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One Input and One Output: A Short-Run Producer Model
Solutions for Microeconomics: An Intuitive Approach
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One Input and One Output: A Short-Run Producer Model 218
11.1 Throughout the text, we used the technology we called more realistic in panel (b) of Graph 11.1. Suppose now that the producer choice set was instead strictly convex everywhere.
(a) Illustrate what such a technology would look like in terms of a production frontier. Answer: This is illustrated in panel (a) of Graph 11.1.
Graph 11.1: Production when Choice Set is Convex
(b) Derive the output supply curve with price on the vertical and output on the horizontal axis (in
graphs analogous to those in Graph 11.9) for this technology.
Answer: In panel (a) of Graph 11.1, two isoprofits corresponding to a low and a high price
level are illustrated tangent to the producer choice set. The dashed one corresponds to the
maximizing production plans at these two prices are then given by A and B . Panel (b) of the
graph then plots the output levels x A and x B from these optimal production plans for the two
price levels pA and pB on the horizontal axis. Panel (c) then simply flips the axes giving us an upward sloping output supply curve.
(c) Derive the labor demand curve for such a technology.
Answer: This is done in panel (d) of Graph 11.1 where the marginal revenue product which is
derived from the slope of the production frontier is downward sloping throughout (because the slope of the production frontier is becoming shallower throughout).
219 One Input and One Output: A Short-Run Producer Model
(d) Now suppose the technology were instead such that the marginal product of labor is always in- creasing. What does this imply for the shape of the producer choice set?
Answer: This implies that the production frontier gets steeper throughout which in turn implies that the producer choice set is non-convex as drawn in Graph 11.2.
Graph 11.2: Increasing Marginal Product of Labor
(e) How much should the firm produce if it is maximizing its profits in such a case? (Hint: Consider
Answer: If the firm is a price taker, then it should produce an infinite amount. This is illustrated in Graph 11.2 where three parallel isoprofits are drawn. The lowest is tangent to the production frontier, but this is not a profit maximizing production plan because we can still go to higher isoprofits that contain production plans which are technologically feasible. In fact, we can keep going to higher and higher isoprofits that continue to intersect the production frontier and thus can keep going up.
One Input and One Output: A Short-Run Producer Model 220
11.2 Below, we will investigate the profit maximizing choice in the two steps that first involve a strict focus on the cost side.
Consider again (as in the previous exercise) a production process that gives rise to a strictly convex pro- ducer choice set.
(a) Derive the cost curve from a picture of the production frontier.
Answer: This is done in panels (a) through (c) of Graph 11.3. Panel (a) illustrates the producer choice set. Panel (b) inverts this, flipping the axes so that x is on the horizontal and l on the vertical. Finally, (c) is derived from (b) by simply multiplying labor input by w to convert the labor input needs of production into the dollar needs for production.
Graph 11.3: Cost Curves and Supply when Choice Set is Convex
(b) Derive the marginal and average cost curves from the cost curve.
Answer: This is done in panel (d) where the MC curve is simply plotting the slope of the cost curve and the average cost curve is simply the total cost divided by x.
(c) Illustrate the supply curve on your graph. How does it change if the wage rate increases? Answer: The supply curve is the part of the MC curve that lies above the AC curve. Since the entire MC curve lies above the AC curve in this case, the supply curve is simply equal to the MC curve. If wages go up, this rotates up the c curve in panel (c) causing an increase in its slope at all output levels. As a result, the MC curve shifts/rotates up which implies the supply curve similarly shifts up.
(d) Now suppose the production process gives rise to increasing marginal product of labor through- out. Derive the cost curve and from it the marginal and average cost curves.
Answer: This is done in Graph 11.4 (next page) and follows steps analogous to those we just went through in Graph 11.3.
One Input and One Output: A Short-Run Producer Model
Graph 11.4: Cost Curves and Supply when Choice Set is Non-Convex
(e) Can you use these curves to derive a supply curve?
Answer: In this case there is no supply curve because the supply curve is the MC curve above
AC, and the MC curve falls below AC throughout.
