000 – 0.5 P = 200 + 0.1 P
P = 1333,33
Q = 333,33
Total revenue is defined as price times quantity, which is in equilibrium 444,444.44
10) MONOPOLY
A monopolist faces the demand function QD = 1000 – 0.5 P and the cost function
TC = 200 000 + 2 Q + 0.1 Q2.
Find the profit maximizing price and output combination.
Profits reach a maximum where marginal revenue equals marginal costs. To get the marginal
revenue function, reformulate the demand function and multiply with Q:
QD = 1000 – 0.5 P
P = 2000 – 2 Q
TR = P * Q = 2000 Q – 2 Q2
The marginal revenue function is the derivative of the total revenue function:
MR = TR`= 2000 – 4 Q
The marginal cost function is the first derivative of the total cost function:
MC = TC`= 2 + 0.2 Q
Equate MR and MC to get the profit maximizing output level:
2000 – 4 Q = 2 + 0.2 Q
Q = 475,71
Plug the optimal output level into the demand function to get the optimal price:
P = 1048,57
Maximal profits are total revenue minus total costs:
Π = 275238,57
11) PERFECT COMPETITION
In a perfectly competitive market the demand function is QD = 2400 – 6 P where QD is
quantity demanded and P is the price.
The firms operating in this market experience the following cost function:
TC = 450 – 10 QS + 2 QS2 where TC is total costs and QS is quantity supplied.
Assume the market is in equilibrium.
a) Calculate the average cost curve and find the minimal average costs?
b) Find the supply curve of a single firm in the market.
c) How many firms operate in this market?
a) To get the average cost curve, just divide the total cost curve by QS
AC = TC / QS
AC = 450 / QS – 10 + 2 QS
To find its minimum, set the dAC / dQS equal to zero.
450 / QS² + 2 = 0
QS = 15
In order to get the average costs at 15 units, insert into the average cost function.
AC = 450 / 15 – 10 + 2 * 15 = 50
b) The supply curve of a company is given by its marginal cost curve; therefore one just has
to find the MC curve. To calculate the MC curve, just derive TC to QS.
dTC / dQS = MC = - 10 + 4 QS
c) Since every firm will produce at its minimum average cost level and there are no profits in
a perfectly competitive market, the price will equal minimum average costs. Therefore one
can calculate the equilibrium quantity with the demand function.
QD = 2400 – 6 * 50 = 2100
Every firm will produce 15 units (that’s the quantity at the minimum average cost level). So
there will be 140 firms in the market (2100/15).
12) PRODUCTION INTER-RELATIONSHIPS
Assume there is a firm in the chemical industry, that produces a main good A and a byproduct B. Per ton produced of good A we get a quarter of a ton of the by-product B. For
simplicity we denote the quantities in units, which is one ton of good A and a quarter ton of
good B. Because it is not possible to differentiate the cost of production for each good, there
is just one cost function. The demand function for the two goods and the total cost function
are the following:
DA: PA = 1,000 – 10 QA
DB: PB = 500 – 8QB
TC: TC = 10,000 + 200 QT +2.5 QT2
DA…demand for good A
DB…demand for good B
TC…total cost, QT…QA+QB
a) What is the function of the marginal revenues?
b) Find the profit maximizing price-quantity combination.
a) The marginal revenue curve is the horizontal sum of the two individual MR curves, as long
as both are positive. If the quantity is big enough and the MR curve of one good is negative,
we use the higher MR curve from that point on.
MRA = 1,000 – 20 QA
MRB = 500 – 16 QB
MRT = 1500 - 36 QT if QB =< 31,25
MRT = 1000 – 20 QT if QB > 31,25
b) MCT = 200 + 5 QT
MCT = MRT
200 + 5 QT = 1,500 – 36 QT
QT = 31.71
Since this quantity is too high to be valid at the first MR curve, it can not be the profit
maximizing quantity.
MCT = MRB
200 + 5 QT = 1,000 – 20 QB
QB = 32
The profit maximizing quantity is 32 units (tons) of good A and 31.25 units (quarter of tons)
of good B. The reason why the firm doesn’t sell as much of good B as it produces, is that the
MR would be negative, so the firm is be better off not to sell those 0.75 units.
13) PARALELL DEVELOPMENT EFFORTS
Bayer pharmaceutical firm can do research on a special drug using either of two independent
processes, each of which has a 25 percent chance of costing $ 10,000,000 and a 75 percent
chance of costing $ 20,000,000. If the firm can determine the true cost of both processes after
spending $ 400,000 on each, what is the minimized expected cost of developing this drug?
