Econ 100a Question Paper
Econ 100A–Midterm 2 solutions. Thursday, March 22, 2012.
True/False
(2 questions, 10 points total)
Answer true or false and explain your answer. Your answer must fit in the space provided. T/F 1. (5 points) Suppose the government wants to place a tax on one of two goods, and suppose that supply is perfectly elastic for both goods. If the government wants to minimize the deadweight loss from a tax of a given size, it should put the tax on whichever good has worse substitutes.
False: If the supply curves are identical, the only factor that determines the amount of deadweight loss is the elasticity of demand. Placing the tax on the good that has the lower elasticity of demand will minimize the deadweight loss of the tax. It is true …show more content…
that, holding all else equal, a good without good substitutes will have more inelastic demand than a good with good substitutes. However, this is not the only factor that determines the elasticity of demand. The goods could also differ in terms of the income effect. If the good with worse substitutes happened to be strongly normal while the good with better substitutes was strongly inferior, then the income effects might overwhelm the substitution effects, causing the good with better substitutes to be more inelastic. T/F 2. (5 points) In a perfectly competitive market with no taxes, if the price consumers are willing to pay for the marginal unit is the same as the price at which producers are willing to produce the marginal unit, then there will be no way to make anyone in the market better off without making someone else worse off.
True. The price consumers are willing to pay for the marginal unit is the height of the inverse demand curve, and the price at which producers are willing to produce the marginal unit is the height of the inverse supply curve. Thus, when these prices are equal, it must be the case that supply is equal to demand, which is to say, the market is in equilibrium. If the quantity firms produce, and consumers consume, is more than the equilibrium quantity, then the firms’ cost of production will be greater than the consumers’ willingness to pay, and either consumers will have to pay more than the units are worth to them, making them worse off, or firms will have to receive less than the units cost them, making them worse off, or both. If the quantity is less than equilibrium, then there will be units not produced or consumed for which the cost of production would have been less than consumers’ willingness to pay, meaning that either firms have given up profitable units, or consumers have given up units that generated consumer surplus, or both. In any case, at least one side of the market will have been made worse off. Thus, from equilibrium there is no way that either firms or consumers can be made better off without someone being made worse off.
1
Short Answer
(2 questions, 20 points total)
Your answer must fit in the space provided.
SA 2. (10 points) Explain what we mean when we say that firms in long-run equilibrium are earning zero profit even though their owners and investors are making an adequate return on their labor and investments.
The statement refers to “economic profit”, which is the difference between revenue and opportunity cost. The opportunity cost of the labor of the owner of a firm is the wage the owner could have earned if he or she chose not to run the firm, but to get a job instead. The opportunity cost of the capital investors invest in a firm is the rate of return they could have earned by investing their capital in some other firm in some other industry. Thus, if the owner of the firm receives an amount just equal to the opportunity cost of their labor, and the investors receive an amount just equal to the opportunity cost of their capital, we do not include those amounts in economic profit, and the firm will be said to be earning zero economic profit, even though an accountant would say that both the owner and the investors are making an “accounting profit”. The accounting profit earned by the owner and the investors is the amount of money that is just adequate to make them choose to put their labor and capital into the firm.
2
Problem Solving
(2 problems, 50 points total)
Problem 1. (26 points total) Consider a perfectly competitive firm with a production technology 1 1 represented by the production function, y = 10 K 2 + L 2 . Let p, r, and w be the price of the firm’s output, the rental rate of capital, and the wage, respectively. (a) (8 points) First let’s consider long-run profit maximization. (i) Set up the firm’s long-run profit maximization problem and compute the firm’s profitmaximizing demand for labor and capital, and profit-maximizing output, as functions of p, r, and w. (ii) Is labor a gross complement or a gross substitute for capital, or neither. Prove your answer mathematically and explain what it means.
The long-run profit maximization problem is, max p · 10
K,L
√ K+
√ L
The first-order conditions are, 5p 5p for L: √L − w = 0 for K: √K − r = 0 Solving these for L and K respectively we get L∗ (p, r, w) = (f rac5pw)2 and K ∗ (p, r, w) = (f rac5pr)2 . Plugging these profit-maximizing levels of capital and labor into the production function we get the profit-maximizing output of the firm, y ∗ (p, w, r) = y(K ∗ , L∗ ) = 10
5p r 2
,
5p w
2
= 50p
r+w rw
.
