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Assume for simplicity that a monopolist has no costs of production and faces a demand curve given by Q = 150 - P.
a. Calculate the profit-maximizing price-quantity combination for this monopolist. Also calculate the monopolist's profit.
b. Suppose instead that there are two firms in the market facing the demand and cost conditions just described for their identical products. Firms choose quantities simultaneously as in the Cournot model. Compute the outputs in the Nash equilibrium. Also compute market output, price, and firm profits.
c. Suppose the two firms choose prices simultaneously as in the Bertrand model. Compute the prices in the Nash equilibrium. Also compute firm output and profit as well as market output.
d. Graph the demand curve and indicate where the market price-quantity combinations from parts (a)-(c) appear on the curve.
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Answer:
The demand curve shows the relation between quantity demanded and prices keeping all other things constant.
a.
The demand curve of the monopolist is given below:
The cost of the monopolist is equal to 0.
The total revenue is the addition of the revenue earned by the firm at each unit.
The formula for the calculation of TR is given below:
The total revenue for the firm is shown below:
The total cost for firm is zero
The profit of the firm is the difference between the total revenue received and total cost incurred.
The profit for the monopolist is calculated below:
Monopolist will maximize his profit. The output level for maximum profit is calculated below:
Equate the differentiation to zero to get the equilibrium level for the output as shown below:
The equilibrium price at this quantity is calculated below:
The profit calculated at this quantity is shown below:
b.
In Cournot competition, there are two firms which choose their profit maximizing quantities taking the quantity of the other firm as given.
The market demand curve is given below:
Q is the total quantity demanded.
.
Where,
Quantity of firm 1
Quantity of firm 2
The inverse demand curve is given below:
The profit function for firm 1 is given below:
Differentiate the profit function and equate it to zero to maximize the profit as shown below:
The profit function for firm 2 is given below:
Differentiate the profit function and equate it to zero to maximize the profit as shown below:
Substitute the value of q 2 in q 1 as shown below:
Put the value of
in equation for
as shown below:
Total quantity is given below:
The profit of the firm 1 is calculated below:
The profit of the firm 1 is calculated below:
c.
When the firms are in Bertrand competition instead of choosing quantity simultaneously as in Cournot competition, they choose price simultaneously. While maximizing profit, each firm takes the price of the other good as fixed while choosing its own price. It is more aggressive competition than Cournot competition because each firm wants to
undercut the other firm to steal the whole market. This leads to the equilibrium price being equal to the marginal cost. As the firms in this excercise do not have any cost, this means that the firms charge zero prices.
The profit maximizing strategy for firm 1 is given below:
There are four possible scenarios as given below:
1.
Firm 1 earns a negative margin on every unit it sells. Since it sells a positive quantity, it must earn negative pro?ts. It could increase its pro?t by deviating to a higher price. Hence this is not Nash Equilibrium.
2.
This cannot be Nash equilibrium either because the firm can increase its price, keeping it lower than that of firm 2 by a marginal amount of
.Until price of firm 1 remain lower than price of firm 2, fir, 1 can increase its profit by increasing the price.
3.
This cannot be Nash equilibrium either because the firm can decrease its price, undercutting firm 2 by a marginal amount
. Then it would capture the entire market and increase its profit.
4.
Now this is Nash equilibrium for both firms as they cannot increase their profit given the strategy of the other firm.
In Nash equilibrium both firm charge the price that is equal to MC. As MC is zero, price will be zero.
Total output supplied by the firms at price equal to zero is calculated below:
As the firms are symmetric
Therefore,
So,
and
The profit of the firm 1 is calculated below:
The profit of the firm 2 is calculated below:
d.
The market demand curve shows the negative relation between quantity demanded and prices.
The market demand curve is shown below.
Figure 1
The X axis shows the quantity and Y axis shows the price. The points A, B and C shows the equilibrium points in different market.
