me2 - final exam-5 - ibp union · page4%of%30% 2000 0 2000 4000 6000 0 1000 2000 3000 4000 5000...

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Managerial Economics II IBP 3 day exam case study

Spring 2013

CTU Counts: 46.077

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Question 1)

This question regards the demand faced by a firm in a particular market structure. First and foremost, it is essential to identify the market structure in which the firm operates, since influences on its demand vary accordingly. The concept of market structure refers to the competitive environment in which the sellers and buyers of the product operates. In this case the market consists of the firm WindPartners, which is the clear market leader. Hence, this assignment assumes that WindPartners is not the sole firm in the industry, but the dominant firm, which means that WindPartners is the price setter. The firm is characterized by a high degree of product differentiation, since they provide the most comprehensive and user-‐friendly software package. In consequence the market structure is that of differentiated oligopoly, which consists of few sellers that produce differentiated products. Additionally, entry into an oligopolistic industry is possible but not easy, which is shown in the few numbers of firms. In connection to this, the industry in which WindPartners operates is characterized by a high degree of technological know-‐how making entry difficult. Thus, this assignment assumes that the market structure is that of oligopoly, where the distinguishing characteristic is that the action of each firm affects the other firms in the industry.

As WindPartners operates in an oligopolistic market the factors which influence its demand can now be identified. The factors influencing a firm’s demand for a commodity arise from the individual demand of the consumers. Individual demand is then determined by the consumers’ possibilities and will to buy the product, which is influenced by several factors related to the demand function faced by a firm.

Firstly, the price of the commodity influences the demand greatly, as the price will determine how much of the commodity consumers are capable and willing to buy. Secondly, the number of consumers in the market will affect demand. In the case of WindPartners the consumers are not private households but manufactories, developers and companies, why WindPartners operates in a business-‐to-‐business market. Hence, WindPartners’ consumer base is relatively limited. Thirdly, the firm’s demand is influenced by the income of its consumers. In terms of WindPartners’ consumers, their income is related to the profit which they generate, and thus if they obtain a large profit, they will demand more of WindPartners products. Fourthly, another important factor which influences demand is the price of related commodities. In this particular case, the demand for WindPartners’ software could change if the price of substitute or complementary products varies. For instance if the price of solar energy went down, thus making it more attractive to buy solar energy, the demand for wind turbines, which is a complementary product to WindPartners, would decrease and consequently shift WindPartners’ demand curve inwards. Additionally, the tastes of consumers influence demand. WindPartners produces comprehensive and user-‐friendly software packages, which most likely must appeal to their consumers making them the market leader.

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Furthermore, there are three other factors which might influence the demand of WindPartners. First, the size of their potential buyers will have an impact on the size of WindPartners’ demand. If they have a large amount of small buyers, these will not be able to put in large orders, thus lowering the demand. In addition, the reputation of WindPartners is essential to their demand. Since the firm produces user-‐friendly software and has more than 20 years of experience, it is assumed that WindPartners has a well renowned reputation. Lastly, legal issues as well as political decisions can have an impact on demand. As many countries focuses on climate friendly technology solutions, some governments promote the use of wind energy, thereby increasing the demand for wind turbines, which will lead to an increase in the demand for WindPartners’ products. Thus, this will lead to an outward shift of the firm’s demand curve.

In conclusion there are many factors which influence the demand of WindPartners directly as well as indirectly.

Question 2)

This question concerns the demand function and the correlating marginal revenue curve as well as the total revenue curve. Consequently, we will define, derive and illustrate the demand function, the MR function and the TR function.

Firstly, we will define the demand function. The demand function explains the relationship between the quantity demanded at a certain price based on the factors influencing demand, which are described in question 1. The slope of the demand function is negative because of the inverse relationship between quantity and price.

The information in the case provides us with two points on the graph, whereby we can derive the demand function: (0;4000) and (2000;2000).

We now derive the slope of the demand function by inserting the two points into the equation below:

Furthermore, we are informed that the maximum price is 4000, why the intersection with the second axis is then 4000. Hence, the demand function will be given as:

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-‐2000

0

2000

4000

6000

0 1000 2000 3000 4000 5000 6000

P

Q

P and MR

P = -‐1Q+4000

MR = -‐2Q+4000

Secondly, we will define the MR function. The term marginal revenue describes the additional revenue for each additional unit sold. From economic theory we know that if the demand function is linear, the MR function has the same intercept with the second axis but with double the slope of the demand function. Thus, the MR function can be written as:

MR = −2Q+ 4000

Consequently, we can illustrate the demand function and the corresponding MR function as below.

Thirdly, we will define total revenue. The TR function describes the total revenue earned at a certain level of output. To derive the TR function we integrate the MR function and get the following:

𝑇𝑅 = −2𝑄 + 4000 𝑑𝑞

TR = −Q! + 4000Q

The TR function is illustrated below:

In conclusion, the above graphs and functions describe the demand side faced by WindPartners.

0 1000000 2000000 3000000 4000000 5000000

0 450

900

1350

1800

2250

2700

3150

3600

TR

Quan6ty

TR=4000Q-‐Q^2

TR=4000Q-‐Q^2

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Question 3)

This question regards price elasticity of demand and its connection to the MR function and the TR function.