To determine whether labor is a gross complement or gross substitute for capital we take the partial derivative of the labor demand function with respect to the rental ∗ rate of capital, ∂L = 0. Since this is zero, labor is neither a gross complement ∂r nor a gross substitute for capital. What this means is that when the price of capital changes, the amount of labor the firm uses will not change. (b) (8 points) Set up the firm’s cost-minimization problem and compute the firm’s conditional demand for labor and capital, as functions of y, r, and w. The firm’s cost minimization problem is, √ min rK + wL
K,L
√ K+ L =y ¯
s.t. 10
Setting up the LaGrangian function, this minimization problem becomes, min rK + wL − λ 10 √ K+ √ L −y ¯ √
K,L,λ
The first-order conditions are,
5 for L: w − λ √L = 0 for K: r − λ √5 = 0 for λ: 10 K the production constraint.
√
K+
L = y , which is just ¯ w 2 L. r
Taking the ratio of the first two conditions we get this into the production constraint we get, 10 3
√ √K = w ⇒ r L √ √ w r L+ L
K=
Plugging
= y ⇒ L∗ (y; r, w) = ¯
y2
r 10(r+w)
2
. Plugging this back into the expression for K that we derived earlier
2
w we get, K ∗ (y; r, w) = y 2 10(r+w) labor and capital respectively.
. These are the firm’s conditional demand for
(c) (10 points) Now let’s consider scale and substitution effects. Assume that initially the price of the firm’s output, p, the rental rate of capital, r, and the wage, w, are all equal to 10. (i) How much labor will the firm use at these prices, and how much output will it produce? (ii) Using only the mathematical results you got in parts (a) and (b), compute effect of an increase in the rental rate to r = 20.
Plugging the given prices into the profit-maximizing labor demand and output supply 2 functions from part (a) we get, L∗ (p, w, r) = 5·10 = 25, and y ∗ (p, w, r) = 50 · 10 10 (f rac10 + 1010 · 10) = 100.
∗
−
you might have plugged the new prices into the firm’s supply function to get y ∗ (10, 10, 20) = 50·10 10+20 = 75. If you then plugged this into the 10·20 firm’s conditional factor demand at the new prices you would get L∗ (75; 10, 20) =
75 20 10 10+20 2
= 25.
4
Problem 2. (24 points total) Consider a perfectly competitive industry with 10 identical firms, each of which has variable costs of 10y 2 and fixed costs of 1000. We will define the short run as the time scale in which firms cannot enter or exit the industry, and cannot avoid their fixed costs. (In other words, in the short run firms must continue to pay their fixed costs even if they produce zero output.) In the long run, firms can enter or exit the industry, and can avoid their fixed costs by shutting down. (a) (8 points) Compute the short-run inverse supply curve of the firm, and the short-run inverse supply curve of the industry, and graph them on the same graph. [Hint: it matters a lot that firms can’t avoid their fixed costs in the short …show more content…
run.]
