Econ 100a Midterm
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 ? t 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 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 di? er in terms of the income e? ect. If the good with worse substitutes happened to be strongly normal while the good with better substitutes was strongly inferior, then the income e? ects might overwhelm the substitution e? cts, 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 o? without making someone else worse o?. 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 ? rms produce, and consumers consume, is more than the equilibrium quantity, then the ? rms’ 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 o? , or ? rms will have to receive less than the units cost them, making them worse o? , 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 ? rms have given up pro? table 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 o?. Thus, from equilibrium there is no way that either ? rms or consumers can be made better o? without someone being made worse o?. 1 Short Answer (2 questions, 20 points total) Your answer must ? t in the space provided. SA 2. 10 points) Explain what we mean when we say that ? rms in long-run equilibrium are earning zero pro? t even though their owners and investors are making an adequate return on their labor and investments. The statement refers to “economic pro? t”, which is the di? erence between revenue and opportunity cost. The opportunity cost of the labor of the owner of a ? rm is the wage the owner could have earned if he or she chose not to run the ? rm, but to get a job instead. The opportunity cost of the capital investors invest in a ? rm is the rate of return they could have earned by investing their capital in some other ? m in some other industry. Thus, if the owner of the ? rm 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 pro? t, and the ? rm will be said to be earning zero economic pro? t, even though an accountant would say that both the owner and the investors are making an “accounting pro? t”. The accounting pro? t 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 ? m. 2 Problem Solving (2 problems, 50 points total) Problem 1. (26 points total) Consider a perfectly competitive ? rm 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 ? rm’s output, the rental rate of capital, and the wage, respectively. (a) (8 points) First let’s consider long-run pro? t maximization. (i) Set up the ? rm’s long-run pro? t maximization problem and compute the ? rm’s pro? tmaximizing demand for labor and capital, and pro? t-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 pro? t maximization problem is, max p · 10 K,L v K+ v L The ? rst-order conditions are, 5p 5p for L: vL ? w = 0 for K: vK ? 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 pro? t-maximizing levels of capital and labor into the production function we get the pro? t-maximizing output of the ? rm, 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 ? rm uses will not change. (b) (8 points) Set up the ? rm’s cost-minimization problem and compute the ? rm’s conditional demand for labor and capital, as functions of y, r, and w. The ? rm’s cost minimization problem is, v min rK + wL K,L K+ L =y ? s. t. 10 Setting up the LaGrangian function, this minimization problem becomes, min rK + wL ? ? 10 v K+ v L ? y ? v K,L,? The ? rst-order conditions are, 5 for L: w ? ? vL = 0 for K: r ? ? v5 = 0 for ? : 10 K the production constraint. v K+ L = y , which is just ? w 2 L. r Taking the ratio of the ? rst two conditions we get this into the production constraint we get, 10 3 v vK = w ? r L v v 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 ? rm’s conditional demand for (c) (10 points) Now let’s consider scale and substitution e? ects. Assume that initially the price of the ? rm’s output, p, the rental rate of capital, r, and the wage, w, are all equal to 10. (i) How much labor will the ? rm 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 e? ect of an increase in the rental rate to r = 20. Plugging the given prices into the pro? t-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 ? rm’s supply function to get y ? (10, 10, 20) = 50·10 10+20 = 75. If you then plugged this into the 10·20 ? rm’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 ? rms, each of which has variable costs of 10y 2 and ? xed costs of 1000. We will de? ne the short run as the time scale in which ? rms cannot enter or exit the industry, and cannot avoid their ? xed costs. In other words, in the short run ? rms must continue to pay their ? xed costs even if they produce zero output. ) In the long run, ? rms can enter or exit the industry, and can avoid their ? xed costs by shutting down. (a) (8 points) Compute the short-run inverse supply curve of the ? rm, and the short-run inverse supply curve of the industry, and graph them on the same graph. [Hint: it matters a lot that ? rms can’t avoid their ? xed costs in the short run. ] Each ? rm’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 ? m 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 ? rm will make negative pro? t and will shut down. However, in this case, in the short run, if a ? rm shuts down it will still have to pay its ? xed 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 ? nd the price below which the ? rm will have lose more than $1000. Pro? t 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 ? rm’s pro? t-maximizing quantity as a function of price, which we get by solving the ? rm’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 ? rm will continue to produce at any positive price rather than shut down and 5 pay its ? xed cost without any revenue. Thus, the ? rm’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 ? rst have to aggregate ? rm supply to industry supply, and to do that we have to have the direct supply curve of the ? m, 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 de? ned 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 de? ition 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 de? nition 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 o? by increasing or decreasing consumption, and ? rms cannot be made better o? 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 o? , and thus have no incentive to change. For ? rms, 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 pro? t. In either case, ? ms are worse o? , 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 ? rms are earning zero pro? t, and to determine that we need to know something about the ? rm’s average cost curve, which is AC = 10y + 1000 . If we minimize this we ? nd y that the ? rms’ minimum average cost is minAC = 200. And since this is equal to the price in the current equilibrium, ? rms’ pro? t is (p ? AC)y = 0y = 0. Long-run equilibrium is de? ned as the point at which ? rms will have no incentive to enter or exit the industry. The reason ? ms enter or exit is in response to pro? ts being either positive or negative, so if pro? ts are zero in the industry there will be no incentive to enter or exit, which is to say, no ? rm will have any incentive to do anything di? erent from what they are currently doing. (c) (10 points) Suppose the government imposes a tax of $50 per unit on the ? rms in the industry. (i) Compute the short-run after-tax equlibrium quantity, price paid by consumers, and price received by ? rms, 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 ? rms. Add this equilibrium to your graph. How many ? rms 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 di? erent from your answer to ii above? To compute the short-run after tax equilibrium we need to ? nd the point at which the quantity demanded by consumers, at the price they pay, is equal to the quantity supplied by ? rms at the price they receive. This is the quantity that solves the equation, pd = ps + t, which is to say, 700 ? Y = 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 ? rms. It includes the ? rms’ share of the tax revenue as well as the part of deadweight loss that comes from ? rms. In the case of linear supply it is the area of a parallelagram with height equal to the di? erence between the pre-tax price and the after-tax price received by ? rms, and bases of Y ? nd 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 ? rms the long-run supply curve is horizontal, which is to say, in long-run equilibrium ? rms will be earning zero pro? t 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 ? rms will be ps = 200. Otherwise ? rms 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 ? rms in the industry we have to know how much output each ? rm will produce when they are operating at their minimum average cost. We computed the direct supply curve of p the ? rm in part (a), y(p) = 20 , which means that at the minimum of their average cost, minAC = 200, each ? rm will produce 200 = 10 units of output.
Since the 20 industry as a whole is producing 90 units, there must be 9 ? rms in the industry. One has exited the industry. Your graph should look like this: In an industry with identical ? rms, by de? nition, the long-run producer surplus is zero. There are two ways to see this. The ? rst 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 de? ition of long-run equilibrium in an industry with identical ? rms, which is that all ? rms are earning zero pro? t. The reason this is di? erent from the answer to ii, above, is that in the long-run ? rms 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