There are four potential outcomes if we use two independent processes.
Process 2
Process 1
$10 Mio.
$20 Mio.
2
0.25*0.75
$10 Mio.
0.25
$20 Mio.
0.75*0.25
0.752
In three out of those four cases, we will have at least one process with $10 Mio. costs. The
probability of the worst case (both processes will costs $20 Mio.) is 0.752. So the expected
costs are:
0,75² * 20 + (1 – 0,75²) * 10 + 0,4 = 16,025 Mio.
14) TECHNOLOGICAL CHANGE AND INDUSTRIAL INNOVATION
The VOEST Alpine company produced 50.000 units in 2008 and 65.000 units in 2009. Given
the data below, calculate the change in total factor productivity?
QUANTITY
PRICE
QUANTITY
PRICE
INPUT
2008
2008
2009
2009
Steel
1000
500
1300
650
Labor
40000
8
55000
9
Coal
1200
2
1400
3.5
50000 / (1000*500 + 40000*8 + 1200*2) = 0,0608
65000 / (1300*500 + 55000*8 + 1400*2) = 0,0595
Prices of the base year are used for all years.
0,0608 Æ 0,0595 Æ 2,17% decrease
15) TECHNOLOGICAL CHANGE AND INDUSTRIAL INNOVATION
You’re the strategic planner at your company and you try to figure out what would be the best
point of time to release your latest invention. The longer you wait, the less innovative it will
be because your competitors work on a similar product. On the other hand, your costs fall the
longer your researchers have time to develop the product perfectly and optimize the
production processes. You estimate the following extrapolation of revenues and costs (t is
measured in weeks):
R(t) = (100 – 2 t)*Q
C(t) = (90/(0.2 t))*Q
What is the profit maximizing point of time to release the product?
dC / dt = dR / dt
It would be best to release the product in 15 weeks.
16) TECHNOLOGICAL CHANGE AND INDUSTRIAL INNOVATION
The unit cost of manufacturing airplanes is subject to a learning curve described by
log C = 7 – 0,4 log Q, where C is the current period's unit cost and Q is the accumulated past
production. How much past output must be accumulated before unit cost falls to 1.000.000 €?
If Q increases by 1% then unit (average) costs for these additional pieces decrease by 0.4%
log 1.000.000 = 7 – 0,4 log Q
6= 7 – 0,4 log Q
log Q = 2,5 Æ Q = 316
17) TECHNOLOGICAL CHANGE AND INDUSTRIAL INNOVATION
Turner Labs (TL) produces personal computers using labor and other technological
components. In 2000 it produced 25,000 PCs using 3,200 workers and 6,000,000 components.
In 2001 TL produced 35,000 computers using 4,000 workers and 7,500,000 components. In
2000 the wage rate was $ 8 per hour and the price of the technological components was $ 0.7
a piece. In 2001 the wage rate was $ 10 per hour and the price per piece decreased to $ 0.5.
Thus, from 2000 to 2001 TL's total factor productivity changed by
25000 / (3200*8 + 6000000*0,7) = 0,00592
35000 / (4000*8 + 7500000*0,7) = 0,00663
0,00592 Æ 0,00663 Æ 12 %
18) MONOPOLY
A monopolist faces the following facts:
QD = 20 – 0.5 P
TC = 50 +2 Q + 0.05 Q²
QD… quantity demanded
P… price
TC… total cost
At which price/quantity combination will this monopolist produce and what will be the profit?
MR = MC
Q = 20 – 0,5 P Æ P = 40 – 2 Q
TR = (40 – 2 Q)* Q Æ MR = 40 – 4 Q
MC = dTC / dQ Æ MC = 2 + 0,1 Q
Π= TR – TC
TR = P*Q
price = 21.46, quantity = 9.27, profit = 126.10
19) ELASTICITY
A monopoly firm has two groups of customers with different price elasticity of demand. Since
the arbitrage problem is solved for their product, the company can perform price
discrimination. The price elasticity of demand of group 1 is – 4 and at group 2 it is -1.5.
Marginal cost for their products is 24 €.
What are the optimal prices p1 and p2?
p1 = $ 32, p2 = $ 72
P = MC * 1 / (1 + 1 /η)
20) ELASTICITY
The elasticity of demand for Wienerberger bricks at a local retail outlet is -1.5 at the current
price of 300€ a ton. Which price should the store manager set, if the marginal costs are 110€?
increase the price to $ 330
P = MC * 1 / (1 + 1 /η)
P = 110 * 1 / (1 + 1 /(-1,5)) = 330