Here,
A: Equilibrium in Monopoly
B: Equilibrium in Cournot Competition
C: Equilibrium in Bertrand Competition
Suppose that firms' marginal and average costs are constant and equal to c and that inverse market demand is given by P = a - bQ, where a, b 0.
a. Calculate the profit-maximizing price-quantity combination for a monopolist. Also calculate the monopolist's profit.
b. Calculate the Nash equilibrium quantities for Cournot duopolists, which choose quantities for their identical products simultaneously. Also compute market output, market price, and firm and industry profits.
c. Calculate the Nash equilibrium prices for Bertrand duopolists, which choose prices for their identical products simultaneously. Also compute firm and market output as well as firm and industry profits.
d. Suppose now that there are n identical firms in a Cournot model. Compute the Nash equilibrium quantities as functions of n. Also compute market output, market price, and firm and industry profits.
e. Show that the monopoly outcome from part (a) can be reproduced in part (d) by setting n = 1, that the Cournot duopoly outcome from part (b) can be reproduced in part (d) by setting n = 2 in part (d), and that letting n approach infinity yields the same market price, output, and industry profit as in part (c).
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Answer:
a) The inverse market demand curve is given by
The average and marginal cost of the monopolist is equal to c. So the cost function is given by
Hence, profit of the monopolist is given by
On differentiating the profit function with respect to quantity we get,
For maximizing profit,
Hence we get that profit maximizing quantity is given as
The price associated with this quantity is equal to
The profit of the monopolist is given by
b) In Cournot competition, firms choose their profit maximizing quantities taking the quantity of the other firm as given.
The inverse market demand curve of is given by
Where
and
are the quantity produced by firm 1 and firm 2 respectively.
When deciding its profit maximizing level of output, firm 1 will take output of firm 2 fixed at
.
Then its profit will be given as
The first order condition for the above profit function is given by
So we get,
...... (1)
As the firms are symmetric, similarly
From the above equation we get,
...... (2)
As in NE equation (1) and (2) should be equal, hence
From the above we get,
As the firms are symmetric,
Hence, market output is equal to
Putting the above value in the inverse-demand function, we get the price
The profit of the firms is then given by
Industry profits are given by
c) When the firms are in Bertrand competition instead of choosing quantity simultaneously as in Cournot competition, they choose price simultaneously. While maximizing profit, each firm takes the price of the other good as fixed while choosing its own price. It is more aggressive competition than Cournot competition because each firm wants to undercut the other firm to capture the whole market. This leads to the equilibrium price being equal to the marginal cost. This can be shown below.
We observe the profit maximizing strategy for firm 1
There are four possible scenarios
(i)
Firm 1 earns a negative margin on every unit it sells. Since it sells a positive quantity, it must earn negative pro?ts. It could increase its pro?t by deviating to a higher price. Hence this is not NE.
(ii)
.
This cannot be NE either because the firm can increase its price , keeping it lower than p 2 by a marginal amount of
. It would not lose the market but increase its profit.
(iii)
This cannot be NE either because the firm can decrease its price, undercutting firm 2 by a marginal amount
. Then it would capture the market and increase its profit.
iv)
Now this is NE for both firms as they cannot increase their profit given the strategy of the other firm.
Output the firms is found by using the demand function
As the firms are symmetric
Hence,
From this we get the quantity produced by each firm
Profit of each firm is given by
The market output is given by
`
d) In Cournot competition, firms choose their profit maximizing quantities taking the quantity of the other firm as given.
The inverse market demand curve of the monopolist is given by
Where
and
are the quantity produced by firm 1 and firm 2 respectively. As the firms are symmetric, firm 1 expects the same output from firm 2 and the remaining (n-2) firms. Adding the output of firm 2 and the remaining (n-2) firms we get (n-1)Q 2.
When deciding its profit maximizing level of output, firm 1 will take output of firm 2 fixed at
.
Then its profit will be given as
The first order condition for the above profit function is given by
So we get,
......(3)
As the firms are symmetric, similarly
From this we get,
As the firms are symmetric,
Hence, market output is equal to
Putting the above value in the inverse-demand function, we get the price
So profit of each firm is given by
Profit of the industry is then given by
e) Setting n=1, we get the following value
Quantity produced by the firm is given by
Price set by the firm is given by
Profit of the firm is given by
The above results are identical with the outcomes of the monopolist in part a). Setting n=2, we get the following value
Quantity produced by firm 1 is given by
Price set by the firm is given by
Profit of the firm is given by
Profit of the industry is given by
The above results are identical with the outcomes of the Cournot duopolist in part b).
