University of Toronto (STG), Department of Economics ECO ...€¦ · In the 3rd group presentation...

1 © ECO 204 2017-2018 TEST 3 SOLUTIONS

-University of Toronto (STG), Department of Economics ECO 204 Microeconomic Theory for Commerce 2017-2018

Ajaz’s Sections Test 3 Solutions

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TEST DETAILS

Duration: Two hours (7 – 9 pm)

Total number of questions: 10

Total number of pages: 20

Maximum number of points: 280

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QUESTION 1 [TOTAL 20 Points] The following table lists the top 5 daily global “wideners” and the top 5 daily global “tighteners” (as of February 6th 2018):

Name Spread in basis points (5 year CDS) % Change (1 Day)

Brazil 153.54 -3.11%

Chile 45.67 -3.18%

Colombia 94.77 -1.67%

Indonesia 86.32 2.87%

Malaysia 62.76 3.27%

Mexico 102.34 -1.49%

New Zealand 15.68 11.30%

Peru 68.35 -1.34%

Philippines 63.64 2.84%

Spain 47.05 3.02%

Source: Wall Street Journal, February 7th, 2018

(a) [TOTAL 10 Points] On February 6th 2018, which country was the top daily widener (i.e. which country’s risk increased the most) and which country was the top daily tightener (i.e. which country’s risk decreased the most)? Explain your answer in one/two sentences. Answer The “spread” is the CDS insurance premium. New Zealand with the largest positive % change is the top daily widener while Chile with the largest negative % change was the top daily tightener. (b) [TOTAL 10 Points] What is the riskiest country’s probability of default? State salient assumptions and show essential calculations. Answer Assuming no risk of recovery in case of default:

𝑃(𝐷𝑒𝑓𝑎𝑢𝑙𝑡) = 𝑆𝑝𝑟𝑒𝑎𝑑 ∗1

1001%

Brazil with the largest “spread” is the riskiest country:

𝑃(Brazil will Default) = 153.54 ∗1

1000.01 = 0.015354

http://www.wsj.com/mdc/public/npage/2_3023_creditdervs.html?mod=topnav_2_3022


QUESTION 2 [TOTAL 15 Points] In the 2nd group presentation (“Stress Test of Financial Portfolios”), all groups (whether or not presenting) were expected to “stress test” a financial portfolio. Every single group that presented this case used McSim to “stress test” their respective financial portfolio by running at least 10,000 rounds of simulations (each round corresponded to one month). In reality, an average investor’s life span is (say) 800 months. Why then did every group conduct at least 10,000 rounds of simulations when in reality the client is going to live for at most 800 months? Explain in one/two sentences. Answer 800 draws would give us a limited and “biased” picture of almost infinite number of possible permutations/combinations. To get a sense of how the portfolio would perform in real life and paint a more complete picture, it is better to have a really “high number” of simulations.


QUESTION 3 [TOTAL 25 Points] In the 3rd group presentation (“Decision Making Under Uncertainty: Drill or Not Drill for Oil”), all groups (whether or not presenting) were expected to solve a decision making problem on whether to drill for oil vs. not drill for oil. The 3rd group presentation was based on the “Gold Claims at Sturgeon Lake” case discussed in the last week of the Fall 2017 semester.

(a) [TOTAL 10 Points] In both the “gold” and “oil” cases, the probability of success 𝑃(𝑆) was uncertain: it could take any value (with equal likelihood) between a “min” and a “max” value. Write down a formula (algebraic or Excel) for how a

computer should select a particular value of 𝑃(𝑆) in Monte Carlo simulations such that min ≤ 𝑃(𝑆) ≤ max. Answer The probability of success is between a lower and an upper bound. To select a probability between the “min” and the “max” we could do:

𝑃(𝑆) = min+ (max−min) ∗ (a # smaller than 1)

Or

𝑃(𝑆) = min+ (max−min) ∗ rand()

From the Gold case, remember that 𝑟𝑎𝑛𝑑( ) generates a value between 0 and 1. (b) [TOTAL 15 Points] Explain how we forecasted gold (or oil) prices. Be sure to provide the rationale behind the forecasting method and the forecasting “algorithm”. Answer You can’t assume that prices have a normal distribution because prices can’t be negative. Instead, we assume that the growth rate of prices have a normal distribution. For simulations, collect historical prices and assume:

growth rate ~ 𝑁(average growth rate, standard deviation of growth rates)