Each firm’s cost function is C(y) = 10y 2 + 1000, and the marginal cost curve is M C = 20y. Normally we say that the inverse supply curve of the firm is the upward sloping part of the marginal cost curve, above the minimum of the average cost curve, because if the price is below the minimum of the average cost curve, the firm will make negative profit and will shut down. However, in this case, in the short run, if a firm shuts down it will still have to pay its fixed cost of $1000. As a result, it will continue to produce output even if it is losing money, as long as it does not lose more than $1000. So we need to find the price below which the firm will have lose more than $1000. Profit is py − 10y 2 − 1000 and we want the price below which this is less than −1000. To do this we have to plug in the firm’s profit-maximizing quantity as a function of price, which we get by solving the firm’s marginal cost curve p p p 2 to get y ∗ = 20 , which gives us p 20 − 10 20 − 1000 = −1000 ⇒ p2 19 = 0 ⇒ p = 0. 40 The firm will continue to produce at any positive price rather than shut down and
5
pay its fixed cost without any revenue. Thus, the firm’s inverse supply curve is simply the entire marginal cost curve, p(y) = 20y. To compute the short-run inverse supply curve of the industry we first have to aggregate firm supply to industry supply, and to do that we have to have the direct supply curve of the firm, which we get by solving the inverse supply curve for y to p p get y(p) = 20 . Short-run industry supply is Y (p) = N yj (p) = 10 20 = f racp2. j=1 Solving for p we get the short-run inverse supply curve of the industry, p(Y ) = 2Y . Your graph should look like this:
(b) (6 points) Suppose the demand for the industry’s product is defined by pd (Y ) = 700 − 5Y . (i) What will be the short-run equilibrium price and quantity for the industry? Illustrate this equilibrium on a graph. (ii) Explain why this market outcome is an equilibrium in the short run. [Be sure to make reference to the general definition of equilibrium in your answer.] (iii) Is this industry in long-run equilibrium? Explain why or why not. [Again, be sure to make reference to the general definition of equilibrium in your answer.]
The short-run market equilibrium is where the quantity demanded at the price paid by consumers is equal to the quantity supplied at the price received by producers, and since, in the absence of a tax, the price paid by consumers is the same as the price paid by producers, we just solve for the intersection of the supply curve and the demand curve: 700 − 5Y = 2Y ⇒ Y ∗ = 100. Plugging that into either the demand or the supply curve we get p(Y ) = 200. Your graph should look like this:
In general, equilibrium means that no individual agent has an incentive to do anything other than what they are currently doing, which means that the system will 6
not move from the point it is at. In the case of short-run market equilibrium this means that at the market price consumers cannot be made better off by increasing or decreasing consumption, and firms cannot be made better off by increasing or decreasing production. This is clearly the case at the market equilibrium we have solved for. If consumers increase consumption they will have to pay more for the additional units of the good than the value of those units, and if they consume less they will be giving up units that are worth more to them than they are required to pay for them. In either case, they are made worse off, and thus have no incentive to change. For firms, roughly the same argument applies. If they produce more, the maximum they will be able to charge will be less than the cost of production, and if they produce less they will be giving up units that they were able to sell at a profit. In either case, firms are worse off, so they have no incentive to change what they were doing. The industry is in long-run equilibrium. To see this we need to know whether firms are earning zero profit, and to determine that we need to know something about the firm’s average cost curve, which is AC = 10y + 1000 . If we minimize this we find y that the firms’ minimum average cost is minAC = 200. And since this is equal to the price in the current equilibrium, firms’ profit is (p − AC)y = 0y = 0. Long-run equilibrium is defined as the point at which firms will have no incentive to enter or exit the industry. The reason firms enter or exit is in response to profits being either positive or negative, so if profits are zero in the industry there will be no incentive to enter or exit, which is to say, no firm will have any incentive to do anything different from what they are currently doing. (c) (10 points) Suppose the government imposes a tax of $50 per unit on the firms in the industry. (i) Compute the short-run after-tax equlibrium quantity, price paid by consumers, and price received by firms, and graph them. (ii) Calculate the change in producer surplus caused by the tax in the short-run. Add it to your graph. (iii) Compute the long-run after-tax equilibrium quanitity, price paid by consumers, and price received by firms. Add this equilibrium to your graph. How many firms will exit the industry? (iv) Calculate the change in producer surplus caused by the tax in the long-run. Why is this the same or different from your answer to ii above?
To compute the short-run after tax equilibrium we need to find the point at which the quantity demanded by consumers, at the price they pay, is equal to the quantity supplied by firms at the price they receive. This is the quantity that solves the equation, pd = ps + t, which is to say, 700 − 5Y = 2Y + 50 ⇒ YtSR = 92.9. Plugging this quantity back into the inverse supply curve we get ps = 2 · YtSR = 185.8, which means the price paid by consumers is pd = ps + t = 185.8 + 50 = 135.8. The change in producer surplus is the area to the left of the supply curve between the pre-tax price and the after-tax price received by firms. It includes the firms’ share of the tax revenue as well as the part of deadweight loss that comes from firms. In the case of linear supply it is the area of a parallelagram with height equal to the difference between the pre-tax price and the after-tax price received by firms, and bases of Y ∗ and YtSR , which is ∆P SS R = (200 − 185.7) 100−92.9 = 1379.2. 2 7
By now your graph should look like this:
In an industry with identical firms the long-run supply curve is horizontal, which is to say, in long-run equilibrium firms will be earning zero profit because entry and exit will always drive the price down (or in this case up) to the point where the price is equal to the minimum average cost.