Setting n=8, we get the following value
Total quantity produced is given by
Price is given by
Profit of the industry is given by
The above results are identical with the outcomes of the Bertrand duopolist in part c).
Let c i be the constant marginal and average cost for firm i (so that firms may have different marginal costs). Suppose demand is given by P = 1 - Q.
a. Calculate the Nash equilibrium quantities assuming there are two firms in a Cournot market. Also compute market output, market price, firm profits, industry profits, consumer surplus, and total welfare.
b. Represent the Nash equilibrium on a best-response function diagram. Show how a reduction in firm 1's cost would change the equilibrium. Draw a representative isoprofit for firm 1.
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Answer:
a) In Cournot competition, firms choose their profit maximizing quantities taking the quantity of the other firm as given.
The inverse market demand curve is given by
Where
and
are the quantity produced by firm 1 and firm 2 respectively.
When deciding its profit maximizing level of output, firm 1 will take output of firm 2 fixed at
.
Cost function for firm 1 is given by
Then its profit will be given as
The first order condition for the above profit function is given by
So we get,
......(1)
Similarly
From this we get,
......(2)
As in NE equation (1) and (2) should be equal, hence
From the above we get,
Similarly,
Hence, market output is equal to
Putting the above value in the inverse-demand function, we get the price
The profit of the firms is then given by
Similarly
Industry profits are given by
Consumer surplus is given the are above the price in the demand curve
The inverse market demand function is given by
Hence, the market demand function is given by
The Cournot Competition price is equal to
The formula for consumer surplus is given by
Thus, the total welfare is given by
Profits + CS =
b) The best NE can be shown on the graph by mapping the best response functions of both the firms and seeing their intersection points. The best response function are given by the following equations.
Similarly
The graph is shown below:
A decrease in the cost of firm 1 shifts its best response curve out. Hence there is a shift in the NE. Production of firm 1 increases, whereas that of firm 2 decreases.
This is shown in the graph below.
An iso-profit curve shows the locus of points at which a firm would get equal profit. A representative iso-profit curve for firm 1 is shown below. It should be noted that the highest point of the iso-profit curve lies on the firm's best response function.
The graph is shown below:
Suppose that firms 1 and 2 operate under conditions of constant average and marginal cost but that firm 1's marginal cost is c 1 = 10 and firm 2's is c 2 = 8. Market demand is Q = 500 - 20P.
a. Suppose firms practice Bertrand competition, that is, setting prices for their identical products simultaneously. Compute the Nash equilibrium prices. (To avoid technical problems in this question, assume that if firms charge equal prices, then the low-cost firm makes all the sales.)
b. Compute firm output, firm profit, and market output.
c. Is total welfare maximized in the Nash equilibrium? If not, suggest an outcome that would maximize total welfare, and compute the deadweight loss in the Nash equilibrium compared with your outcome.
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Consider the following Bertrand game involving two firms producing differentiated products. Firms have no costs of production. Firm 1's demand is
q 1 = 1 - p 1 + bp 2 ,
where b 0. A symmetric equation holds for firm 2's demand.
a. Solve for the Nash equilibrium of the simultaneous price-choice game.
b. Compute the firms' outputs and profits.
c. Represent the equilibrium on a best-response function diagram. Show how an increase in b would change the equilibrium. Draw a representative isoprofit curve for firm 1.
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Recall Example 15.6, which covers tacit collusion. Suppose (as in the example) that a medical device is produced at constant average and marginal cost of $10 and that the demand for the device is given by
Q = 5,000 - 100P:
The market meets each period for an infinite number of periods. The discount factor is d.
a. Suppose that n firms engage in Bertrand competition each period. Suppose it takes two periods to discover a deviation because it takes two periods to observe rivals' prices. Compute the discount factor needed to sustain collusion in a subgame-perfect equilibrium using grim strategies.
b. Now restore the assumption that, as in Example 15.7, deviations are detected after just one period. Next, assume that n is not given but rather is determined by the number of firms that choose to enter the market in an initial stage in which entrants must sink a one-time cost K to participate in the market. Find an upper bound on n. Hint: Two conditions are involved.