Price elasticity of demand (EP) can be defined as the change in the quantity demanded of a commodity to a change in its price. A linear demand curve is elastic above the midpoint, unitary elastic at the midpoint and inelastic below the midpoint. In correlation to this TR will act differently depending on whether or not demand is elastic, unitary elastic or inelastic. Assuming there is a price decline, TR will increase if demand is elastic meaning that IEPI>1. Opposite, TR will decline when demand is inelastic and IEPI<1. TR is then maximized when demand is unitary elastic and IEPI=1.

Furthermore, there is a relation between MR, TR and EP. As long as demand is elastic, TR will be increasing and since MR is defined as the change in TR divided by the change in Q, MR will be positive until the point where TR is maximized. When TR is maximized, at the point of unitary elasticity, MR will be 0 as there is no additional revenue to be added. Hereafter demand becomes inelastic, TR declines and MR will be negative, as every additional unit sold will bring a loss in revenue.

Related to the case, this above defined correlation of EP, MR and TR is illustrated below.

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-‐2000

0

2000

4000

6000

0 1000 2000 3000

P

Q

MR and MC

P = -‐1Q+4000

MR = -‐2Q+4000

MC = 2Q

From the demand function’s intersection with the second axis to the midpoint of the curve, which is at Q=2000, EP is elastic. At this interval, MR is positive and TR is increasing, which means that every additional software package sold will add additional revenue. At the midpoint of WindPartners’ demand curve, EP is unitary elastic, MR=0 and TR is maximized, meaning that additional sales will no longer add additional revenue. From Q=2000 until the demand function intersects the first axis at Q=4000, EP is inelastic and TR is declining. This means that every additional software package sold provides a loss in revenue, since MR is negative. From this point, WindPartners lose money by selling additional output.

In conclusion, the demand function of WindPartners is unitary elastic at Q=2000 when TR is maximized and MR is 0. Question 4)

This question considers the optimal condition for profit maximization.

The optimal quantity of software packages sold is found where MR=MC. According to marginal analysis, the firm will continue to produce as long as there is an additional revenue to be made (i.e. MR>MC). Hence, the firm will maximize its profit when marginal revenue equals marginal cost.

Firstly, we derive MC by differentiating TC = 500+ Q!.

As to find the optimal output level, we equate MC with the MR function from question 2:

MR = MC ↔ −2Q+ 4000 = 2Q ↔ 4000 = 4Q ↔ Q = 1000

We now find the optimal price by inserting Q=1000 into the demand function of WindPartners:

P = −Q+ 4000 = −1000+ 4000 = €3000

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In conclusion, the optimal quantity of software packages is 1000 units and the optimal price is then €3000.

Question 5)

This question concerns the difference between what consumers would have been willing to pay and what they actually pay for the product in question, which can be defined as consumer surplus.

Hence, consumer surplus explains the benefit that consumers obtain from purchasing a product, which has a lower asking price than what the consumer is willing to pay. The maximum amount a consumer is willing to pay is reflected in the demand function’s intercept with the second axis, which is then subtracted by the actual market price, and thus the benefit from engaging in the transaction is found.

When calculating the consumer surplus (CS) we use the demand function and thereby calculate the red area on the graph below using the following equation:

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Where Qe is the equilibrium quantity (1000), Pe is the equilibrium price (€3000) and Pmax is the maximum price that consumers are willing to pay (€4000). Inserting these values into the equation we obtain the consumer surplus in the case of WindPartners:

𝐶𝑆 = 𝑄! (𝑃!"# − 𝑃!)

2= 1000 (€4000 − €3000)

2= €500,000

Hereby it is shown that WindPartners fail to benefit from the consumer surplus thereby forgoing €500,000.

Obviously, WindPartners would like to capture as much consumer surplus as possible. They will be capable of doing so, if they are in an advantageous bargaining position, or if they are in a situation where they can exercise price discrimination. By differentiating prices, WindPartners are in a position where they can charge different prices for different consumers, and thus capture more consumer surplus.

As a result price discrimination can be very beneficial if the concerned company is in a situation where it is possible to differentiate prices.

Question 6)

This question regards the charging of different prices in different markets whereby it concerns price discrimination.

If a firm has a Uniform Pricing Policy, where it charges all consumers the same price, then it faces a constraint on how high a price it can charge consumers. The firm will ideally want to charge a higher price to consumers with a high willingness to pay and a lower price to consumers with a low willingness to pay. But if the firm is able to charge different prices to different consumers, the firm can extract larger surplus from consumers with a high willingness to pay, while still being able to serve consumers with a low willingness to pay. Hence, when the firm charges two different prices in two markets it exercises price discrimination.

There are three different types of price discrimination: first, second and third degree. The firm can increase its profits by capturing all or part of the consumer’s surplus by practicing any type of the above mentioned price discrimination practices. In this case WindPartners wishes to practice third-‐degree price discrimination, as it refers to the charging of different prices for the same product in different markets until the marginal revenue of the last unit of the product sold in each market equals the marginal cost of producing the product.

In the case of WindPartners we will now find the optimal quantity and price in each market. Since market 1 has the most elastic curve we anticipate the price in this market to be lower than the price in market 2.

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At first we use point price elasticity to determine Qmax, in order to obtain two points from where we can derive the slope of the demand function.