Thus, the after-tax price received by firms will be ps = 200. Otherwise firms would be losing money and would have an incentive to leave the industry, and the industry would not be in long-run equilibrium. Thus, we know that the tax will be passed on entirely to consumers, which means that the price paid by consumers will be pd = ps + t = 200 + 50 = 250. Setting the inverse demand curve equal to that price, we can compute the long-run after-tax equilibrium quantity, 250 = 700 − 5Y ⇒ YtLR = 90. To determine the number of firms in the industry we have to know how much output each firm will produce when they are operating at their minimum average cost. We computed the direct supply curve of p the firm in part (a), y(p) = 20 , which means that at the minimum of their average cost, minAC = 200, each firm will produce 200 = 10 units of output. Since the 20 industry as a whole is producing 90 units, there must be 9 firms in the industry. One has exited the industry. Your graph should look like
this:
In an industry with identical firms, by definition, the long-run producer surplus is zero. There are two ways to see this. The first is that the long-run supply curve is horizontal, which means that in long-run equilibrium the price is the same as the height of the supply curve, and since producer surplus is the area between the price line and the supply curve, there clearly can be no producer surplus. The other way to see it is to refer to the definition of long-run equilibrium in an industry with identical firms, which is that all firms are earning zero profit. The reason this is different from the answer to ii, above, is that in the long-run firms can escape the burden of the tax by leaving the industry and going into some other industry that is not taxed. We know that the burden of a tax always falls most heavily on the side of the market that is less able to change it’s behavior to escape the tax, which is to say, the side of the market that is most inelastic. In the long-run, the supply side of the industry is perfectly elastic, and thus bears none of the burden of the tax.
8
True/False
(2 questions, 10 points total)
Answer true or false and explain your answer. Your answer must fit in the space provided. T/F 1. (5 points) Suppose the government wants to place a tax on one of two goods, and suppose that supply is perfectly elastic for both goods. If the government wants to minimize the deadweight loss from a tax of a given size, it should put the tax on whichever good has worse substitutes.
False: If the supply curves are identical, the only factor that determines the amount of deadweight loss is the elasticity of demand. Placing the tax on the good that has the lower elasticity of demand will minimize the deadweight loss of the tax. It is true …show more content…
that, holding all else equal, a good without good substitutes will have more inelastic demand than a good with good substitutes. However, this is not the only factor that determines the elasticity of demand. The goods could also differ in terms of the income effect. If the good with worse substitutes happened to be strongly normal while the good with better substitutes was strongly inferior, then the income effects might overwhelm the substitution effects, causing the good with better substitutes to be more inelastic. T/F 2. (5 points) In a perfectly competitive market with no taxes, if the price consumers are willing to pay for the marginal unit is the same as the price at which producers are willing to produce the marginal unit, then there will be no way to make anyone in the market better off without making someone else worse off.
True. The price consumers are willing to pay for the marginal unit is the height of the inverse demand curve, and the price at which producers are willing to produce the marginal unit is the height of the inverse supply curve. Thus, when these prices are equal, it must be the case that supply is equal to demand, which is to say, the market is in equilibrium. If the quantity firms produce, and consumers consume, is more than the equilibrium quantity, then the firms’ cost of production will be greater than the consumers’ willingness to pay, and either consumers will have to pay more than the units are worth to them, making them worse off, or firms will have to receive less than the units cost them, making them worse off, or both. If the quantity is less than equilibrium, then there will be units not produced or consumed for which the cost of production would have been less than consumers’ willingness to pay, meaning that either firms have given up profitable units, or consumers have given up units that generated consumer surplus, or both. In any case, at least one side of the market will have been made worse off. Thus, from equilibrium there is no way that either firms or consumers can be made better off without someone being made worse off.