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Assume as in Problem 15.1 that two firms with no production costs, facing demand Q = 150 - P, choose quantities q 1 and q 2.
a. Compute the subgame-perfect equilibrium of the Stackelberg version of the game in which firm 1 chooses q 1 first and then firm 2 chooses q 2.
b. Now add an entry stage after firm 1 chooses q 1. In this stage, firm 2 decides whether to enter. If it enters, then it must sink cost K 2 , after which it is allowed to choose q 2. Compute the threshold value of K 2 above which firm 1 prefers to deter firm 2's entry.
c. Represent the Cournot, Stackelberg, and entry-deterrence outcomes on a best-response function diagram.
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Recall the Hotelling model of competition on a linear beach from Example 15.5. Suppose for simplicity that ice cream stands can locate only at the two ends of the line segment (zoning prohibits commercial development in the middle of the beach). This question asks you to analyze an entry-deterring strategy involving product proliferation.
a. Consider the subgame in which firm A has two ice cream stands, one at each end of the beach, and B locates along with A at the right endpoint. What is the Nash equilibrium of this subgame? Hint: Bertrand competition ensues at the right endpoint.
b. If B must sink an entry cost KB, would it choose to enter given that firm A is in both ends of the market and remains there after entry?
c. Is A's product proliferation strategy credible? Or would A exit the right end of the market after B enters? To answer these questions, compare A's profits for the case in which it has a stand on the left side and both it and B have stands on the right to the case in which A has one stand on the left end and B has one stand on the right end (so B's entry has driven A
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Herfindahl index of market concentration
One way of measuring market concentration is through the use of the Herfindahl index, which is defined as
where s t = q i /Q is firm i's market share. The higher is H, the more concentrated the industry is said to be. Intuitively, more concentrated markets are thought to be less competitive because dominant firms in concentrated markets face little competitive pressure. We will assess the validity of this intuition using several models.
a. If you have not already done so, answer Problem 15.2d by computing the Nash equilibrium of this n-firm Cournot game. Also compute market output, market price, consumer surplus, industry profit, and total welfare. Compute the Herfindahl index for this equilibrium.
b. Suppose two of the n firms merge, leaving the market with n n? 1 firms. Recalculate the Nash equilibrium and the rest of the items requested in part (a). How does the merger affect price, output, profit, consumer surplus, total welfare, and the Herfindahl index?
c. Put the model used in parts (a) and (b) aside and turn to a different setup: that of Problem 15.3, where Cournot duopolists face different marginal costs. Use your answer to Problem 15.3a to compute equilibrium firm outputs, market output, price, consumer surplus, industry profit, and total welfare, substituting the particular cost parameters c 1 = c 2 = 1/ 4. Also compute the Herfindahl index.
d. Repeat your calculations in part (c) while assuming that firm 1's marginal cost c 1 falls to 0 but c 2 stays at 1/4. How does the cost change affect price, output, profit, consumer surplus, total welfare, and the Herfindahl index?
e. Given your results from parts (a)-(d), can we draw any general conclusions about the relationship between market concentration on the one hand and price, profit, or total welfare on the other?
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where s t = q i /Q is firm i's market share. The higher is H, the more concentrated the industry is said to be. Intuitively, more concentrated markets are thought to be less competitive because dominant firms in concentrated markets face little competitive pressure. We will assess the validity of this intuition using several models.
a. If you have not already done so, answer Problem 15.2d by computing the Nash equilibrium of this n-firm Cournot game. Also compute market output, market price, consumer surplus, industry profit, and total welfare. Compute the Herfindahl index for this equilibrium.
b. Suppose two of the n firms merge, leaving the market with n n? 1 firms. Recalculate the Nash equilibrium and the rest of the items requested in part (a). How does the merger affect price, output, profit, consumer surplus, total welfare, and the Herfindahl index?