(f ) The typical production process is one that has increasing marginal product initially but eventu- ally turns to one where marginal product is diminishing. Can you see how the two cases consid- ered in this exercise combine to form the typical case?
Answer: The case graphed in Graph 11.4(d) represents the initial part of the typical production process the part where production is getting easier and easier because marginal product is increasing. The case graphed in Graph 11.3 represents the later part of the typical production process the part where production is getting harder and harder because the marginal prod- uct is decreasing. If you were to combine the (d) panels of the two graphs, you would then get precisely the U-shaped marginal and average cost curves that we find in the typical production process.
One Input and One Output: A Short-Run Producer Model 222
11.3 Consider a profit maximizing firm.
Explain whether the following statements are true or false:
(a) For price-taking, profit maximizing producers, the constraint is determined by the technologi- cal environment in which the producer finds herself while the tastes are formed by the economic environment in which the producer operates.
Answer: The physical constraint that a producer cannot get around arises from the simple laws of physics you can only get so much x out of the inputs you use. The more sophisticated the technology available to the producer, the more x she can squeeze out and thus the tech- nology creates the production constraint that tells the producer which production plans are feasible and which are not. In terms of tastes, we typically assume that producers simply care about profit and that more profit is better than less. To form indifference curves for produc- ers who simply care about profit, we have to find production plans that all result in the same amount of profit. And how easy it is to make profit depends on how high the output price is and how low the input prices are relative to the output price. Thus, the production plans that pro- duce the same level of profit will differ depending on the economic environment depending on how much the producer can get for her output and how much she has to pay for the inputs in her production plans.
(b) Every profit maximizing producer is automatically cost-minimizing.
Answer: This is true. If you are maximizing profit, you must be producing whatever you are pro-
ducing without wasting any inputs i.e. you must be producing the output at the minimum cost possible.
(c) Every cost-minimizing producer is automatically profit maximizing.
Answer: This is not true. You could be producing some arbitrary quantity without wasting any inputs i.e. in the least cost way. But that does not mean you are producing the right quantity. Cost minimization is only part of profit maximization it only takes into account input prices as they relate to the cost of production. Only when you profit maximize do you also take into account the output price and what that implies for how much you should produce in a cost minimizing way.
(d) Price taking behavior makes sense only when marginal product diminishes at least at some point.
Answer: This is true. If marginal product always increases, then it is getting cheaper and cheaper to produce additional units of output. And if I can sell all my output at the same price (i.e. if I am a price taker), then I should keep producing and drive down my average cost.
223 One Input and One Output: A Short-Run Producer Model
11.4 In this exercise, we will explore how changes in output and input prices affect output supply and input demand curves.
Suppose your firm has a production technology with diminishing marginal product throughout.
(a) With labor on the horizontal axis and output on the vertical, illustrate what your production frontier looks like.
Answer: This is illustrated in panel (a) of Graph 11.5 an initially steep slope to the produc- tion frontier (representing initially high marginal product of labor) which becomes shallower as labor input increases.
Graph 11.5: Output Supply and Labor Demand
(b) On your graph, illustrate your optimal production plan for a given p and w. True or False: As long as there is a production plan at which an isoprofit curve is tangent, it is profit maximizing to produce this plan rather than shut down.
Answer: This is also illustrated in panel (a) where the isoprofit is tangent at the profit maxi- mizing production plan A. As long as such a tangency exists, the isoprofit will have positive intercept which implies that profit will be positive. Therefore it is better to produce at the tangency than not at all. It is also the case that only one such tangency can exist under this shape of the production frontier so no other potentially profit maximizing production plan can exist. (Note: It is technically possible for price to be so low or wage to be so high that the
One Input and One Output: A Short-Run Producer Model 224
optimal production plan is at the origin but in this case, there is no tangency at a production plan with positive output.)
(c) Illustrate what your output supply curve looks like in this case.
Answer: The output supply curve illustrates the relationship between output price on the ver- tical axis and output quantity on the horizontal. This is illustrated in panel (b) of Graph 11.5. This curve must be upward sloping because an increase in p causes the isoprofits to become shallower which in turn causes the tangency with the production frontier to move to the right on the production frontier.