1
Short Answer
(2 questions, 20 points total)
Your answer must fit in the space provided.
SA 2. (10 points) Explain what we mean when we say that firms in long-run equilibrium are earning zero profit even though their owners and investors are making an adequate return on their labor and investments.
The statement refers to “economic profit”, which is the difference between revenue and opportunity cost. The opportunity cost of the labor of the owner of a firm is the wage the owner could have earned if he or she chose not to run the firm, but to get a job instead. The opportunity cost of the capital investors invest in a firm is the rate of return they could have earned by investing their capital in some other firm in some other industry. Thus, if the owner of the firm receives an amount just equal to the opportunity cost of their labor, and the investors receive an amount just equal to the opportunity cost of their capital, we do not include those amounts in economic profit, and the firm will be said to be earning zero economic profit, even though an accountant would say that both the owner and the investors are making an “accounting profit”. The accounting profit earned by the owner and the investors is the amount of money that is just adequate to make them choose to put their labor and capital into the firm.
2
Problem Solving
(2 problems, 50 points total)
Problem 1. (26 points total) Consider a perfectly competitive firm with a production technology 1 1 represented by the production function, y = 10 K 2 + L 2 . Let p, r, and w be the price of the firm’s output, the rental rate of capital, and the wage, respectively. (a) (8 points) First let’s consider long-run profit maximization. (i) Set up the firm’s long-run profit maximization problem and compute the firm’s profitmaximizing demand for labor and capital, and profit-maximizing output, as functions of p, r, and w. (ii) Is labor a gross complement or a gross substitute for capital, or neither. Prove your answer mathematically and explain what it means.
The long-run profit maximization problem is, max p · 10
K,L
√ K+
√ L
The first-order conditions are, 5p 5p for L: √L − w = 0 for K: √K − r = 0 Solving these for L and K respectively we get L∗ (p, r, w) = (f rac5pw)2 and K ∗ (p, r, w) = (f rac5pr)2 . Plugging these profit-maximizing levels of capital and labor into the production function we get the profit-maximizing output of the firm, y ∗ (p, w, r) = y(K ∗ , L∗ ) = 10
5p r 2
,
5p w
2
= 50p
r+w rw
.
To determine whether labor is a gross complement or gross substitute for capital we take the partial derivative of the labor demand function with respect to the rental ∗ rate of capital, ∂L = 0. Since this is zero, labor is neither a gross complement ∂r nor a gross substitute for capital. What this means is that when the price of capital changes, the amount of labor the firm uses will not change. (b) (8 points) Set up the firm’s cost-minimization problem and compute the firm’s conditional demand for labor and capital, as functions of y, r, and w. The firm’s cost minimization problem is, √ min rK + wL
K,L
√ K+ L =y ¯
s.t. 10
Setting up the LaGrangian function, this minimization problem becomes, min rK + wL − λ 10 √ K+ √ L −y ¯ √
K,L,λ
The first-order conditions are,
5 for L: w − λ √L = 0 for K: r − λ √5 = 0 for λ: 10 K the production constraint.
√
K+
L = y , which is just ¯ w 2 L. r
Taking the ratio of the first two conditions we get this into the production constraint we get, 10 3
√ √K = w ⇒ r L √ √ w r L+ L
K=
Plugging
= y ⇒ L∗ (y; r, w) = ¯
y2
r 10(r+w)
2
. Plugging this back into the expression for K that we derived earlier
2
w we get, K ∗ (y; r, w) = y 2 10(r+w) labor and capital respectively.
. These are the firm’s conditional demand for
(c) (10 points) Now let’s consider scale and substitution effects. Assume that initially the price of the firm’s output, p, the rental rate of capital, r, and the wage, w, are all equal to 10. (i) How much labor will the firm use at these prices, and how much output will it produce? (ii) Using only the mathematical results you got in parts (a) and (b), compute effect of an increase in the rental rate to r = 20.
Plugging the given prices into the profit-maximizing labor demand and output supply 2 functions from part (a) we get, L∗ (p, w, r) = 5·10 = 25, and y ∗ (p, w, r) = 50 · 10 10 (f rac10 + 1010 · 10) = 100.