c. Put the model used in parts (a) and (b) aside and turn to a different setup: that of Problem 15.3, where Cournot duopolists face different marginal costs. Use your answer to Problem 15.3a to compute equilibrium firm outputs, market output, price, consumer surplus, industry profit, and total welfare, substituting the particular cost parameters c 1 = c 2 = 1/ 4. Also compute the Herfindahl index.
d. Repeat your calculations in part (c) while assuming that firm 1's marginal cost c 1 falls to 0 but c 2 stays at 1/4. How does the cost change affect price, output, profit, consumer surplus, total welfare, and the Herfindahl index?
e. Given your results from parts (a)-(d), can we draw any general conclusions about the relationship between market concentration on the one hand and price, profit, or total welfare on the other?
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Inverse elasticity rule
Use the first-order condition (Equation 15.2) for a Cournot firm to show that the usual inverse elasticity rule from Chapter 11 holds under Cournot competition (where the elasticity is associated with an individual firm's residual demand, the demand left after all rivals sell their output on the market). Manipulate Equation 15.2 in a different way to obtain an equivalent version of the inverse elasticity rule:
where s i =q i /Q is firm i's market share and e Q, P is the elasticity of market demand. Compare this version of the inverse elasticity rule with that for a monopolist from the previous chapter..
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where s i =q i /Q is firm i's market share and e Q, P is the elasticity of market demand. Compare this version of the inverse elasticity rule with that for a monopolist from the previous chapter..
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Competition on a circle
Hotelling's model of competition on a linear beach is used widely in many applications, but one application that is difficult to study in the model is free entry. Free entry is easiest to study in a model with symmetric firms, but more than two firms on a line cannot be symmetric because those located nearest the endpoints will have only one neighboring rival, whereas those located nearer the middle will have two.
To avoid this problem, Steven Salop introduced competition on a circle. As in the Hotelling model, demanders are located at each point, and each demands one unit of the good. A consumer's surplus equals v (the value of consuming the good) minus the price paid for the good as well as the cost of having to travel to buy from the firm. Let this travel cost be td, where t is a parameter measuring how burdensome travel is and d is the distance traveled (note that we are here assuming a linear rather than a quadratic travel-cost function, in contrast to Example 15.5). Initially, we take as given that there are n firms in the market and that each has the same cost function C i = K + cq i , where K is the sunk cost required to enter the market [this will come into play in part (e) of the question, where we consider free entry] and c is the constant marginal cost of production. For simplicity, assume that the circumference of the circle equals 1 and that the n firms are located evenly around the circle at intervals of 1/n. The n firms choose prices p i simultaneously.
a. Each firm i is free to choose its own price (p i ) but is constrained by the price charged by its nearest neighbor to either side. Let p * be the price these firms set in a symmetric equilibrium. Explain why the extent of any firm's market on either side (x) is given by the equation
b. Given the pricing decision analyzed in part (a), firm i sells qi = 2x because it has a market on both sides. Calculate the profit-maximizing price for this firm as a function of p _ , c , t , and n.
c. Noting that in a symmetric equilibrium all firms' prices will be equal to p * , show that p i = p * = c + t/n. Explain this result intuitively.
d. Show that a firm's profits are t/n 2 - K in equilibrium.
e. What will the number of firms n * be in long-run equilibrium in which firms can freely choose to enter?
f. Calculate the socially optimal level of differentiation in this model, defined as the number of firms (and products) that minimizes the sum of production costs plus demander travel costs. Show that this number is precisely half the number calculated in part (e). Hence this model illustrates the possibility of over differentiation.
b. Given the pricing decision analyzed in part (a), firm i sells qi = 2x because it has a market on both sides. Calculate the profit-maximizing price for this firm as a function of p _ , c , t , and n.
c. Noting that in a symmetric equilibrium all firms' prices will be equal to p * , show that p i = p * = c + t/n. Explain this result intuitively.
d. Show that a firm's profits are t/n 2 - K in equilibrium.
e. What will the number of firms n * be in long-run equilibrium in which firms can freely choose to enter?