𝑔 ~ 𝑁(��, 𝑠𝑔) To forecast prices:

𝑃𝑡+1 = 𝑃𝑡(1 + randomly chosen growth rate)

On each round, the growth rate is chosen by first generating a random number between 0 and 1 (call it 𝑝) and then having

Excel give a particular growth rate 𝑔∗ such that:

𝑃(𝑔 ≤ 𝑔∗) = 𝑝 In Excel:

𝑃𝑡+1 = 𝑃𝑡 (1 + norm. inv(𝑝, ��, 𝑠𝑔))

The rationale for this algorithm is that it generates “volatile” price forecasts.


QUESTION 4 [TOTAL 25 Points] In the “Futures and Options” portion of the lectures on “Uncertainty”, we studied the case of a California farmer who wanted to purchase a put option to hedge against the risk of “CA Mailbox Price” falling below $12.50. We learned that (at that time) the Chicago Mercantile Exchange (CME), did not sell option contracts on CA Mailbox Prices. However, the CME did sell options contracts on Class III milk, Class IV milk, Butter, and Non-Fat Dairy Milk (NFDM). (a) [TOTAL 5 Points] Use the following table to explain (in one sentence) why the California farmer should hedge against Mailbox price falling below $12.50 by purchasing a put option on Class III milk:

Answer Because, of all the dairy products listed, Class III prices has the highest correlation with Mailbox prices.


(b) [TOTAL 20 Points] Suppose we regress Class III prices on Mailbox Prices:

Class III Price = −1.56 + 1.18 Mailbox Price

Given that the CME trades options in increments of $0.25, what is the strike price of a Class III put option such that it has a 50% chance of being in the money whenever Mailbox Prices dip below $12.50? State all salient assumptions and show all essential calculations. Answer When the Mailbox Prices is $12.50, the predicted Class III price is:

Class III Price = −1.56 + 1.18 ∗ 12.50 = 13.19

This means that when the mailbox price is $12.50, then 50% of the time, the actual Class III price will be above $13.19 and 50% of the time, the actual Class III price will be less than or equal to $13.19:

Class III Price

Mailbox Price$12.50

$13.19

50%

Thus, a put option of $13.25 on Class III milk will have an approximately 50% chance of “being in the money” (remember a

put option is in the money when the actual price is ≤ strike price (assuming a negligible cost for the put option).


QUESTION 5 [TOTAL 60 Points] (a) [TOTAL 10 Points] Which of the following “input expansion paths” can be used to characterize a company’s “returns to scale” (explain your answer in one/two sentences)? Based on your answer, what are the general properties of “input expansion paths” that can be used to characterize a company’s “returns to scale”?

Answer To explore returns to scale, we need to examine what happens to output as we (say) double all inputs. The only input expansion path which allows us to always double all inputs is the one on the bottom right. In general, returns to scale can be

gauged only along linear expansion paths with positive, finite 𝐾

𝐿 ratios.


(b) [TOTAL 15 Points] Suppose that a company produces output according to the Cobb-Douglas production function 𝑄 =

𝐴 𝛼 𝛽. When will such a company exhibit increasing, or constant, or decreasing returns to scale? Please show essential calculations. Answer The output with an arbitrary input mix is:

𝑄( , ) = 𝐴 𝛼 𝛽 The output with twice as many inputs as before is:

𝑄(2 , 2 ) = 𝐴(2 )𝛼(2 )𝛽 = 2𝛼+𝛽𝑄( , ) Thus:

Doubling inputs always more than doubles output whenever 𝛼 + 𝛽 > 1 (increasing returns to scale).

Doubling inputs always doubles output whenever 𝛼 + 𝛽 = 1 (constant returns to scale).

Doubling inputs always less than doubles output whenever 𝛼 + 𝛽 < 1 (decreasing returns to scale).