𝐸! = 𝑄 − 𝑄!"#

𝑄 ↔ 𝑄!!" = 𝑄 − 𝐸! ∙ 𝑄 = 50 − −1 ∙ 50 = 100

We now have two points: (0,10000) and (100,0). Using these we can obtain the slope:

𝛼 = 𝑃! − 𝑃!𝑄! − 𝑄!

= 0 − 10000100 − 0

= −100

Knowing Pmax (intersection with the second axis) to be 10000, the demand function will be:

𝑃! = −100𝑄 + 10000

Then we use this demand function of market 2 as well as the demand function of market 1 obtained from question 2 to differentiate prices on the two markets.

𝑃! = −𝑄 + 4000

The two MR functions are:

𝑀𝑅! = −2𝑄! + 4000

𝑀𝑅! = −200𝑄! + 10000

We then isolate Q in each market’s MR function, and thereafter add them horizontally:

𝑀𝑅! = −2𝑄! + 4000 ↔ 𝑄! = −12𝑀𝑅! + 2000

𝑀𝑅! = −200𝑄! + 10000 ↔ 𝑄! = −1200

𝑀𝑅! + 50

𝑄!"!#$ = 𝑄! + 𝑄! = −12𝑀𝑅 + 2000 + −

1200

𝑀𝑅 + 50 = 2050 − 0.505𝑀𝑅

𝑄!"!#$ = 2050 − 0.505𝑀𝑅 ↔ 𝑀𝑅!"!#$ = −1.9802𝑄 + 4059.41

In order to derive the value of Q where the total MR curve kinks, we insert 4000 (Pmax in MR1) into MR2

𝑀𝑅! = −200𝑄! + 10000 ↔ 4000 = −200𝑄 + 10000 ↔ 𝑄 = 30

So in the interval 0<Q<30 MR2 prevails, and in the interval 30<Q<2050 MRtotal prevails.

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As clearly shown, by the above graph, MC intersects the MR curve in the interval 30<Q<2050. Therefore, in order to find optimal Q and P in each market, we must equilibrate MC=2Q and MRtotal

𝑀𝑅!"!#$ = 𝑀𝐶 ↔ −1.9802𝑄 + 4059.41 = 2𝑄 ↔ 𝑄 ≈ 1020

The quantity and price in each market is then:

Market 1

𝑀𝐶 = 𝑀𝑅! ↔ 2𝑄 = −2𝑄 + 4000 ↔ 2 ∙ 1020 = −2𝑄 + 4000 ↔ 𝑄! = 980

The price is then:

𝑃! = −𝑄 + 4000 = −980 + 4000 = €3020

Market 2 𝑀𝐶 = 𝑀𝑅! ↔ 2𝑄 = −200𝑄 + 10000 ↔ 2 ∙ 1020 = −200𝑄 + 10000 ↔ 𝑄! = 40

The price is then:

𝑃! = −100𝑄 + 10000 = −100 ∙ 40 + 10000 = €6000

Conclusively, WindPartners will sell 980 software packages at a price of €3020 in market 1 and 40 software packages at a price of €6000 in market 2.

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Question 7)

This question is about the possible profit which is to be gained by practicing third-‐ degree price discrimination.

To find the additional gain in profit from differentiating prices we must first find the profit from having a Uniform Pricing Policy. If we hold the same price in both markets we must still add the MR curves horizontally, why we will obtain the same MR function as in question 6.

To derive the total demand function for the interval 30<Q<2050, we halve the slope of the total MR function.

𝑀𝑅!"!#$ = −1.9802𝑄 + 4059.41

𝑃!"!#$ = −1.9802𝑄

2+ 4059.41 = −0.9901𝑄 + 4059.41

As MRtotal and MC remains the same, the output level where they intersect will be the same as in question 6, thus Qtotal = 1020. Thereby we can derive the price on the two markets without price discrimination:

𝑃!"!#$ = −0.9901𝑄 + 4059.41 = −0.9901 ∙ 1020 + 4059.41 = €3049.508

The profit without price discrimination will then be:

𝜋!"!#$ = 𝑇𝑅 − 𝑇𝐶 = 1020 ∙ 3049.508 − 500 + 1020! = €2,069,598.16

The profit with price discrimination:

𝜋! = 980 ∙ 3020 − 500 + 980! = €1,998,700

𝜋! = 40 ∙ 6000 − 500 + 40! = €237,900

𝜋!"!#$ = 𝜋! + 𝜋! = 1,998,700 + 237,900 = €2,236,600

The gain in profit is then:

𝑃𝑟𝑜𝑓𝑖𝑡 𝑔𝑎𝑖𝑛 = 2,236,600 − 2,069,598.16 = €167,001.84

In conclusion, WindPartners will obtain an additional profit by differentiating pricing in the two markets.

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Question 8)

This question deals with whether third-‐degree price discrimination is profitable for WindPartners.

As to perform third-‐degree price discrimination, the firm in question needs to fulfill five conditions. Consequently, it is needed to perceive whether WindPartners meets these criteria, as this assignment cannot advice them to do so unless the below conditions are fulfilled.