∗
−
you might have plugged the new prices into the firm’s supply function to get y ∗ (10, 10, 20) = 50·10 10+20 = 75. If you then plugged this into the 10·20 firm’s conditional factor demand at the new prices you would get L∗ (75; 10, 20) =
75 20 10 10+20 2
= 25.
4
Problem 2. (24 points total) Consider a perfectly competitive industry with 10 identical firms, each of which has variable costs of 10y 2 and fixed costs of 1000. We will define the short run as the time scale in which firms cannot enter or exit the industry, and cannot avoid their fixed costs. (In other words, in the short run firms must continue to pay their fixed costs even if they produce zero output.) In the long run, firms can enter or exit the industry, and can avoid their fixed costs by shutting down. (a) (8 points) Compute the short-run inverse supply curve of the firm, and the short-run inverse supply curve of the industry, and graph them on the same graph. [Hint: it matters a lot that firms can’t avoid their fixed costs in the short …show more content…
run.]
Each firm’s cost function is C(y) = 10y 2 + 1000, and the marginal cost curve is M C = 20y. Normally we say that the inverse supply curve of the firm is the upward sloping part of the marginal cost curve, above the minimum of the average cost curve, because if the price is below the minimum of the average cost curve, the firm will make negative profit and will shut down. However, in this case, in the short run, if a firm shuts down it will still have to pay its fixed cost of $1000. As a result, it will continue to produce output even if it is losing money, as long as it does not lose more than $1000. So we need to find the price below which the firm will have lose more than $1000. Profit is py − 10y 2 − 1000 and we want the price below which this is less than −1000. To do this we have to plug in the firm’s profit-maximizing quantity as a function of price, which we get by solving the firm’s marginal cost curve p p p 2 to get y ∗ = 20 , which gives us p 20 − 10 20 − 1000 = −1000 ⇒ p2 19 = 0 ⇒ p = 0. 40 The firm will continue to produce at any positive price rather than shut down and
5
pay its fixed cost without any revenue. Thus, the firm’s inverse supply curve is simply the entire marginal cost curve, p(y) = 20y. To compute the short-run inverse supply curve of the industry we first have to aggregate firm supply to industry supply, and to do that we have to have the direct supply curve of the firm, which we get by solving the inverse supply curve for y to p p get y(p) = 20 . Short-run industry supply is Y (p) = N yj (p) = 10 20 = f racp2. j=1 Solving for p we get the short-run inverse supply curve of the industry, p(Y ) = 2Y . Your graph should look like this:
(b) (6 points) Suppose the demand for the industry’s product is defined by pd (Y ) = 700 − 5Y . (i) What will be the short-run equilibrium price and quantity for the industry? Illustrate this equilibrium on a graph. (ii) Explain why this market outcome is an equilibrium in the short run. [Be sure to make reference to the general definition of equilibrium in your answer.] (iii) Is this industry in long-run equilibrium? Explain why or why not. [Again, be sure to make reference to the general definition of equilibrium in your answer.]