(d) What happens to your supply curve if w increases? What happens if w falls?
Answer: When w increases, the isoprofits become steeper which means that the tangency with the production frontier moves to the left even as p remains the same. Thus, each point on the output supply curve shifts to the left. The reverse happens when w decreases.
(e) Illustrate what your marginal product of labor curve looks like and derive the labor demand curve.
Answer: The marginal product of labor curve is simply the slope of the production frontier. Since the production frontier starts steep, the marginal product of labor is high for initial labor units, and since the production frontier becomes shallower as labor increases, the marginal product of labor falls. This downward sloping MPl curve is illustrated in panel (c) of Graph 11.5. The labor demand curve is then simply derived from the marginal revenue product curve which in turn is simply the marginal product curve multiplied by output price p. This is illustrated in panel (d) of the graph.
(f ) What happens to your labor demand curve when p increases? What happens when p decreases? Answer: When p increases, the isoprofit curve becomes shallower which implies the tan- gency of the isoprofit with the production frontier moves to the right, resulting in more labor input. Thus, when p increases and w stays the same, more labor is hired which means the labor supply curve shifts to the right. This can also be seen by simply recognizing that pMPl increases as p increases. The reverse happens when p decreases.
225 One Input and One Output: A Short-Run Producer Model
11.5 When we discussed optimal behavior for consumers in Chapter 6, we illustrated that there may be two optimal solutions for consumers whenever there are non-convexities in either tastes or choice sets. We can now explore conditions under which multiple optimal production plans might appear in our producer model.
Consider only profit maximizing firms whose tastes (or isoprofits) are shaped by prices.
(a) Consider first the standard production frontier that has initially increasing marginal product of labor and eventually decreasing marginal product of labor. True or False: If there are two points at which isoprofits are tangent to the production frontier in this model, the lower output quantity cannot possibly be part of a truly optimal production plan.
Answer: This is true. Panel (a) of Graph 11.6 illustrates this with two tangencies at A and B. Since the isoprofit tangent at A has a negative intercept, profit at the production plan A is neg- ative.
Graph 11.6: Profit Maximizing under Different Production Sets
(b) Could it be that neither of the tangencies represents a truly optimal production plan?
Answer: Yes. This is illustrated in panel (b) of Graph 11.6 where the two tangencies occur at production plans C and D. In this case, both isoprofit curves have negative intercepts and both therefore involve negative profit. In this case, the profit maximizing production plan is (0,0) i.e. no labor input is bought and no output is produced.
(c) Illustrate a case where there are two truly optimal solutions where one of these does not occur at a tangency.
Answer: This is illustrated in panel (c) of Graph 11.6. The production plan F occurs at a tan- gency and involves zero profit because the isoprofit curve intersects at the origin. Thus, profit at F is the same as profit at E = (0,0) where no labor is purchased and no output is produced. Thus both E and F are truly profit maximizing production plans.
One Input and One Output: A Short-Run Producer Model 226
(d) What would a production frontier have to look like in order for there to be two truly optimal production plans which both involve positive levels of output? (Hint: Consider technologies that involve multiple switches between increasing and decreasing marginal product of labor.)
Answer: This is illustrated in panel (d) of Graph 11.6 where both G and H are tangencies that lie on the same isoprofit curve (and thus result in the same amount of positive profit).
(e) True or False: If the producer choice set is convex, there can only be one optimal production plan. Answer: This is true. In panel (e) of Graph 11.6, a strictly convex production frontier is illus- trated with only I emerging as a profit maximizing production plan.
(f) Where does the optimal production plan lie if the production frontier is such that the marginal product of labor is always increasing?
Answer: This is illustrated in panel (f) of Graph 11.6 where the increasing marginal product of labor results in an ever steepening production frontier. The only tangency occurs at J but J actually results in negative profit since the isoprofit has negative intercept. The producer can do better by moving to higher isoprofits that intersect the production frontier as those that
intersect at K and L. But its always possible to go to an even higher isoprofit and mover higher on the production frontier. Thus, the profit maximizing quantity is infinite (which, of course, does not make sense in a world of scarcity but neither does an ever increasing marginal product of labor.)