First and foremost, the firm must have some control over the price, as not to obstruct competition. In question 1 this assignment assumed that WindPartners is the market leader and thereby the price setter of the industry, why the firm meets the first criteria. Secondly, when performing a Uniform Pricing Policy the price elasticity must vary on the two different markets. By doing simple calculus it can be shown that this is the case of WindPartners, whereby this condition is fulfilled. Thirdly, it must be possible to separate the markets, as to prevent resale. WindPartners will separate their markets between financial institutions and its other consumers. As financial institutions, namely banks, are distinct from e.g. project developers and engineering companies, the two markets can be clearly separated. Fourthly, price discrimination must be legal, ethic and politically correct when markets are separated. This case does not mention any legal issues related to the separation of the two markets, why this assignment assumes that this criterion is meet. Lastly, price discrimination must not be substantiated in cost differences. WindPartners is tempted to differentiate prices on the two markets, since the financial institutions have a relatively price inelastic demand. Hence this condition is fulfilled. In conclusion, WindPartners meets the five above mentioned conditions, therefore they are able to practice price discrimination.

In the short run, WindPartners will obtain a larger profit by differentiating prices on the two markets, as calculated in question 7. Opposite, the hazzle of implementing price discrimination might obstruct the full benefit of a larger profit. In connection to this, the theory of the firm states that the main objective of a firm is to increase its profit, why this assignment advices WindPartners to differentiate prices.

In the long run, the act of practicing price discrimination might lead to a poorer reputation of WindPartners, as the high paying consumers might feel cheated. As stated in question 1 a poor reputation will influence the demand of the firm by shifting the demand curve inwards and thereby lowering demand. On the other hand, WindPartners separates its markets very distinctively, whereby it can be argued that the firm might not suffer from a worse reputation since the two markets relate to two different sectors.

Conclusively, this assignment advice WindPartners to differentiate prices, why it will benefit from a larger profit in the short run and the long run effect is highly unlikely to occur.

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Question 9)

This question focuses on other types of pricing strategies that are relevant to WindPartners.

Previously it has been showed that WindPartners is in a position where it has the possibility to exercise third-‐degree price discrimination as an advantageous pricing practice. Several other pricing strategies can be used, depending on product, industry and consumers.

Firstly, as WindPartners produces differentiated software packages, it would be relevant to use tying. This is a pricing strategy where the seller requires that if the consumer purchases one product, they will need to buy another product connected to the first. In the case of WindPartners, it could be that if a wind turbine manufacturer buys a software package, they would have to buy a connected software package in order to get full use of the product.

Secondly, WindPartners is a well established company with many years of experience and a good reputation which gives WindPartners the possibility to exercise the pricing strategy of prestige pricing. This method is a way of attracting prestige-‐oriented consumers by setting a higher price for the product. Some consumers expect that a higher price corresponds to higher quality, and thus they may be attracted by the higher price, making WindPartners earn higher profits. WindPartners is in a situation where prestige pricing could be implemented, as their product is the most comprehensive and user-‐friendly software package on the market, making it a quality product.

Thirdly, since WindPartners produce a differentiated product and keep developing their product line, they are in a situation where skimming could be a beneficial type of pricing strategy. Skimming refers to a pricing practice where the seller starts out by charging a higher price when a product is introduced and then regularly decreasing the price. In terms of WindPartners, this strategy is relevant when it introduces a new software package, as the firm could be in doubt of the demand for the product and thus making sure that the price it charges is high enough. By observing that consumers gradually do not purchase as much of the new software package, WindPartners will slowly decrease the price and thereby earn higher profits from the consumers that have a lower willingness to pay.

In conclusion, WindPartners has several options of pricing strategies, including third-‐degree price discrimination, tying, prestige pricing and skimming.

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Question 10)

This question explains a strategic market analysis for WindPartners, including the reactions of their competitors, thus concerning strategic behavior and game theory.

Strategic behavior is the act that an oligopolist will take after having considered the behavior of its competitors. The reasoning for these strategic moves is that in a market structure with only a few suppliers, the reactions of others are of crucial importance. Earlier it was assumed that WindPartners is an oligopolist, and therefore it is extremely relevant to look at possible future pay-‐offs and evaluate the reaction of competitors.

Game theory refers to the choice of the best strategy in a situation with multiple outcomes in terms of the reaction of the competitor. An oligopolist can exploit the possibilities of game theory by getting a competitive advantage over a rival, or protecting itself from a likely strategic move made by the competitor.

In the case of WindPartners, we observe a situation where both WindPartners and their competitors have the possibility of either maintaining their prices or reducing their prices. If WindPartners decide to reduce their price they gain either 12 or 20 depending on whether the competitors reduce or maintain their prices respectively. If WindPartners decide to maintain their price they gain either 4 or 16 depending on whether the competitors reduce or maintain their prices respectively. This means that WindPartners will reduce its price independently of what its competitors do, and is therefore said to have a dominant strategy of reducing prices. A dominant strategy is the best choice for the company, independently of the reaction of the rival. Furthermore, it can be stated that the dominant strategies of the competitors are to reduce prices as well.

Besides the dominant strategy WindPartners and its competitors are in a situation where they could both do better and thus capture higher future pay-‐offs. Since they both have a dominant strategy of reducing their prices, they will only earn 12 each, but if instead both had decided to maintain their prices they could each have earned 16. This situation is known as the prisoners’ dilemma, where each player could have earned higher profits by cooperating, but instead they follow their own interests and thereby lose profits.

