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ECO 204, 2016-2017 (AJAZ), Test 3

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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)

∎ Department of Economics ∎ ECO 204 ∎ Microeconomic Theory for Commerce ∎ 2016-2017 (Ajaz)

Test 3 Solutions

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EXAM DETAILS

Duration: 2 hours

Total number of questions: 6

Total number of pages: 15 (including title page)

Total number of points: 140

Please answer all questions. To earn credit you must show all calculations.

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ECO 204, 2016-2017 (AJAZ), Test 3


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QUESTION 1 [TOTAL 10 Points] Recall from the Prestige Telephone Company case that:

𝐶𝑝𝑜𝑤𝑒𝑟 = 𝑇𝐹𝐶𝑝𝑜𝑤𝑒𝑟 + 𝑇𝑉𝐶𝑃𝑜𝑤𝑒𝑟 = 179.51 + 4(𝑞𝑖 + 𝑞𝑐 + 𝑞𝑠)

𝐶𝑜𝑝𝑠 = 𝑇𝐹𝐶𝑜𝑝𝑠 + 𝑇𝑉𝐶𝑜𝑝𝑠 = 21,600 + 24(𝑞𝑖 + 𝑞𝑐)

𝑇𝐹𝐶𝑃𝐷𝑆 = 𝑇𝐹𝐶𝑝𝑜𝑤𝑒𝑟⏟

179.51

+ 𝑇𝐹𝐶𝑜𝑝𝑠⏟ 21,600

+ 𝑇𝐹𝐶𝑎𝑙𝑙 𝑜𝑡ℎ𝑒𝑟⏟ 191,200

≈ $212,800

𝑇𝑉𝐶𝑃𝐷𝑆 = 𝑇𝑉𝐶𝑝𝑜𝑤𝑒𝑟 + 𝑇𝑉𝐶𝑜𝑝𝑠 = 28𝑞𝑐 + 28𝑞𝑖 + 4𝑞𝑠

𝐶𝑃𝐷𝑆 = 𝑇𝐹𝐶𝑃𝐷𝑆 + 𝑇𝑉𝐶𝑃𝐷𝑆 ≈ 212,800⏟

𝑇𝐹𝐶𝑃𝐷𝑆

+ 28𝑞𝑐 + 28𝑞𝑖 + 4𝑞𝑠⏟ 𝑇𝑉𝐶𝑃𝐷𝑆

All costs are reported on a monthly basis (in dollars) where 𝑞𝑐 = Commercial hours billed per month, 𝑞𝑖 = Intercompany hours billed per month, and 𝑞𝑠 = “service” hours each month. Denote the amount of the “variable power input” by 𝑃, the amount of the “variable operations input” by 𝐿, and all fixed inputs (including “fixed” power and “fixed” operations) by 𝑘. Suppose that PDS’s short run “commercial output” production function is:

𝑞𝑐 = 𝐴𝑃𝛼𝐿𝛽 𝑘𝛾

Assume 𝛼, 𝛽, 𝛾 > 0. (a) [5 Points] Given the information above, what can we say about the values of the parameters 𝛼, 𝛽 (or any combination thereof)? State all salient assumptions and show all essential calculations. Hint: What is the equation of 𝐶(𝑞𝑐)? Answer: Taking the hint we have:

𝐶(𝑞𝑐) = 𝑇𝐹𝐶𝑐 + 𝑇𝑉𝐶(𝑞𝑐) = 𝑇𝐹𝐶𝑐 + 28𝑞𝑐 Note that the total cost function, or for that matter the total variable cost function is linear. Thus, the underlying production function has “constant returns to variable inputs” in the short run. This means that doubling all variable inputs will double output. The output with an arbitrary level of variable power, variable labor, and fixed capital is:

𝑞𝑐(𝑃, 𝐿, 𝑘) = 𝐴𝑃𝛼𝐿𝛽 𝑘𝛾

The output with double the variable inputs is:

𝑞𝑐(2𝑃, 2𝐿, 𝑘) = 𝐴(2𝑃)𝛼(2𝐿)𝛽 𝑘𝛾 = 2𝛼+𝛽𝐴𝑃𝛼𝐿𝛽 𝑘𝛾 = 2𝛼+𝛽𝑞𝑐(𝑃, 𝐿, 𝑘)

There are constant returns in the short run as long as 𝛼 + 𝛽 = 1.