(g) Finally, suppose that the marginal product of labor is constant throughout. What production plans might be optimal in this case?
Answer: This would result in a production frontier that is simply a straight line. If the ratio w /p happens to be the same as the slope of this production frontier, then every production plan on the production frontier lies on the same isoprofit curve which in turn intersects at the origin. Thus, all production plans on the production frontier yield zero profit and all are therefore optimal. If the isoprofits are steeper than the production frontier, then all production plans on the frontier other than (0,0) result in negative profit and only (0,0) is optimal. If, on the other
hand, the isoprofits are shallower than the production frontier, the optimal output quantity is infinite for reasons similar to what we described in part (f) where we considered a production frontier with increasing marginal product of labor throughout.
227 One Input and One Output: A Short-Run Producer Model
11.6 This exercise explores in some more detail the relationship between production technologies and marginal product of labor.
We often work with production technologies that give rise to initially increasing marginal product of labor that eventually decreases.
(a) True or False: For such production technologies, the marginal product of labor is increasing so long as the slope of the production frontier becomes steeper as we move toward more labor input. Answer: This is true. The marginal product of labor is in fact equal to the slope of the pro- duction frontier so if the slope of the production frontier is increasing, this must mean the marginal product of labor is increasing.
(b) True or False: The marginal product of labor becomes negative when the slope of the production frontier begins to get shallower as we move toward more labor input.
Answer: This is false. When the slope of the production frontier becomes shallower, it is still positive but is now declining. Thus, at that point the marginal product of labor is declining as more labor is hired, but it is still positive.
(c) True or False: The marginal product of labor is positive so long as the slope of the production frontier is positive.
Answer: This is true since the slope of the production frontier and the marginal product of labor is the same thing, one being positive implies the other must be positive.
(d) True or False: If the marginal product of labor ever becomes zero, we know that the production frontier becomes perfectly flat at that point.
Answer: This is true as the production frontier becomes flat, its slope approaches zero, which means the marginal product of labor (which is the same thing as the slope of the production frontier) approaches zero.
(e) True or False: A negative marginal product of labor necessarily implies a downward sloping production frontier at that level of labor input.
Answer: True a downward slope is the same thing as a negative slope. Since the slope of the production frontier is the marginal product of labor, a negative marginal product of labor must imply a downward slope of the production frontier.
One Input and One Output: A Short-Run Producer Model 228
11.7 We have shown that there are two ways in which we can think of the producer as maximizing profits: Either directly, or in a two-step process that begins with cost minimization.
This exercise reviews this equivalence for the case where the production process initially has increasing marginal product of labor but eventually reaches decreasing marginal product. Assume such a produc- tion process throughout.
(a) Begin by plotting the production frontier with labor on the horizontal and output on the vertical axis. Identifyinyourgraphtheproductionplan A=(lA,xA)atwhichincreasingreturnsturns to decreasing returns.
Answer: This is illustrated in panel (a) of Graph 11.7 where A lies at the point at which the production frontier stops increasing at an increasing rate and starts increasing at a decreasing rate. Put differently, at A the slope stops increasing and starts decreasing.
Graph 11.7: 2 Ways to Maximize Profit
(b) Suppose wage is w = 1. Illustrate in your graph the price p0 at which the firm obtains zero profit by using a production plan B . Does this necessarily lie above or below A on the production frontier?
Answer: This is also illustrated in panel (a) where B lies on an isoprofit that is tangent to the production frontier at B and intersects the origin (which implies zero profit). The slope of the isoprofit is 1/p0 since w = 1. It is apparent from the graph that B must lie above A on the
229 One Input and One Output: A Short-Run Producer Model
production frontier i.e. it must be the case that the zero-profit price results in production on the decreasing marginal product of labor portion of the production frontier.
(c) Draw a second graph next to the one you have just drawn. With price on the vertical axis and output on the horizontal, illustrate the amount the firm produces at p0.
Answer: This is illustrated in panel (b) of Graph 11.7 where the point (xB ,p0) illustrates the lowest price at which the firm produces positive output.