Conclusively, WindPartners and its competitors are in a situation where they are both facing a dominant strategy of reducing prices, and additionally they are caught in a prisoners’ dilemma.

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Question 11)

This question concerns production theory in the short run, since capital is fixed.

First, we will briefly define the terms TPL, APL and MPL.

The total product of labor (TPL) indicates how many units of output we can produce with different units of labor. It is important to remember that capital is held constant. The marginal product of labor (MPL) is the change in total output per change in labor used, while the average product of labor (APL) indicates how much output every labor unit can produce on average.

In order to calculate APL and MPL we use the following equations in an excel sheet.

𝐴𝑃! = !"! 𝑀𝑃! =

∆!"∆!

The graph depicting TPL is shown below

Number of Staff (L)

Problems Solved (TP)

AP(L) MP(L)

1 4 4 2 8 4 4 3 16 5,33 8 4 26 6,5 10 5 32 6,4 6 6 36 6 4 7 39 5,57 3 8 42 5,25 3 9 44 4,89 2

0

10

20

30

40

50

0 2 4 6 8 10

Total outpu

t of p

rodu

ct

Quan6ty of Labor



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The graph depicting APL and MPL is shown below:

In conclusion, the above calculations show APL, MPL and TPL for WindPartners.

Question 12)

This question concerns the 3 stages of production in the interpretation of the graphs from question 11.

In production theory there are 3 stages of production. These stages are connected to the TPL, MPL and APL.

In the first stage, APL, MPL and TPL start by rising. MPL will then reach its maximum, and start to decline. At this point the law of diminishing returns is reflected. This means that as we use additional units of labor, our output level will continue to rise, but with a smaller percentage than before this point.

The second stage begins, where the MPL curve intersects the APL curve from above, which is also the point where APL is maximized. In this stage, both the APL and MPL curves will decline. The TPL curve will still increase, but with even more diminishing returns. Stage 2 continues until MPL is 0, which is also the point where TPL is maximized.

Stage 3 begins after the point where MPL reaches 0, and thereby becomes negative. A rational producer will never produce in this stage, as TPL will now start declining, meaning that every additional unit of labor will yield declining output, so that greater units of output could be produced with less units of labor.

0

2

4

6

8

10

12

0 2 4 6 8 10

Total outpu

t of p

rodu

ct

Quan6ty of Labor

AP(L) and MP(L)

AP(L)

MP(L)

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From the above given it is clear that the producer will want to produce in stage 2.

The graphs below show the relationship between TPL, APL and MPL in the case of WindPartners. In these graphs the law of diminishing returns is shown as TPL increases at a decreasing rate. It is clear, that the maximum TPL is soon reached as MPL is declining towards 0 and TPL reaches its maximum when MPL equals 0.

The stages of production are evident in the graphs and are shown by the rigid line. However, stage 3 cannot be depictured in the graph with the information available, as stage 3 of labor occurs when the MPL curve has intersected the first axis. Stage 2 is only half present as it begins when MPL intersects APL from above at approximately 4 units of labor and until MPL intersects the first axis. Since the data given does not inform where MPL equals zero, there is no way of knowing when stage 2 of production ends or when TPL is maximized. Stage 1 is fully depicted from the beginning of the graphs and until MPL intersects APL from above.

Thereby the above graphs have been interpreted.

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Question 13)

This question regards the optimal number of staff.

As shown in the graphs from the previous question WindPartners can keep hiring staff as we have not yet reached the maximum TPL so the firm will still benefit from employing additional workers to its Customer Assistance Unit. However we do not have any information about cost, so we are not able to recommend a certain number of employees. All we know is that to gain the most output, WindPatners should hire as to stay put in stage 2 of production.

Question 14)

This question deals with isoquants, since it comprehends the relation between two input factors.

First, it can be stated that this question is related to production theory in the long run, since both input factors are variable. An isoquant shows the various combinations of the two input factors, which the firm can use to produce a specific level of output. Since inputs are not free, the firm would only want to produce in the economic region of the isoquant. The economic region is determined by the negatively sloped portion of the isoquant, which is encircled by ridge lines. This furthermore relates to stage 2 of production, where MPL is declining but positive. Producers will never want to operate outside this region, since it in this positively sloped part of the isoquant could produce the same level of output with less capital and less labor, thereby lowering its costs.

The absolute value of the slope of the isoquant is called the marginal rate of technical substitution. For a movement down along an isoquant, the marginal rate of technical substitution of labour for capital is given by

LKMRTS

ΔΔ−

=

We then multiply by -‐1 in order to express the MRTS as a positive number. Consequently, if the firm wants to reduce the quantity of capital that it uses in production, while still remaining on the same isoquant, it must increase the quantity of labour.

Relating the theory to the case, we use excel to make a power regression for the isoquant Q = 26. Hence, we get a function describing capital use, K, as a function of labor use, L. This is shown in the graph below. We have chosen power regression, as an isoquant will never intersect with the first or second axis, which a power function will also never do.

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Thus, the function can be written as:

𝐾 = 4𝐿!!

Conclusively, the above function combines the given values of K and L.

Question 15)

This question concerns the optimal combination of capital and labor.