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(b) [5 Points] Suppose that doubling all “commercial” inputs in the long run always reduces long run 𝐴𝐶 by 50%. What is

the value of 𝛾 in the commercial production function 𝑞𝑐 = 𝐴𝑃𝛼𝐿𝛽 𝐾𝛾? State all salient assumptions and show all

essential calculations. Answer: Doubling all inputs in the long run always doubles long run cost. This means that for the long run 𝐴𝐶 to fall by 50% the long run output must’ve had to quadruple. Now, for what value of 𝛾 will doubling all inputs in the long run quadruple output? In the long run the output with an arbitrary level of variable power, variable labor, and variable capital is:

𝑞𝑐(𝑃, 𝐿, 𝐾) = 𝐴𝑃𝛼𝐿𝛽 𝐾𝛾

The output with double the inputs is:

𝑞𝑐(2𝑃, 2𝐿, 2𝐾) = 𝐴(2𝑃)𝛼(2𝐿)𝛽 (2𝐾)𝛾 = 2𝛼+𝛽+𝛾𝐴𝑃𝛼𝐿𝛽 𝐾𝛾

𝑞𝑐(2𝑃, 2𝐿, 2𝐾) = 2𝛼+𝛽+𝛾𝑞𝑐(𝑃, 𝐿, 𝐾)

If doubling all inputs quadruples output then:

𝑞𝑐(2𝑃, 2𝐿, 2𝐾) = 4𝑞𝑐(𝑃, 𝐿, 𝐾)

This implies that 2𝛼+𝛽+𝛾 = 4 = 22 = 21+1. Since 𝛼 + 𝛽 = 1 it means that 𝛾 = 1.


ECO 204, 2016-2017 (AJAZ), Test 3


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QUESTION 2 [TOTAL 10 Points] Answer the following questions independently of each other:

(a) [5 Points] Suppose that a firm’s short run production function is 𝑄 = 𝐴𝐿𝛼𝑘𝛽 and that 𝑑𝑀𝐶𝑠ℎ𝑜𝑟𝑡 𝑟𝑢𝑛

𝑑𝑞< 0. True or false:

the long run 𝐴𝐶 declines with output? State all salient assumptions and provide a brief explanation. Answer:

True. If 𝑑𝑀𝐶𝑠ℎ𝑜𝑟𝑡 𝑟𝑢𝑛

𝑑𝑞< 0 (i.e. 𝑀𝐶𝑠ℎ𝑜𝑟𝑡 𝑟𝑢𝑛 falls with output) it means that that the firm has increasing returns in the short

run, i.e. there are increasing returns to (say) labor (i.e. 𝛼 > 1). Another way to see this is that since 𝑀𝐶𝑠ℎ𝑜𝑟𝑡 𝑟𝑢𝑛 =𝑑𝑇𝑉𝐶

𝑑𝑞 that

𝑑𝑀𝐶𝑠ℎ𝑜𝑟𝑡 𝑟𝑢𝑛

𝑑𝑞=𝑑2𝑇𝑉𝐶

𝑑𝑞2 or that the 𝑇𝑉𝐶 function is strictly concave. Since 𝛽 > 0 it means that 𝛼 + 𝛽 > 1 so that

the company has increasing returns to scale in the long run which means that the long run 𝐴𝐶 must always fall with output.

(b) [5 Points] Suppose that a firm’s short run production function is 𝑄 = 𝐴𝐿𝛼𝑘𝛽 and that 𝑑𝐴𝑉𝐶

𝑑𝑞> 0. True or false: the

long run 𝐴𝐶 rises with output? State all salient assumptions and provide a brief explanation. Answer:

False. If 𝑑𝐴𝑉𝐶

𝑑𝑞> 0 it means that 𝐴𝑉𝐶 increases with output which means that in the short run there are decreasing

returns to (say) labor (i.e. 𝛼 < 1). This means that depending on the value of 𝛽, we can have 𝛼 + 𝛽 ⋛ 1, i.e. the

company can have increasing, or constant, or decreasing returns to scale in the long run which means that long run 𝐴𝐶 can fall, stay constant, or rise with output.