(d) Suppose price rises above p0. What changes on your graph with the production frontier and how does that translate to points on the supply curve in your second graph?
Answer: If price rises above p0, the isoprofit lines become shallower which implies the new optimal quantity lies at a tangency higher on production frontier than B . The isoprofit that is tangent at the new profit maximizing production plan also has positive intercept on the vertical axis implying profit will be positive. Thus, output increases as p rises above p0 leading to an upward sloping supply curve (as illustrated in panel (b).)
(e) What if price falls below p0?
Answer: If price falls below p0, the isoprofit curves become steeper implying tangencies to the left of B . At those tangencies, however, the intercept on the vertical axis will be negative implying negative profit. Thus, the firm is better off not producing at all which is why the supply curve in panel (b) is vertical at zero output level up to the price p0.
(f ) Illustrate the cost curve on a graph below your production frontier graph. What is similar about the two graphs and what is different around the point that corresponds to production plan A.
Answer: The cost curve, as illustrated in panel (c) of Graph 11.7, has the inverse shape from the production frontier because when each additional labor unit increases production more than the last (on the increasing marginal product part of the production frontier), the cost of increasing output rises at a decreasing rate (and vice versa). Around A the point correspond- ing to A in panel (a), the cost curve switches from increasing at a decreasing rate to increasing at an increasing rate (just as the switch from increasing at an increasing rate to increasing at a decreasing rate happens on the production frontier.)
(g) Next to your cost curve graph, illustrate the marginal and average cost curves. Which of these reaches its lowest point at the output quantity x A ? Which reaches its lowest point at x B ? Answer: This is illustrated in panel (d) of Graph 11.7. The marginal cost (MC) curve is the slope of the cost curve so it reaches its lowest point at the output level x A where the slope of the cost curve begins to get steeper. The average cost curve reaches its lowest point where the MC curve crosses it which is also where the supply curve begins. This occurs at xB .
(h) Illustrate the supply curve on your graph and compare it to the one you derived in parts (c) and (d).
Answer: The supply curve is the part of the MC curve that lies above the AC curve with output of zero below that. This is highlighted in panel (d) of Graph 11.7 with the resulting supply curve being identical to what we derived before in panel (b).
One Input and One Output: A Short-Run Producer Model 230
11.8 Everyday Application: Workers as Producers of Consumption: We can see some of the connections between consumer and producer theory by re-framing models from consumer theory in producer lan- guage.
Suppose we modeled a worker as a producer of consumption who can sell leisure of up to 60 hours per week at a wage w.
(a) On a graph with labor as the input on the horizontal axis and consumption as the output on the vertical, illustrate what the producer choice set faced by such a producer would look like. Answer: This is illustrated in panel (a) of Graph 11.8.
Graph 11.8: Workers as Producers of Consumption
(b) How is this fundamentally different from the usual producer case where the producer choice set
has nothing to do with prices in the economy?
Answer: Typically, the producers choice set is determined by the state of the technology how easy it is to turn inputs into outputs. It has nothing to do with prices. In this case, however, the way that the worker produces consumption is by selling his labor and thus the market wage is the defining characteristic of the technology used by the worker.
(c) What does the marginal product of labor curve look like for this producer? Answer: This is depicted in panel (b) of Graph 11.8.
(d) On the graph you drew for part (a), illustrate what producer tastes for this producer would look like assuming the workers tastes over consumption and leisure satisfy the usual five assump- tions for tastes we developed in Chapter 4. How is this fundamentally different from the usual producer case where the producers indifference curves are formed by prices in the economy? Answer: Three indifference curves are illustrated in panel (a) of the graph. The worker becomes better off in the direction of the arrows i.e. as he moves to the north-west in the graph. These indifference curves are derived strictly from the internal tastes of the worker and have noth- ing to do with market prices. This is different from the typical producer case where indiffer- ence curves are simply isoprofits and where the producer is equally well off as long as profit does not change. Since profit is determined by subtracting costs from revenues and since costs and revenues are impacted by market prices, these isoprofits typically depend on market prices (with the slope equal to a ratio of input over output prices).