In the previous question, isoquants were described as different combinations of input that gives the same output. In order to find the optimal combination of capital and labor, isoquants will be used along with the concept of isocost lines. An isocost line shows the different combinations of input that gives the same costs. Therefore the relationship between isoquants and isocost lines can be used to obtain the optimal input combination for minimizing costs or maximizing output. The optimal input combination is found where the isoquant is tangent to the related isocost line. The slope of the isocost line corresponds to the ratio of input prices (wage and rent), and by making MRTS tangent to the isocost line, we find the optimal combination of inputs:

rwMRTS =

This equation can be used in the case of WindPartners to find the price ratio as the numbers for price of labor PL and price of capital PK are given.

𝑤𝑟 =

100150 =

23

y = 4x-‐1 R² = 1

0 0,5 1

1,5 2

2,5 3

3,5 4

4,5

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5

Capital

Labor

L and K

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At the optimal combination of K and L, MRTS should then equal !!, which is the slope of the isocost

line.

Using the function we derived in question 14, we calculate further combinations of K and L, by inserting other values of L into the equation than those given in the case. We then calculate the

corresponding values of MRTS using the equation 𝑀𝑅𝑇𝑆 = !∆!∆!

to determine the optimal

combination of K and L.

From the table above we can see, that the optimal combination of K and L is L = 2 and K = 2 as the MRTS equals the slope of the isocost line at this point.

Question 16)

This question explains what is happening when the price of labor is decreased and thereby how it affects the optimal combination of inputs.

As the price of labor has changed to 50, we have calculated a new price ratio.

𝑤𝑟 =

50150 =

13

We then compare the new price ratio to the table from question 15, to see which combination of K and L has the MRTS with the same value as the new price ratio.

L K MRTS

5 0,8

4 1 0,2

3 1,333333 0,333333

2 2 0,666667

1 4 2

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L K MRTS

5 0,8

4 1 0,2

3 1,333333 0,333333

2 2 0,666667

1 4 2

Here the optimal combination of capital and labor is L = 3 and K = 1.33333. Thus, when the price of labor is decreased WindPartners will hire more labor and rent less capital than before, as to lower its costs. This can also be seen in the graph below, which shows the intersection of the isoquant, which yields an output of 26, with the different isocost lines that reflects the two costs of labor.

In conclusion, when there is a decrease in the price of labor then capital is substituted for labor, why the isocost line becomes less steep.

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Question 17)

This question concerns the applied discount rate that fits the situation of WindPartners.

First and foremost the theory of discounting will be explained as this is important background knowledge for understanding the use of the discount rate.

Discounting is the principle of translating all future net cash flows back to present time. These cash flows must be made comparable, which can be done by applying a discount rate.

The discount rate can be seen as the cost of capital which is defined as the opportunity costs of using money for the opportunity at hand. The discount rate is a fixed reflection of positive time preference, risk, inflation, market rate and taxes. The discount rate consists of two elements which are the cost of debt and the cost of equity. The cost of debt is set by the market, while the cost of equity is the opportunity cost that shareholders forego by investing in the project in progress. To find the discount rate, we calculate the weighted average cost of capital (WACC) which can be found by using the following equation:

In the case of WindPartners, we assume that equity will constitute for 60 % and debt will constitute for 40 %. Since we know that cost of equity must be higher than cost of debt, we assume that the cost of equity is 20 % and the cost of debt is 10 %. We have assumed that the cost of debt is lower than the cost of equity since financial institutes will have security in the capital stock of the firm. The cost of equity has to be higher than the cost of debt since there is no security in the capital stock and thus a higher risk premium is needed.

Now we can apply the assumed numbers in the equation for finding WACC.

𝑊𝐴𝐶𝐶 = 60 % ∙ 20 % 𝑝.𝑎.+40 % ∙ 10 % 𝑝.𝑎.= 14 % 𝑝.𝑎.

In conclusion the proposed discount rate is 14 % p.a.

Question 18)

This question consists of a financial evaluation, a possible scenario analysis and a real option analysis in the case of WindPartners.

Prior to the evaluation, we will begin by assuming that no taxes and no inflation prevail in the case. Besides we assume that this is a single investment, and thus we can observe the net present value (NPV), the internal rate of return (IRR) and the annuity in order to determine whether or not the investment is profitable.

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Firstly, we will determine NPV, which must be positive, or else the investment is not profitable. NPV can be found by using the following equation or by using excel (see appendix 1):

𝑁𝑃𝑉! = 𝐶𝐹! ∙ (1+ 𝐼)!!!

!!!

NPV0= €1,570,049.1

Thus NPV was found to be positive at €1,570,049.1 and thereby the investment is solid.

Secondly, we will determine IRR, which is the internal rate of return that yields a NPV equal to 0. IRR has to be higher than the applied discount rate which was earlier determined at 14 %, otherwise the investment will not be profitable.

IRR can be found by using the following equation or by using excel, as this assignment has done (see appendix 1):

𝑁𝑃𝑉! = 0 = 𝐶𝐹! ∙ (1+ 𝐼𝑅𝑅)!!!!!

IRR= 22 %

Thus IRR was found to be greater than the proposed discount rate, and thereby the investment is solid.