ECO 204, 2016-2017 (AJAZ), Test 3


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QUESTION 3 [TOTAL 10 Points] (a) [5 Points] Show that the 𝑇𝑅𝑆 of the production function 𝑄 = 2𝐴𝐿0.18𝐾0.33 is identical to the 𝑇𝑅𝑆 of the production function 𝑄 = 𝐴𝐿0.18𝐾0.33. State all salient assumptions and provide a brief explanation. Answer: The 𝑇𝑅𝑆 of 𝑄 = 𝐴𝐿0.18𝐾0.33 is:

𝑇𝑅𝑆 = −𝑀𝑃𝐿

𝑀𝑃𝐾= −

𝐴 0.18 𝐿0.18−1𝐾0.33

𝐴 0.33 𝐿0.18𝐾0.33−1= −

0.18

0.33

𝐾

𝐿

The 𝑇𝑅𝑆 of 𝑄 = 2𝐴𝐿0.18𝐾0.33 is:

𝑇𝑅𝑆 = −𝑀𝑃𝐿

𝑀𝑃𝐾= −

2𝐴 0.18 𝐿0.18−1𝐾0.33

2𝐴 0.33 𝐿0.18𝐾0.33−1= −

0.18

0.33

𝐾

𝐿

(b) [5 Points] True or false: since the 𝑇𝑅𝑆 of the production function 𝑄 = 2𝐴𝐿0.18𝐾0.33 is identical to the 𝑇𝑅𝑆 of the production function 𝑄 = 𝐴𝐿0.18𝐾0.33, these two production functions model the same “input-output” process. Answer: False. While the two production functions have identical 𝑇𝑅𝑆 they represent two different “input-output” processes – notice that for an arbitrary amount of labor and capital, 𝑄 = 2𝐴𝐿0.18𝐾0.33 produces twice as much output as 𝑄 = 𝐴𝐿0.18𝐾0.33.


ECO 204, 2016-2017 (AJAZ), Test 3


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QUESTION 4 [TOTAL 15 Points] Evaluate the following statement:

“The introduction of the ‘New Coke’ was a stroke of marketing genius. Faced with declining sales of ‘Classic Coke’, Coca-Cola’s executives decided to withdraw ‘Classic Coke’ from the market and introduce the ‘New Coke’ anticipating that the American Public would decry the loss of an American icon and would push Coca-Cola to re-introduce ‘Classic Coke’ , which would revive sales of ‘Classic Coke’. In fact, that’s exactly what happened: there was a backlash when ‘Classic Coke’ was “retired” leading Coca-Cola to re-introduce ‘Class Coke’ resulting in record high sales. In retrospect, Coca-Cola executives were, to put it mildly, prophetic geniuses.”

Answer: This statement is a bunch of hog wash. The Original Coke (late 1800s) used sugar as an ingredient and the original

production technology and product formulation didn’t permit other sweeteners as inputs. When Sugar prices increased

in the 1970s, Coke was unable to re-configure its production process or re-formulate Coke’s secret formula to use other

sweeteners and produce the same taste as before. Unable to use other sweeteners, in the 1980s Coke “reformulated”

Coca-Cola Classic into New Coke . Consumers pushed back against New Coke and sales of Coca-Cola classic (ironically)

increased. Eventually, Coke re-formulated formula and adopted a technology allowing either/or sugar and fructose to be

used as inputs (permitted substitution and zero sugar)


http://en.wikipedia.org/wiki/Coca-Cola

http://www.thecoca-colacompany.com/heritage/cokelore_newcoke.html

http://en.wikipedia.org/wiki/New_Coke

ECO 204, 2016-2017 (AJAZ), Test 3


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QUESTION 5 [TOTAL 30 Points]

Suppose that a company produces output according to the Cobb-Douglas production function 𝑄 = 𝐴𝐿𝛼𝑘𝛽 where 𝐿 = labor, 𝑘 = fixed capital and the parameters 𝐴, 𝛼, 𝛽 > 0. Denote the wage rate by 𝑃𝐿 > 0 and the price of leasing capital (or, the user cost of owned capital) by 𝑃𝐾 > 0. (a) [20 Points] Setup and solve the following Cost Minimization Problem for the optimal amount of labor needed to produce the target output 𝑞 > 0 and any Lagrange multipliers in this constrained optimization problem:

min𝐿𝑇𝑉𝐶 = 𝑃𝐿𝐿 𝑠. 𝑡. 𝐴𝐿

𝛼𝑘𝛽 = 𝑞

Interpret any Lagrange multipliers in this CMP. State salient assumptions and show essential calculations. Answer: The CMP can re-written as:


𝛼𝑘𝛽 = 𝑞

Notice we are NOT told to have 𝐿 ≥ 0.

max 𝐿−𝑇𝑉𝐶 = −𝑃𝐿𝐿 𝑠. 𝑡. 𝐴𝐿

𝛼𝑘𝛽 − 𝑞 = 0

max𝐿, 𝜆1

L = −[𝑃𝐿𝐿] − 𝜆1[𝐴𝐿𝛼𝑘𝛽 − 𝑞]

max𝐿, 𝜆1

L = −[𝑃𝐿𝐿] − 𝜆1[𝐴𝐿𝛼𝑘𝛽 − 𝑞]

The FOCs are:

𝜕L

𝜕𝐿= −𝑃𝐿 − 𝜆1𝑀𝑃𝐿 = 0

𝜕L

𝜕𝜆1= −[𝐴 𝐿𝛼𝑘𝛽 − 𝑞] = 0

The second FOC implies:

𝑞 = 𝐴𝐿𝛼𝑘𝛽 ⇒ 𝐿∗ = (𝑞

𝐴𝑘𝛽)

1

𝛼

We can find 𝜆1

∗ by either solving the FOCs or using the envelope theorem to see that:

𝑑L *

𝑑𝑞=𝑑(−𝑇𝑉𝐶∗)

𝑑𝑞= −𝑀𝐶 = 𝜆1

∗


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(b) [10 Points] The following problem was solved in Excel:


𝛼𝑘𝛽 = 𝑞

max𝐿−𝑇𝑉𝐶 = −5𝐿 𝑠. 𝑡. (1)𝐿0.2(10)0.8 = 10

Here is the Solver solution and portions of the “sensitivity” report:

alpha = 0.20

beta = 0.80

k = 10.00

A = 1.00

P_L = $5.00

TVC = $50.00

-TVC = $(50.00) << Objective

Output produced = 10.00

Target output q = 10.00

Optimal Labor = 10.00 << Decision Variable

Constraints

Cell Name Final Value Lagrange Multiplier

$B$9 Output produced 10.00 -25.00

What is the 𝑇𝑉𝐶 equation? State salient assumptions and show essential calculations. Answer: The 𝑇𝑉𝐶 function is:

𝑇𝑉𝐶 = 𝑃𝐿𝐿 = 𝑃𝐿 (𝑞

𝐴𝑘𝛽)

1

𝛼= 𝑃𝐿 (

1

𝐴𝑘𝛽)

1

𝛼

𝑞1

𝛼 = 5 (1

100.8)

1

0.2

𝑞1

0.2 =1

2,000𝑞5

We can confirm whether this is correct [Not required in student’s answer]

𝑀𝐶 =𝑑𝑇𝑉𝐶

𝑑𝑞 =

5

2,000𝑞4

At the output of 10 we have:

𝑀𝐶 =𝑑𝑇𝑉𝐶

𝑑𝑞 =

5

2,000104 = 25

Which is the −1 ∗Lagrange multiplier (as it should be).


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QUESTION 6 [TOTAL 65 Points]

A risk averse agent must decide whether to invest in “Project A” or in “Project B”. Both projects are characterized by

uncertainty where:

Project 𝐴 = {$7,800,−$300; 0.6, 0.4}

Project 𝐵 = {$9,700,−$200; 0.3, 0.7}

(a) [10 Points] Suppose this decision maker’s utility function is 𝑈($𝑋) = (𝑎 + 𝑋)1

𝑏 where 𝑎, 𝑏 > 0 are parameters and 𝑋 is measured in dollars. Show that 𝑎 = 300 and 𝑏 = 2 and use these parameter values throughout this question (i.e.

assume 𝑈($𝑋) = (300 + 𝑋)1

2) . State all salient assumptions and show all essential calculations. Answer The utility function has the form:

𝑈($𝑋) = (𝑎 + 𝑋)1

𝑏 Since utility is an ordinal number, we can assign utility values to the highest and lowest outcomes as follows:

𝑈(Highest outcome in the decision problem) = 𝑈($9,700) = 100

𝑈(Lowest outcome in the decision problem) = 𝑈(−$300) = 0 Now:

𝑈($𝑋) = (𝑎 + 𝑋)1

𝑏

𝑈(−$300) = (𝑎 − 300)1

𝑏 = 0 ⇒ 𝑎 = 300 Next:

𝑈($9,700) = (300 + 9,700)1

𝑏 = 100

⇒ 10,000 = 100𝑏

⇒ ln10,000 = 𝑏 ln 100

⇒ 𝑏 =ln10,000

ln 100= 2 (after all, 1002 = 10,000)

Thus:

𝑈($𝑋) = (300 + 𝑋)1

2


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(b) [5 Points] True or false: -$200 is the certainty equivalence to the hypothetical “gamble”: {Highest outcome in the decision making problem, Lowest outcome in the decision making problem; 0.1, 0.9}?

State all salient assumptions and show all essential calculations. Answer

From the utility function 𝑈($𝑋) = (300 + 𝑋)1

2 we see that:

𝑈(−$200) = (300 − 200)1

2 = 10 Now, consider the following hypothetical gamble: {Highest outcome in the decision making problem, Lowest outcome in the decision making problem;0.1, 0.9}

The 𝐸𝑈 of this hypothetical gamble is: 𝐸𝑈 = 0.1 𝑈(Highest outcome in the decision making problem) + 0.9 𝑈(Lowest outcome in the decision making problem)

𝐸𝑈 = 0.1 ∗ 100 + 0.9 ∗ 0 = 10

Since 𝑈(−$200) = 𝐸𝑈 we see that −$200 is in fact the 𝐶𝐸 to the hypothetical “gamble”: {Highest outcome in the decision making problem, Lowest outcome in the decision making problem;0.1, 0.9}

Thus, true.


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(c) [15 Points] Graph the decision tree below and solve the decision making under uncertainty problem to show that Project A is the optimal choice. State all salient assumptions and show all essential calculations. Hint: Remember that

𝑈($𝑋) = (300 + 𝑋)1

2 Answer The risk averse agent must decide whether to invest in “Project A” or in “Project B” where:

Project 𝐴 = {$7,800,−$300; 0.6, 0.4}

Project 𝐵 = {$9,700,−$200; 0.3, 0.7} The decision tree is (oval nodes represent uncertainty):

Choose:

Project A Nature$7,800

-$300

Project B Nature$9,700

-$200

𝑃 𝑆 = 0.6

𝑃 𝑆 = 0.3

𝑃 𝐹 = 0.4

𝑃 𝐹 = 0.7

Since the agent is risk averse, she will choose the project with the highest expected utility. To do this, we need to compute the utilities of all outcomes. We already know that:

𝑈($9,700) = 100

𝑈(−$300) = 0

From 𝑈($𝑋) = (300 + 𝑋)1

2 we have:

𝑈($7,800) = (300 + 7,800)1

2 = 90

𝑈(−$200) = (300 − 200)1

2 = 10 Thus:

𝐸𝑈(𝑃𝑟𝑜𝑗𝑒𝑐𝑡 𝐴) = 0.6𝑈($7,800) + 0.4𝑈(−$300) = 0.6 ∗ 90 + 0.4 ∗ 0 = 54


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𝐸𝑈(𝑃𝑟𝑜𝑗𝑒𝑐𝑡 𝐵) = 0.3𝑈($9,700) + 0.7𝑈(−$200) = 0.3 ∗ 100 + 0.7 ∗ 10 = 37

The agent will choose Project A over Project B (following tree is not required in the answer):

Chose Project A

Project A Nature

$7,800

-$300

Project B Nature

$9,700

-$200

𝑃 𝑆 = 0.6

𝑃 𝑆 = 0.3

𝑃 𝐹 = 0.4

𝑃 𝐹 = 0.7


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(d) [10 Points] Calculate and interpret the “discount due to risk” of Project 𝐴 = {$7,800,−$300; 0.6, 0.4}. State all

salient assumptions and show all essential calculations. Hint: Remember that 𝑈($𝑋) = (300 + 𝑋)1

2 Answer The discount due to risk is equal to 𝐸𝑉 − 𝐶𝐸. First, consider Project 𝐴 = {$7,800,−$300; 0.6, 0.4}.

𝐸𝑉(𝑃𝑟𝑜𝑗𝑒𝑐𝑡 𝐴) = 0.6 ∗ 7,800 + 0.4 ∗ (−300) = $4,560

𝐸𝑈(𝑃𝑟𝑜𝑗𝑒𝑐𝑡 𝐴) = 0.6𝑈($7,800) + 0.4𝑈(−$300) = 54 The 𝐶𝐸 of Project A is:

𝑈(𝐶𝐸) = (300 + 𝐶𝐸)1

2 = 𝐸𝑈(𝑃𝑟𝑜𝑗𝑒𝑐𝑡 𝐴) = 54

𝐶𝐸 = 542 − 300 = $2,616 (You should confirm that 𝑈(𝐶𝐸) = 𝐸𝑈) The discount due to risk for project A is:

Project 𝐴 discount due to risk = 𝐸𝑉 − 𝐶𝐸 = $4,560 − $2,616 = $1,944 Here’s the interpretation of the discount due to risk. Project A’s 𝐸𝑉 is $4,560. Now, if the agent invests in project A, her 𝐸𝑈 will be 54. Now, would this agent prefer the 𝐸𝑉 for sure over facing the uncertain situation? Yes, because (as you can show) the 𝑈(𝐸𝑉) > 𝐸𝑈. What about $4,559: will the agent prefer this or the uncertain situation? As you can show, she will prefer $4,559 over the uncertain situation even though it is $1 less than the 𝐸𝑉 of the project. That is, she is willing to “sacrifice” some of the 𝐸𝑉 to avoid uncertainty and gain peace of mind. What’s the maximum amount she’ll “sacrifice”? That is, of course, the 𝐶𝐸 (at which point the utility of the 𝐶𝐸 is the same as 𝐸𝑈). Thus, the maximum discount due to risk is 𝐸𝑉 − 𝐶𝐸.


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(e) [25 Points] In part (c) you showed that the agent should choose Project A over Project B. Now suppose she can purchase an actuarially fair insurance policy to eliminate all risk from investing in Project A. Prove that she will purchase full insurance, calculate the insurance premium, and show that she will remove all risk from investing in Project A. State all salient assumptions and show all essential calculations. Answer Suppose the agent can purchase actuarially fair insurance. Now, consider project A:

Project 𝐴 = {$7,800,−$300; 0.6, 0.4} One can interpret this as 𝑊 = Wealth before loss = $7,800 and 𝑊 − 𝐿 = Wealth after loss = −$300, i.e. the loss is 𝐿 = $8,100 where the probability of loss is 𝑝 = 0.4.

Agent buys insurance and pays insurance premium =

Nature

If there is no loss:

𝑈 𝑊 − nsurance Premium = 𝑈 𝑊 −

If there is a loss:

𝑈 𝑊 − 𝐿 − nsurance Premium+ Pa out = 𝑈 𝑊 − 𝐿 − +

The risk averse agent chooses the optimal insurance policy to:

max𝐼𝐸𝑈 = (1 − 𝑝)𝑈(𝑊 − ) + 𝑝𝑈(𝑊 − 𝐿 − + )

Here, = price per dollar of insurance so that = Insurance Premium. Now:

𝑑𝐸𝑈

𝑑 = (1 − 𝑝)𝑈′(𝑊 − )(− ) + 𝑝𝑈′(𝑊 − 𝐿 − + )(− + 1) = 0

(1 − 𝑝)𝑈′(𝑊 − ) = 𝑝𝑈′(𝑊 − 𝐿 − + )(1 − )

Under actuarially fair insurance = 𝑝:

𝑈′(𝑊 − 𝑝 ) = 𝑈′(𝑊 − 𝐿 − 𝑝 + ) Since the agent is risk averse, we see that:


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𝑊 − 𝑝 = 𝑊 − 𝐿 − 𝑝 +

= 𝐿 Thus, the agent will buy full insurance, i.e. = 𝐿 = $8,100. The insurance premium will be = 𝑝 = 0.4 ∗ $8,100 =$3,240. Finally, note that in the event of no loss (i.e. project A is a success):

Net Wealth if project 𝐴 is a success with probabilit of 60% = 𝑊 − = $7,800 − $3,240 = $4,560 In the event of a loss (i.e. project A is a failure):

Net Wealth if project 𝐴 is a failure with probabilit of 40% = 𝑊 − − 𝐿 + = $7,800 − $3,240 − $8,100 + $8,100 = $4,560

Whether the project succeeds or fails, the investor’s net wealth is always $4,560.


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