f. Calculate the socially optimal level of differentiation in this model, defined as the number of firms (and products) that minimizes the sum of production costs plus demander travel costs. Show that this number is precisely half the number calculated in part (e). Hence this model illustrates the possibility of over differentiation. Essay
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Signaling with entry accommodation
This question will explore signaling when entry deterrence is impossible; thus, the signaling firm accommodates its rival's entry. Assume deterrence is impossible because the two firms do not pay a sunk cost to enter or remain in the market. The setup of the model will follow Example 15.4, so the calculations there will aid the solution of this problem. In particular, firm i's demand is given by
where ai is product i's attribute (say, quality). Production is costless. Firm 1's attribute can be one of two values: either a 1 = 1, in which case we say firm 1 is the low type, or a 1 = 2 , in which case we say it is the high type. Assume there is no discounting across periods for simplicity.
a. Compute the Nash equilibrium of the game of complete information in which firm 1 is the high type and firm 2 knows that firm 1 is the high type.
b. Compute the Nash equilibrium of the game in which firm 1 is the low type and firm 2 knows that firm 1 is the low type.
c. Solve for the Bayesian-Nash equilibrium of the game of incomplete information in which firm 1 can be either type with equal probability. Firm 1 knows its type, but firm 2 only knows the probabilities. Because we did not spend time this chapter on Bayesian games, you may want to consult Chapter 8 (especially Example 8.7).
d. Which of firm 1's types gains from incomplete information? Which type would prefer complete information (and thus would have an incentive to signal its type if possible)? Does firm 2 earn more profit on average under complete information or under incomplete information?
e. Consider a signaling variant of the model chat has two periods. Firms 1 and 2 choose prices in the first period, when firm 2 has incomplete information about firm 1's type. Firm 2 observes firm 1's price in this period and uses the information to update its beliefs about firm 1's type. Then firms engage in another period of price competition. Show that there is a separating equilibrium in which each type of firm 1 charges the same prices as computed in part (d). You may assume that, if firm 1 chooses an out-of-equilibrium price in the first period, then firm 2 believes that firm 1 is the low type with probability 1. Hint: To prove the existence of a separating equilibrium, show that the loss to the low type from trying to pool in the first period exceeds the second-period gain from having convinced firm 2 that it is the high type. Use your answers from parts (a)-(d) where possible to aid in your solution.
where ai is product i's attribute (say, quality). Production is costless. Firm 1's attribute can be one of two values: either a 1 = 1, in which case we say firm 1 is the low type, or a 1 = 2 , in which case we say it is the high type. Assume there is no discounting across periods for simplicity.
a. Compute the Nash equilibrium of the game of complete information in which firm 1 is the high type and firm 2 knows that firm 1 is the high type.
b. Compute the Nash equilibrium of the game in which firm 1 is the low type and firm 2 knows that firm 1 is the low type.
c. Solve for the Bayesian-Nash equilibrium of the game of incomplete information in which firm 1 can be either type with equal probability. Firm 1 knows its type, but firm 2 only knows the probabilities. Because we did not spend time this chapter on Bayesian games, you may want to consult Chapter 8 (especially Example 8.7).
d. Which of firm 1's types gains from incomplete information? Which type would prefer complete information (and thus would have an incentive to signal its type if possible)? Does firm 2 earn more profit on average under complete information or under incomplete information?
e. Consider a signaling variant of the model chat has two periods. Firms 1 and 2 choose prices in the first period, when firm 2 has incomplete information about firm 1's type. Firm 2 observes firm 1's price in this period and uses the information to update its beliefs about firm 1's type. Then firms engage in another period of price competition. Show that there is a separating equilibrium in which each type of firm 1 charges the same prices as computed in part (d). You may assume that, if firm 1 chooses an out-of-equilibrium price in the first period, then firm 2 believes that firm 1 is the low type with probability 1. Hint: To prove the existence of a separating equilibrium, show that the loss to the low type from trying to pool in the first period exceeds the second-period gain from having convinced firm 2 that it is the high type. Use your answers from parts (a)-(d) where possible to aid in your solution. Essay
Answer:
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