Thirdly, we will determine the annuity, which is cash flows consisting of equal amounts payable at fixed time intervals. Annuity has to be a positive number in order for the investment to be profitable.

Annuity can be found by using the following equation or by using excel, as this assignment has done (see appendix 1):

𝐴 =𝑁𝑃𝑉!

𝑃𝑉𝐼𝐹𝐴!,!= 𝑁𝑃𝑉! ∙ 𝑃𝑉𝐼𝐹𝐴!,!!!

Annuity= €300,999.671

Thus the annuity was found to be positive, and thereby the investment is solid.

Finally it can be concluded that this investment is profitable as NPV is positive, IRR is greater than the applied discount rate and the annuity is positive.

Now we will turn to the possible scenario analysis, which will be done by observing a Worst and Best-‐case scenario.

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A Worst and Best-‐case scenario can be described as a method that relates to uncertainty, with a starting point in the investment’s base case. Since the base case is already calculated above, we make a conservative estimate that there is a 50 % probability of this scenario to occur.

We will now determine the scenario of the best and the worst case. As WindPartners is the market leader, and thus has a large market share, it is not likely that the best case change in the percentage of sales will exceed 10 %. In addition this assignment assumes that the probability of this scenario is 30 % because WindPartners has the most comprehensive and user-‐friendly software package. The assumption behind the worst case scenario is determined by the fact that wind turbines are a complementary product to WindPartners’ software packages. This relationship makes WindPartners exposed to changes in the demand for wind turbines, which is in close competition with solar energy, why WindPartners will suffer a loss in sales if solar energy gains ground in the clean energy industry. Thus we assume that the worst-‐case percentage change in sales is 20 % and is likely to occur with a probability of 20 %. The reasoning for not setting the probability higher is that WindPartners is the market leader and have a twenty year experience in the industry, thereby having a large consumer base.

In the below table an overview of the Worst and Best-‐case scenario is stated:

Change in % of sales Probability Acc. NPV IRR Annuity

Best Case + 10 % 30 % 2,259,700.22 25 % 433,215.1305

Base Case 0 % 50 % 1,570,049.1 22 % 300,999.671

Worst Case -‐ 20 % 20 % 190,746.84 15 % 36,568.7521

After the scenario analysis the total NPV can be calculated as to decide whether or not the investment should be made, although two uncertainty scenarios are included.

NPV0=€190,746.84·∙20 % + €1,570,049.1·∙50 % + €2,259,700.22·∙30 %

NPV0=€1,501,083.981

Thus the investment is still profitable even if a worst case scenario is included.

Another way of managing the risks of an investment is through the use of real options. Real options refer to a link between strategic planning and investment theory, allowing the firm to reduce its risk by investing capital in a sequence of independent stages. By dividing the investments into stages the firm has the possibility to improve its use of the collected information. This minimizes the risk of going onwards with the investment. The firm should only continue investing in the next stage if the investment has fulfilled the goals of the firm at the previous stage.

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A beneficial way of evaluating an investment is by using a decision tree. WindPartners wishes to launch the new software package next year why it is not possible to postpone the investment as to gather more information and thereby reducing the risk. Thus we assume that WindPartners will invest in a Pilot Project. When doing a pilot project, one estimates a good case, an acceptable case and a bad case. From there, NPV and probability of worst and best case is calculated in order to determine whether the investment is profitable.

Due to the limited amount of information, this assignment has assumed the numbers presented in the decision tree below. First and foremost we have assumed that the overall investment cannot exceed €2,500,000 since this is the given limit in the case. With foundation in the base case we have assumed a qualified estimate of the below NPV0 values and probabilities.

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Hence, we can calculate the accumulated NPV0 of the real options scenario.

𝑁𝑃𝑉!!!!! = −1,000,000

+ 65% ∙ (−2,000,000 + 4,000,000 ∙ 85 % + 750,000 ∙ 15% ) ∙ 1,14!!

+ 30% ∙ (−1,000,000 + 2,000,000 ∙ 85 % + 750,000 ∙ 15% ) ∙ 1,14!!

+ 10 % ∙ 0

= €76206,1

Since the total accumulated NPV0 is positive, the investment is profitable and should be undertaken.

Finally, after having made a financial evaluation, a Worst and Best-‐case scenario and real options scenario we can conclude that, even when taking uncertainty into account, the investment should still be pursued, since it is profitable.

Question 19)

When perceiving the long run, which relates to investments, the time horizon implies strategic uncertainty. The longer the span of the investment project, the less certain the results of the connected probabilities become, since a long time horizon implies a greater range of outcomes.

The concept of critical values can be used to determine the uncertainty as well as the sensitivity of a given investment project, why it provides the firm with information that reduces the risks of the investment at hand. It determines the amount at which an input value in the investment analysis can deviate before the investment becomes unattractive for the firm. As to find the critical values, the firm must start out by defining the net present value, which it will accept as the minimum present value of future cash flows related to the project. The found critical values facilitate a better possibility for the firm to perceive the risks of the investment project and the possibility that actual business results will deviate from the calculated values used for estimating the investment analysis.

In consequence the critical values can be found by using the below equation or by using excel, as this assignment has done (see appendix 2 and 3):

𝑁𝑃𝑉! = 𝑓(𝐶𝐹, 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑓𝑎𝑐𝑡𝑜𝑟)! ∙ (1+ 𝐼)!!

