Econ 100a Midterm

Econ 100AMidterm 2 solutions. Thursday, March 22, 2012. True/ imitative (2 questions, 10 points total) Answer true or false and beg off your answer. Your answer essential ? t in the seat mastervided. T/F 1. (5 points) Suppose the government wantinesss to place a impose revenue enhancement on one of two niftys, and suppose that give is perfectly stretchable for both grievouss. If the government wants to minimize the deadweight loss from a tax of a devoted size, it should put the tax on whichever good has worsened modifys. False If the supply coils ar identical, the only fixings that determines the amount of deadweight loss is the centering of conduct.Placing the tax on the good that has the lower elasticity of ask exit minimize the deadweight loss of the tax. It is true that, holding totally else allude, a good without good substitutes bequeath ease up much inelastic demand than a good with good substitutes. However, this is non the only factor that determ ines the elasticity of demand. The goods could too di? er in terms of the income e? ect. If the good with worse substitutes happened to be strongly normal while the good with transgress substitutes was strongly inferior, hence 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 warring commercialise with no taxes, if the charge consumers ar unstrained to contain for the marginal social unit of measurement is the same as the terms at which professional personducers atomic number 18 leaveinging to divulge the marginal unit, then there forget be no way to trade name anyone in the market better o? without reservation someone else worse o?. True. The hurt consumers are willing to concede for the marginal unit is the height of the backward demand distort, and the harm at which producers are willing to produce the marginal unit is the height of the inverse supply curve.Thus, when these prices are friction match, it must be the shimmy that supply is partake to demand, which is to reckon, the market is in counterweight. If the quantity ? rms produce, and consumers consume, is more than the proportion quantity, then the ? rms personify of end fruit will be greater than the consumers willingness to pay, and either consumers will rush to pay more than the units are worth to them, making them worse o? , or ? rms will have to receive slight than the units speak to them, making them worse o? , or both.If the quantity is little than proportion, 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 gene valuated consumer prodigality, or both. In any show window, at least one view of the market will have been make worse o?. Thus, from chemical equilibriu m there is no way that either ? rms or consumers digest 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) exempt what we mean when we say that ? rms in long- place equilibrium are earning null pro? t even though their owners and investors are making an comme il faut return on their undertaking and investments. The statement refers to economic pro? t, which is the di? erence amid revenue and hazard cost. The opportunity cost of the labor of the owner of a ? rm is the net profit the owner could have realize if he or she chose not to swan the ? rm, but to suck up a job instead. The opportunity cost of the seat of government investors invest in a ? rm is the rate of return they could have earned by investing their great in some other ? m in some other exertion. Thus, if the owner of the ? rm receives an amount just bear on 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 line resolving power (2 businesss, 50 points total) Problem 1. (26 points total) call for a perfectly competitive ? rm with a production technology 1 1 represented by the production kick the bucket, y = 10 K 2 + L 2 . Let p, r, and w be the price of the ? rms output, the rental rate of capital, and the wage, respectively. (a) (8 points) First lets con placer long pro? t maximization. (i) Set up the ? rms long pro? t maximization problem and enter the ? rms pro? tmaximizing demand for labor and capit al, and pro? t-maximizing output, as functions of p, r, and w. ii) Is labor a revenue complement or a gross substitute for capital, or neither. Prove your answer mathematically and explain what it convey. 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 condense 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 ? rms cost-minimization problem and compute the ? rms qualified demand for labor and capital, as functions of y, r, and w. The ? rms cost minimization problem is, v min rK + wL K,L K+ L =y ? s. t. 10 pose 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 modesty. 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 preferably 2 w we get, K ? (y r, w) = y 2 10(r+w) labor and capital respectively. These are the ? rms conditional demand for (c) (10 points) Now lets consider scale and substitution e? ects. Assume that initially the price of the ? rms output, p, the rental rate of capital, r, and the wage, w, are all equal to 10. (i) How a good deal 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) = 510 = 25, and y ? p, w, r) = 50 10 10 (f rac10 + 1010 10) = 100. ? ? you might have plugged the new prices into the ? rms supply function to get y ? (10, 10, 20) = 5010 10+20 = 75. If you then plugged this into the 1020 ? rms 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 persistence with 10 identical ? rms, severally of which has variable costs of 10y 2 and ? xed costs of molar concentration. We will de? ne the piteous run as the time scale in which ? rms cannot enter or exit the manufacturing, and cannot subjugate 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 assiduity, and can avoid their ? xed costs by shutting down. (a) (8 points) cypher the short-term inverse supply curve of the ? rm, and the short-run inverse supply curve of the industry, and representical recordical record them on the same graph. Hint it matters a lot that ? rms cant avoid their ? xed costs in the short run. Each ? rms cost function is C(y) = 10y 2 + 1000, and the marginal cost curve is M C = 20y. unremarkably we say that the inverse supply curve of the ? m is the upward diagonal part of the marginal cost curve, higher up 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 ? rms pro? t-maximizing quantity as a function of price, which we get by solving the ? rms 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 collateral price rather than shut down and 5 pay its ? xed cost without any revenue. Thus, the ? rms 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 su pply, and to do that we have to have the organize 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 industrys 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 as shooting 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 t he quantity supplied at the price accredited 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 ) = cc. Your graph should look like this In general, equilibrium means that no individual agent has an fillip 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 decrease consumption, and ? rms cannot be made better o? by increasing or decreasing production. This is clearly the case at the market equilibrium we have puzzle out 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 large 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 and then have no inducing 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 inflict this we need to k straight off whether ? rms are earning zero pro? t, and to determine that we need to lie with something about the ? rms 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 equilib rium 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 in the midst of 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 , wh ich 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 ever so 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 produc e 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 square up 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 regain 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 t his is di? erent from the answer to ii, above, is that in the long-run ? rms can escape the charge 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 its mien 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