(c) [TOTAL 20 Points] The following table shows estimates of 𝛼 and 𝛽 for various US sectors (assuming that each sector

produces output according to the Cobb-Douglas production function 𝑄 = 𝐴 𝛼 𝛽):

𝑄 = 𝐴 𝛼 𝛽

SECTOR 𝛼 𝛽 𝛼 + 𝛽

TOBACCO PRODUCTS 0.18 0.33 0.51

FOOD AND KINDRED PRODUCTS 0.43 0.48 0.91

TRANSPORTATION EQUIPMENT 0.44 0.48 0.92

APPAREL AND OTHER TEXTILES 0.70 0.31 1.01

FURNITURE AND FIXTURES 0.62 0.40 1.02

ELECTRONICS AND ELECTRIC EQUIPMENT 0.49 0.53 1.02

PAPER AND ALLIED PRODUCTS 0.44 0.65 1.09

PETROLEUM AND COAL PRODUCTS 0.30 0.88 1.18

PRIMARY METAL 0.51 0.73 1.24

FIGURES ARE FOR SELECT US SECTORS (2014)

If your student ID # ends in 0, 1, 3, 5, 7, or 9: choose any sector with decreasing returns to scale.

If your student ID # ends in 2, 4, 6, or 8: choose any sector with increasing returns to scale.

Denote the price of using labor by 𝑃𝐿 and the price of using capital by 𝑃𝐾. Derive expressions for the optimal (cost-minimizing) mix of inputs that solve the long run CMP:

min𝐿,𝐾

𝐶 = 𝑃𝐿 + 𝑃𝐾 𝑠. 𝑡. 𝐴 𝛼 𝛽 = 𝑞 where , ≥ 0 and 𝑞 > 0

Please show all essential calculations. Answer

Regardless of which sector you choose, the optimal mix of inputs is found by solving the long run CMP:

minAll Inputs

𝐶 = 𝑃𝐿 + 𝑃𝐾 𝑠. 𝑡. 𝐴 𝛼 𝛽⏟ 𝑄

= 𝑞⏟>0

, ≥ 0, ≥ 0

Borrowing intuition from consumer theory, we know the optimal mix of inputs is where the iso-quant curve is tangent to the iso-cost line:

Now suppose this is the target output level 𝑞 > 0

Optimal inputs to produce target output level 𝑞 > 0


Now, the slope of the iso-quant curve must be equal to the slope of the iso-cost line so that:

𝑇𝑅𝑆 = −𝑃𝐿𝑃𝐾

−𝑀𝑃

𝑀𝑃 = −

𝑃𝐿𝑃𝐾

𝛼𝐴 𝛼−1 𝛽

𝛽𝐴 𝛼 𝛽−1 = 𝑃𝐿𝑃𝐾

𝛼

𝛽 = 𝑃𝐿𝑃𝐾

At the same time, we require that:

𝐴 𝛼 𝛽⏟ 𝑄

= 𝑞⏟>0

We solve these two simultaneous equations:

𝛼

𝛽 = 𝑃𝐿𝑃𝐾 and 𝑞 = 𝐴 𝛼 𝛽

(Here is one way to solve these): From:

𝛼

𝛽 = 𝑃𝐿𝑃𝐾→ =

𝛼

𝛽

𝑃𝐾𝑃𝐿

Sub in 𝑞 = 𝐴 𝛼 𝛽:

𝑞 = 𝐴 𝛼 𝛽 = 𝐴 (𝛼

𝛽

𝑃𝐾𝑃𝐿 )

𝛼

𝛽 = 𝐴 (𝛼

𝛽

𝑃𝐾𝑃𝐿 )𝛼

𝛼+𝛽

𝐴 (𝛼

𝛽

𝑃𝐾𝑃𝐿 )𝛼

𝛼+𝛽 = 𝑞 → 𝛼+𝛽 =𝑞

𝐴 (𝛼𝛽𝑃𝐾𝑃𝐿 )𝛼 =

𝑞

𝐴 [𝛽

𝛼

𝑃𝐿𝑃𝐾]𝛼

𝛼+𝛽 =𝑞

𝐴 [𝛽

𝛼

𝑃𝐿𝑃𝐾]𝛼

∗ = (𝑞

𝐴)

1𝛼+𝛽

[𝛽

𝛼

𝑃𝐿𝑃𝐾]

𝛼𝛼+𝛽

= Ψ𝐾𝑞1

𝛼+𝛽

Subbing into =𝛼

𝛽

𝑃𝐾

𝑃𝐿 yields:


∗ = (𝑞

𝐴)