!

!!!

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Our calculations from excel can be shown in the tables below.

Table 1 describes the exact amount of units sold in order to achieve a NPV of 0.

Table 1

Year Estimated Units Sold Critical Value Critical Value %

1 50 39 22.77 %

2 100 77 22.77 %

3 300 232 22.77 %

4 500 386 22.77 %

5 600 463 22.77 %

6 500 386 22.77 %

7 400 309 22.77 %

8 300 232 22.77 %

9 100 77 22.77 %

10 50 39 22.77 %

Table 2 describes the exact price of the product which yields a NPV of 0

Table 2

Year Estimated Price Critical Value Critical Value %

1 4000 3089.37 22.77 %

2 4000 3089.37 22.77 %

3 5000 3861.71 22.77 %

4 5000 3861.71 22.77 %

5 5000 3861.71 22.77 %

6 5000 3861.71 22.77 %

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7 4000 3089.37 22.77 %

8 4000 3089.37 22.77 %

9 4000 3089.37 22.77 %

10 3000 2317.03 22.77 %

In conclusion, if the quantity or price decreases by more than 22.77 % WindPartners should not proceed with the investment.

Question 20)

This question shows the cash flows of the annuity loan over a 7 year period.

In the case of WindPartners we are considering an annuity loan, which is a loan with equal payment of principal and interest. This type of loan is one of the easiest to manage, since the payments per period are always equal.

The cash flows of the annuity loan are calculated in excel, which is shown in appendix 4.

The below illustration shows the installment and interest schedule for the annuity loan:

0,00

20000,00

40000,00

60000,00

80000,00

100000,00

120000,00

140000,00

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Installment

Interest

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Conclusively, the illustration shows that in the beginning of the loan period a larger amount of the quarterly payment ratio consists of interest. However, the more payments WindPartners make, the share of interest will decrease as the share of installment increases.

Question 21)

This question concerns the effective cost of debt.

The cost of capital is a key aspect when evaluating the financing of an investment. Thus, it becomes important to perceive the effective cost of debt, which is the actual interest rate that is paid as to maintain the debt. In connection to this, the equation below explains the effective annual interest rate.

𝑁𝑒𝑡𝑃𝑟𝑜𝑐𝑒𝑒𝑑𝑠! = 𝐼𝑛𝑠𝑡𝑎𝑙𝑙𝑚𝑒𝑛𝑡! ∙ (1+ 𝑅)!!!

!!!

On the left hand side of the equation, the term of net proceeds refers to the amount that the debtor of the loan in fact receives as a cash disbursement. Thus, the future payments on the loan contain the actual loan repayments as well as interest, fees and administration costs, etc. Furthermore, the equation specify that the effective annual interest of debt is the precise interest rate which creates an equilibrium between the discounted values of future installment payment and the loan’s net proceeds.

In the case of WindPartners, we calculate the IRR of the loan’s cash flows in excel, since the investment’s IRR is similar to the effective cost of debt. We thereby obtain a quarterly effective interest rate of 4.47 %. As to find the effective annual interest rate we simply multiply the quarterly effective interest rate by 4, and obtain 17.88 %. Comparing this to the nominal interest rate of 15 % p.a. it can be observed that WindPartners in effect pay 2.88 percentage points more than the nominal interest rate.

In conclusion, the effective annual interest rate is 17.88 %.

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Question 22)

This question concerns other issues of consideration with regards to the loan in question.

When taking a loan there are several aspects to consider in addition to the effective annual interest rate.

Firstly, the ratio of financing must be taking into consideration as the relationship between debt and equity will affect the discount rate, and thereby the profit that the firm is able to obtain. If the ratio between debt and equity is decreased, then the financial risk will decrease as well, thereby leading to a decrease in the risk premium. If on the other hand the ratio is increased the company will be higher leveraged and thus increase financial gearing.

Secondly, as WindPartners is a global player, obtaining the loan in another currency could be a possibility why WindPartners should be aware of fluctuations in the concerned currency. Fluctuations in the currency could occur for several reasons such as financial disturbances and political instability. These fluctuations could affect the concrete payback amount and thereby differ from the expected payback amount.

Thirdly, when deciding whether or not to obtain the loan, the company must consider its horizontal balance structure, which refers to comparison of cash flow streams. By comparing cash flow streams of assets and financing it is possible to secure that resources are matched in the most optimal way and thereby ensuring better liquidity.

Fourthly, the stability or rating of the bank from which the firm wants to obtain a loan is of rather great importance. The financial situation of the bank will first and foremost affect the applied interest rate, which then surely affects the cost of the concerned loan. This aspect includes the loan terms that the bank offers, which is important to consider.

Lastly the type of loan that the firm chooses to engage in is of crucial importance. In the case of WindPartners we are considering an annuity loan, which is more beneficial since the investment is stable, meaning that the payments are even throughout the period. This type of loan is one of the easiest to manage, since the payments per period are equal, but still it is not a very flexible type of loan for the same reason. Other types of loans which WindPartners could consider include: bullet loans, serial loans and loans with grace periods.

Conclusively, there are several factors to be identified when considering whether to obtain a loan. Additionally the type of loan that the firm chooses is of great importance as well.

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