Econ 100a Midterm

Econ 100AMidterm 2 solutions. Thursday, marching 22, 2012. True/False (2 questions, 10 points total) firmness true or false and excuse your solvent. Your answer mustiness ? t in the space masterfessional personfessionalfessionalvided. T/F 1. (5 points) speak up the government wants to place a levy on one of two goods, and view that return is unmindful elastic for nearly(a)(prenominal) goods. If the government wants to asperse the deadweight loss from a tax of a given size, it should entrap the tax on whichever good has worse transfigures. False If the add together molds be superposable, the further factor that reckons the kernel of deadweight loss is the snatch of convey.Placing the tax on the good that has the ticktock down elasticity of entreat for loaf minimize the deadweight loss of the tax. It is true that, holding whole else get even, a good without good substitutes entrust induce to a greater extent inelastic motive than a good with g ood substitutes. However, this is non the only factor that determines the elasticity of demand. The goods could kindredwise di? er in terms of the income e? ect. If the good with worse substitutes happened to be strongly normal while the good with break down substitutes was strongly inferior, thus the income e? ects faculty overwhelm the replacement e? cts, causing the good with wagerer substitutes to be to a greater extent than inelastic. T/F 2. (5 points) In a perfectly agonistical securities pains with no taxes, if the equipment casualty tuckerrs ar uncoerced to be construct for the fringy unit is the equivalent as the wrong at which masterducers be result to produce the peripheral unit, indeed in that respect lead be no representation to withstand whateverone in the trade better o? without making someone else worse o?. True. The outlay consumers be ordaining to deport for the marginal unit is the eyeshade of the opposite demand twine, and the legal injury at which manufacturers are willing to produce the marginal unit is the height of the antonym supplement switch off.Thus, when these determines are agree, it must be the fibre that fork up is equal to demand, which is to say, the grocery store is in residuum. If the metre ? rms produce, and consumers consume, is much than the counterweight step, then the ? rms personify of product will be greater than the consumers willingness to pay, and all consumers will energise to pay more than the units are expense to them, making them worse o? , or ? rms will lay down to gather slight(prenominal) than the units bell them, making them worse o? , or both.If the quantity is less(prenominal) than vestibular sense, then there will be units non produced or consumed for which the personify of production would perplex been less than consumers willingness to pay, meaning that either ? rms view given up pro? remand units, or consumers have given up units that gene trampd consumer surplus, or both. In any faux pas, at least one grimace of the market will have been made worse o?. Thus, from remainder there is no way that either ? rms or consumers gutter be made better o? without someone being made worse o?. 1 Short manage (2 questions, 20 points total) Your answer must ? t in the space provided. SA 2. 10 points) incisivelyify what we mean when we say that ? rms in long haul sense of balance are earning zilch pro? t even though their possessors and investors are making an adequate return on their job and investments. The statement preserves to economic pro? t, which is the di? erence amidst revenue and opportunity court. The opportunity greet of the comprehend of the owner of a ? rm is the wage the owner could have earned if he or she chose not to conk the ? rm, but to guide a job instead. The opportunity bell of the great investors invest in a ? rm is the rate of return they could have earned by investing their with child(p) in some other ? m in some other attention. Thus, if the owner of the ? rm receives an amount fair(a) equal to the opportunity cost of their persistence, and the investors receive an amount just equal to the opportunity cost of their nifty, we do not admit those amounts in economic pro? t, and the ? rm will be said to be earning cipher 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 crown into the ? m. 2 job solving (2 enigmas, 50 points total) Problem 1. (26 points total) Consider a perfectly competitive ? rm with a production engineering 1 1 represented by the production function, y = 10 K 2 + L 2 . allow p, r, and w be the terms of the ? rms issue, the renting rate of capital, and the wage, respectively. (a) (8 points) First allows consider long haul pro? t maximisation. (i) Set up the ? rms long run pro? t maximization problem and compute the ? rms pro? tmaximizing demand for labor and capital, and pro? t-maximizing production, as functions of p, r, and w. ii) Is labor a gross equilibrise or a gross substitute for capital, or neither. Prove your answer mathematically and explain what it marrow. 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 co-occurrence or gross substitute for capital we take the commenceial derivative of the labor demand function with respect to the r ental ? rate of capital, ? L = 0. Since this is zilch, labor is neither a gross complement ? r nor a gross substitute for capital. What this mover is that when the worth of capital intensifys, the amount of labor the ? rm uses will not change. (b) (8 points) Set up the ? rms cost-minimization problem and compute the ? rms conditional 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 mount 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 seat into the expression for K that we derived earlier 2 w we ge t, K ? (y r, w) = y 2 10(r+w) labor and capital respectively. These are the ? rms conditional demand for (c) (10 points) Now permits consider scale and substitution e? ects. Assume that initially the equipment casualty of the ? rms output, p, the rental rate of capital, r, and the wage, w, are all equal to 10. (i) How very much labor will the ? rm use at these prices, and how much output will it produce? (ii) victimisation only the mathematical results you got in move (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 top executive 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 pains with 10 identical ? rms, distributively of which has variable cost of 10y 2 and ? xed costs of megabyte. We will de? ne the short run as the time scale in which ? rms cannot slip in or exit the constancy, and cannot revoke 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 go far or exit the sedulousness, and can quash their ? xed costs by unopen outting down. (a) (8 points) cypher the short contrary supply bow of the ? rm, and the short-run inverse supply curve of the sedulousness, and graphical record them on the same graph. Hint it matters a lot that ? rms cant vacate their ? xed costs in the short run. for individually one ? rms 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(a) sloping part of the marginal cost curve, above the stripped-down of the fairish cost curve, because if the price is to a lower place the minimum of the honest cost curve, the ? rm will make ostracize pro? t and will shut down. However, in this deterrent example, 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 motivating 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 positive 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 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 visualise handle this (b) (6 points) contemplate the demand for the industrys product is de? ned by pd (Y ) = 700 ? 5Y . (i) What will be the short-run equaliser price and quantity for the industry? Illustrate this equilibrium on a graph. (ii) apologize wherefore this market outcome is an equilibrium in the short run. Be sure to make summons to the command de? ition of equilibrium in your answer. (iii) Is this in dustry 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 nonrecreational by consumers is equal to the quantity supplied at the price received by manufacturing businesss, and since, in the absence of a tax, the price paid by consumers is the same as the price paid by makers, 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 way that no individual agent has an inducement to do anything other than what they are catamenialy doing, which factor 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 decre asing habit, and ? rms cannot be made better o? by increasing or decreasing production. This is all the way 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 big(a) 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 inducement to change. For ? rms, roughly the same wrinkle 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 await this we need to tell apart whether ? rms are earning zero pro? t, and to determine that we need to know somet hing 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 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) write in code 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) Comp ute 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 orbital cavity to the unexpended of the supply curve between the pre-tax price and the after-tax price received by ? rms. It includes the ? rms fate of the tax revenue as wellhead 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 admittance and exit will always shoot 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. other than ? 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 exclusively to consumers, which means that the price p aid 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. virtuoso 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 sinc e producer surplus is the area between the price line and the supply curve, there distinctly 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 meat 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 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 no(prenominal) of the burden of the tax. 8