1𝛼+𝛽

[𝛼

𝛽

𝑃𝐾𝑃𝐿]

𝛽𝛼+𝛽

= Ψ𝐿𝑞1

𝛼+𝛽


(d) [TOTAL 15 Points] Use your answer to parts (a) – (c) to answer this question: [If your student ID # ends in 0, 1, 3, 5, 7, or 9] True or false: your sector always exhibits “diseconomies of scale” (for

example, it’s impossible for your sector to have a U-shaped 𝐴𝐶 curve). If “true” explain your answer; if “false”, explain your answer by a graph showing how a sector with decreasing returns to scale can have economies of scale. [If your student ID # ends in 2, 4, 6, or 8] True or false: your sector always exhibits “economies of scale” (for example, it’s

impossible for your sector to have a U-shaped 𝐴𝐶 curve). If “true” explain your answer; if “false”, explain your answer by a graph showing how a sector with increasing returns to scale can have diseconomies of scale. Answer

The coordinates of the optimal input expansion path are:

( ∗, ∗) = (Ψ𝐿𝑞1

𝛼+𝛽 , Ψ𝐾𝑞1

𝛼+𝛽)

Notice that along the optimal input expansion path:

=Ψ𝐾𝑞

1𝛼+𝛽

Ψ𝐿𝑞1

𝛼+𝛽

= constant > 0

The optimal input expansion path is a linear expansion path with positive, finite 𝐾

𝐿 ratio. Thus for a Cobb-Douglas

production function:

Increasing returns to scale ⇔ Economies of Scale (𝑖. 𝑒. cannot have a U − shaped 𝐴𝐶 curve)

Decreasing returns to scale ⇔ Disconomies of Scale (𝑖. 𝑒. cannot have a U − shaped 𝐴𝐶 curve)

Thus, true.


QUESTION 6 [TOTAL 25 Points]

The following graph shows the 𝐴𝐶 ($

1,000 𝐾𝑊𝐻) vs. output (Billions of KWH) of electricity power generators (Source:

Christensen & Greene, Economies of Scale in U.S. Electric Power Generation, Journal of Political Economy, 1976):

(a) [TOTAL 10 Points] In one/two sentences, explain the size distribution of firms (in particular, why very few firms operate

at 𝑀𝐸𝑆).

Answer Most companies operate below the MES because there’s insufficient demand to justify building a plant at MES.

http://www.jstor.org/stable/pdfplus/1831326.pdf


(b) [TOTAL 15 Points] Assume electricity is generated using labor and capital according to the Cobb-Douglas production

function 𝑄 = 𝐴 𝛼 𝛽. True or false: the graph above shows that:

𝛼 + 𝛽 {> 1 from 0 to ≈ 30 billion KWH

1 at ≈ 30 billion KWH< 1 from ≈ 30 billion KWH and above

If “true”: explain your answer (be sure to explain the link between 𝛼 + 𝛽 and “Returns to Scale”). If “false”: offer an alternative explanation. Answer The following statement:

𝛼 + 𝛽 {> 1 from 0 to ≈ 30 billion KWH

1 at ≈ 30 billion KWH< 1 from ≈ 30 billion KWH and above

Is equivalent to stating that:

𝑅𝑒𝑡𝑢𝑟𝑛𝑠 𝑡𝑜 𝑆𝑐𝑎𝑙𝑒 {Increasing from 0 to ≈ 30 billion KWH

Constant at ≈ 30 billion KWHDecreasing from ≈ 30 billion KWH and above

This is impossible because returns to scale is a global property. What explains the U-shaped AC curve? There are economies of scale (arising for example because initiually input prices decline with output) followed by diseconomies of scale (arising for example because eventually input prices rise with output).


QUESTION 7 [TOTAL 20 Points] In one-two paragraphs, explain why in recent years, beer companies have acquired US (but not European) wineries/vineyards.

Answer US wineries use steel tanks to ferment grapes – since these tanks can be used to ferment beer and wine, this shared input is one of the elements of the economies of scope. The others are that it’s cheaper for one company to purchase bottles, distribute and market products than for two companies.