231 One Input and One Output: A Short-Run Producer Model
11.9 Everyday Application: Studying for an Exam: Consider the problem you face as a student as you determine how much to study for an exam by modeling yourself as a producer of an exam score between 0 and 100.
Suppose that the marginal payoff to studying for the initial hours you study increases but that this marginal payoff eventually declines as you study more.
(a) Illustrate, on a graph with hours studying for the exam as an input on the horizontal axis and exam score (ranging from 0 to 100) as an output on the vertical axis, what your production frontier will look like.
Answer: The production frontier is graphed as the (dark) frontier plotted in panel (a) of Graph 11.9.
Graph 11.9: Studying for an Exam
(b) Now suppose that your tastes over leisure time (i.e. non-study time) and exam scores satisfies the usual five assumptions about tastes that we outlined in Chapter 4. What will your producer tastes look like. (Be careful to recognize that the producer picture has hours studying and not leisure hours on the horizontal axis.)
Answer: Three indifference curves are graphed in panel (a) of Graph 11.9. You become better off as you move to the north-west in the graph fewer hours of studying, higher exam grades.
(c) Combining your production frontier with graphs of your indifference curves, illustrate the opti- mal number of hours you will study.
Answer: Bundle A in panel (a) of the graph is your optimal production plan resulting in h A hours of studying and an exam grade of x A .
(d) Suppose that you and your friend differ in that your friends marginal rate of substitution at every possible production plan is shallower than yours. Who will do better on the exam?
Answer: Your friend will do better, as illustrated in panel (b) of the graph. At your optimal production plan A, your friends indifference curve cuts the producer choice set in such a way that all the better plans (that lie to the north-west) involve greater effort than h A . Your friends optimal production plan is B .
One Input and One Output: A Short-Run Producer Model 232
(e) Notice that the same model can be applied to anything we do where the amount of effort is an input and how well we perform a task is the output. As we were growing up, adults often told us: Anything worth doing is worth doing well. Is that really true?
Answer: Not really. Both you and your friend could have done very well on the exam even scoring 100 by putting more time in. But you also have other priorities in life which is why you would prefer to devote less time to studying. It would not be optimal to devote all your time to one dimension to doing well on this exam.
233 One Input and One Output: A Short-Run Producer Model
11.10 Business Application: Optimal Response to Labor Regulations: Governments often impose costs on businesses in direct relation to how much labor they hire. They may, for instance, require that businesses provide certain benefits like health insurance or retirement plans.
Suppose we model such government regulations as a cost c per worker in addition to the wage w that is paid directly to the worker. Assume that you face a production technology that has the typical property of initially increasing marginal product of labor that eventually diminishes.
(a) Illustrate the isoprofits for this firm and include both the explicit labor cost w as well as the implicit cost c of the regulation.
Answer: Three isoprofits are illustrated in panel (a) of Graph 11.10. The only difference from the usual case is that we must include c as part of the labor cost to the firm thus causing the slope of the isoprofit curves to be (w +c)/p.
Graph 11.10: Increasing Regulatory Labor Costs
(b) Illustrate the profit maximizing production plan. Answer: This is illustrated as A in panel (a) of the graph.
(c) Assuming that it continues to be optimal for your firm to produce, how does your optimal pro- duction plan change as c increases?
Answer: When c increases to c, the slope of the isoprofits get steeper. Thus, the optimal pro- duction plan in panel (b) of Graph 11.10 changes from A to B less labor input and lower output.
(d) Illustrate a case where an increase in c is sufficiently large to cause your firm to stop producing.
Answer: This is illustrated in panel (c) where the new (dashed) isoprofit has become sufficiently steep as a result of an increase in c such that the optimal production plan C lies on an isoprofit with negative vertical intercept and thus negative profit. As a result, this firm would not produce at C but would simply choose to hire no labor and produce no output.
(e) True or False: For firms that make close to zero profit, additional labor regulations might cause large changes in behavior.
Answer: This is true. When profit is high, the regulatory cost associated with labor can be large without causing profit to fall to zero. While this would still cause a change in firm behavior (as illustrated in panel (b)), it would be a marginal change somewhat less labor and somewhat less output. But if profit initially is close to zero, then even a small increase in regulatory la-
bor costs can cause the firm to shut down completely and thus cause a dramatic change in behavior.