The short-run market equilibrium is where the quantity demanded at the price paid by consumers is equal to the quantity supplied at the price received by producers, and since, in the absence of a tax, the price paid by consumers is the same as the price paid by producers, we just solve for the intersection of the supply curve and the demand curve: 700 − 5Y = 2Y ⇒ Y ∗ = 100. Plugging that into either the demand or the supply curve we get p(Y ) = 200. Your graph should look like this:
In general, equilibrium means that no individual agent has an incentive to do anything other than what they are currently doing, which means that the system will 6
not move from the point it is at. In the case of short-run market equilibrium this means that at the market price consumers cannot be made better off by increasing or decreasing consumption, and firms cannot be made better off by increasing or decreasing production. This is clearly the case at the market equilibrium we have solved for. If consumers increase consumption they will have to pay more for the additional units of the good than the value of those units, and if they consume less they will be giving up units that are worth more to them than they are required to pay for them. In either case, they are made worse off, and thus have no incentive to change. For firms, roughly the same argument applies. If they produce more, the maximum they will be able to charge will be less than the cost of production, and if they produce less they will be giving up units that they were able to sell at a profit. In either case, firms are worse off, so they have no incentive to change what they were doing. The industry is in long-run equilibrium. To see this we need to know whether firms are earning zero profit, and to determine that we need to know something about the firm’s average cost curve, which is AC = 10y + 1000 . If we minimize this we find y that the firms’ minimum average cost is minAC = 200. And since this is equal to the price in the current equilibrium, firms’ profit is (p − AC)y = 0y = 0. Long-run equilibrium is defined as the point at which firms will have no incentive to enter or exit the industry. The reason firms enter or exit is in response to profits being either positive or negative, so if profits are zero in the industry there will be no incentive to enter or exit, which is to say, no firm will have any incentive to do anything different from what they are currently doing. (c) (10 points) Suppose the government imposes a tax of $50 per unit on the firms in the industry. (i) Compute the short-run after-tax equlibrium quantity, price paid by consumers, and price received by firms, and graph them. (ii) Calculate the change in producer surplus caused by the tax in the short-run. Add it to your graph. (iii) Compute the long-run after-tax equilibrium quanitity, price paid by consumers, and price received by firms. Add this equilibrium to your graph. How many firms will exit the industry? (iv) Calculate the change in producer surplus caused by the tax in the long-run. Why is this the same or different from your answer to ii above?
To compute the short-run after tax equilibrium we need to find the point at which the quantity demanded by consumers, at the price they pay, is equal to the quantity supplied by firms at the price they receive. This is the quantity that solves the equation, pd = ps + t, which is to say, 700 − 5Y = 2Y + 50 ⇒ YtSR = 92.9. Plugging this quantity back into the inverse supply curve we get ps = 2 · YtSR = 185.8, which means the price paid by consumers is pd = ps + t = 185.8 + 50 = 135.8. The change in producer surplus is the area to the left of the supply curve between the pre-tax price and the after-tax price received by firms. It includes the firms’ share of the tax revenue as well as the part of deadweight loss that comes from firms. In the case of linear supply it is the area of a parallelagram with height equal to the difference between the pre-tax price and the after-tax price received by firms, and bases of Y ∗ and YtSR , which is ∆P SS R = (200 − 185.7) 100−92.9 = 1379.2. 2 7
By now your graph should look like this:
In an industry with identical firms the long-run supply curve is horizontal, which is to say, in long-run equilibrium firms will be earning zero profit because entry and exit will always drive the price down (or in this case up) to the point where the price is equal to the minimum average cost.
Thus, the after-tax price received by firms will be ps = 200. Otherwise firms would be losing money and would have an incentive to leave the industry, and the industry would not be in long-run equilibrium. Thus, we know that the tax will be passed on entirely to consumers, which means that the price paid by consumers will be pd = ps + t = 200 + 50 = 250. Setting the inverse demand curve equal to that price, we can compute the long-run after-tax equilibrium quantity, 250 = 700 − 5Y ⇒ YtLR = 90. To determine the number of firms in the industry we have to know how much output each firm will produce when they are operating at their minimum average cost. We computed the direct supply curve of p the firm in part (a), y(p) = 20 , which means that at the minimum of their average cost, minAC = 200, each firm will produce 200 = 10 units of output. Since the 20 industry as a whole is producing 90 units, there must be 9 firms in the industry. One has exited the industry. Your graph should look like
this:
In an industry with identical firms, by definition, the long-run producer surplus is zero. There are two ways to see this. The first is that the long-run supply curve is horizontal, which means that in long-run equilibrium the price is the same as the height of the supply curve, and since producer surplus is the area between the price line and the supply curve, there clearly can be no producer surplus. The other way to see it is to refer to the definition of long-run equilibrium in an industry with identical firms, which is that all firms are earning zero profit. The reason this is different from the answer to ii, above, is that in the long-run firms can escape the burden of the tax by leaving the industry and going into some other industry that is not taxed. We know that the burden of a tax always falls most heavily on the side of the market that is less able to change it’s behavior to escape the tax, which is to say, the side of the market that is most inelastic. In the long-run, the supply side of the industry is perfectly elastic, and thus bears none of the burden of the tax.
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