Land Vines Crushing Fermentation Bottling Distribution Marketing

Land Hops … Fermentation Bottling Distribution Marketing

Beer Production Process


QUESTION 8 [TOTAL 35 Points] (a) [TOTAL 15 Points] Explain how, between 1995 and 1998, Dell Computers reduced its inventory period from (approximately) 32 days to 7 days.

Answer Please see the following self explanatory diagrams (students may answer this question without the diagrams):

1. Order

2. Order

PC factory

Monitor factory

2. Order

3. Ship

3. Ship

4.Call5.

Pick

6. Ship

Before Learning by Doing

1. Order

2. OrderPC factory

Monitor factory

2. Order

Dell reduced inventory period from 32 days in 1995 to 7 days in 1998

Source: Matching Dell, HBS case

After Learning by Doing


(b) [TOTAL 20 Points] True or False: Learning by Doing refers to the idea of getting “better” with experience (“doing”): as such, total cumulative experience is the sole determinant of becoming more efficient over time. Explain your answer in one/two paragraphs. Answer False. The following diagram shows that regardless of total experience, the “pace of activity” also determines efficiency gains from learning by doing:

L-1011

Right scale

Left scale

“Forgetting”

“Slowing down”

Notice that when the production rate slows down (around the 100th aircraft) that production becomes inefficient and when the production rates speeds up (around the 175th aircraft) production becomes efficient.


QUESTION 9 [TOTAL 15 Points] What was the main takeaway (vis a vis a company’s choice of technology) from our discussion of the “Classic Coca-Cola” and “New Coke” mini-case? Explain your answer in one/two paragraphs (and, if you wish, a graph).

Answer That companies must, if possible, use inputs as substitutes. Coke’s original production process and product formulation did not allow for sweeteners, other than sugar, as inputs to soda. Consequently, when sugar prices spiked up, Coke was unable to offset higher prices by using other sweeteners.


QUESTION 10 [TOTAL 40 Points] (a) [TOTAL 15 Points] Prove “actuarially fair” insurance rates. Feel free to assume any market structure for insurance companies. State all salient assumptions and show essential calculations. Answer

Scenario # 1: Suppose the Insurance Industry is Regulated

Each insurance company is required to have sufficient capital to cover expected payout. Insurance premium collected from a policy holder = Expected payout to policy holder

𝜋𝐼 = (1 − 𝑝)0 + 𝑝𝐼

𝝅 = 𝒑

Scenario # 2: Suppose the Insurance Industry is Competitive

Each company’s expected profit (𝐸[Π]) will be zero:

𝐸[Π] = (1 − 𝑝)(Insurance Premium) + 𝑝(Insurance Premium – Payout) = 0

𝐸[Π] = (1 − 𝑝)𝜋𝐼 + 𝑝(𝜋𝐼 − 𝐼) = 0

𝝅 = 𝒑


(b) [TOTAL 25 Points] Assuming “actuarially fair” insurance rates, derive the optimal amount of insurance purchased by a risk averse agent. State all salient assumptions and show essential calculations. Answer

A Risk averse policy holder chooses the optimal 𝐼 by solving her “UMP”:

max𝐼𝐸𝑈 = (1 − 𝑝)𝑈(𝑊 − 𝜋𝐼) + 𝑝𝑈(𝑊 − − 𝜋𝐼 + 𝐼)

Risk averse agents have strictly concave utility functions

max𝐼𝐸𝑈 = (1 − 𝑝)𝑈(𝑊 − 𝜋𝐼) + 𝑝𝑈(𝑊 − − 𝜋𝐼 + 𝐼)

𝑑𝐸𝑈

𝑑𝐼= (1 − 𝑝)𝑈′(𝑊 − 𝜋𝐼)(−𝜋) + 𝑝𝑈′(𝑊 − − 𝜋𝐼 + 𝐼)(−𝜋 + 1) = 0

(1 − 𝑝)𝑈′(𝑊 − 𝜋𝐼)𝜋 = 𝑝𝑈′(𝑊 − − 𝜋𝐼 + 𝐼)(1 − 𝜋)

Since 𝜋 = 𝑝:

𝑈′(𝑊 − 𝑝𝐼) = 𝑈′(𝑊 − − 𝑝𝐼 + 𝐼)

Invoking strict concavity:

𝑊 − 𝑝𝐼 = 𝑊 − − 𝑝𝐼 + 𝐼

𝑰 = 𝑳

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