One Input and One Output: A Short-Run Producer Model 234
11.11 Business Application: Technological Change in Production: Suppose you and your friend Bob are in the business of producing baseball cards.
Both of you face the same production technology which has the property that the marginal product of labor initially increases for the first workers you hire but eventually decreases. You both sell your cards in a competitive market where the price of cards is p, and you hire in a competitive labor market were the wage is w.
(a) Illustrate your profit maximizing production plan assuming that p and w are such that you and Bob can make a positive profit.
Answer: This is shown in the solid lines of panel (a) of Graph 11.11 where A is the profit maxi- mizing production plan given the tangency of the (solid) isoprofit line at A.
Graph 11.11: Technological Innovation
(b) Now suppose you find a costless way to improve the technology of your firm in a way that un- ambiguously expands your producer choice set. As a result, you end up producing more than Bob (who has not found this technology). Illustrate how the new technology might have changed your production frontier.
Answer: Two possibilities are illustrated with the dashed portions of panels (a) and (b) of Graph 11.11. In panel (a), your new profit maximizing production plan is B which results in more baseball cards. In panel (b), your new profit maximizing production plan is B which also re- sults in more baseball cards.
(c) Can you necessarily tell whether you will hire more or less labor with the new technology? Answer: No, it is not clear. In panel (a), you end up hiring fewer workers while in panel (b)
you end up hiring more workers.
(d) Can you say for sure that adopting the new technology will result in more profit?
Answer: Yes, profit increases unambiguously in both panels (as seen in the higher intercept of the dashed isoprofit curve relative to the solid one.)
(e) Finally, suppose p falls. Illustrate how it might now be the case that Bob stops producing but you continue to stay in the business.
Answer: A drop in p causes the slope of the isoprofits (i.e. w/p) to increase. Panel (c) then illustrates a case where Bobs optimal production plan C is not actually optimal because the isoprofit that is tangent at C has negative intercept and thus involves negative profit. You, on the other hand, maximize profit at D along an isoprofit with positive intercept (and thus posi- tive profit). So Bob shuts down and you remain open.
235 One Input and One Output: A Short-Run Producer Model
11.12 Policy Application: Politicians as Producers of Good Feelings: Consider a politician who has to determine how much effort he will exert in his re-election campaign.
We can model such a politician as a producer of good feelings among voters.
(a) Begin with a graph that puts effort on the horizontal axis and the good voter feelings on the
vertical axis. Assume that the marginal payoff from exerting effort initially increases with addi- tional effort but eventually declines. Illustrate this politicians feasible production plans. Answer: This is illustrated in panel (a) of Graph 11.12.
Graph 11.12: Politicians as Producers
(b) Suppose that the politician dislikes expending effort but likes the higher probability of winning re-election that results from good voter feelings. Assume that tastes are rational, continuous and convex. Illustrate what indifference curves for this politician will look like.
Answer: This is illustrated in panel (b) of Graph 11.12.
(c) Combining your two graphs, illustrate the optimal level of effort expended by a politician during
his re-election campaign.
Answer: This is done in panel (c) where the optimal level of effort for the politician whose indifference curves are solid (as opposed to those that are dashed in the graph) is indicated by e A at the optimal production plan A.
(d) Now suppose that the politicians opponent in the campaign has the same production technol- ogy. Suppose further that, at any production plan in the model, the opponents indifference curve has a shallower slope than the incumbent. Assuming the candidate who has produced more good voter feelings will win, will the incumbent or the challenger win this election? Answer: This second politicians tastes therefore have indifference curves that look like the dashed ones in panel (c) of the graph. The dashed indifference curve that goes through A but has a shallower slope necessarily gives rise to production plans that are feasible and that this politician would prefer. The highest indifference curve he can get to is tangent at B , giving rise to effort level e B . Since e B > e A , this challenger defeats the incumbent.
One Input and One Output: A Short-Run Producer Model 236
11.13 Policy Application: Determining Optimal Class Size: Public policy makers are often pressured to reduce class size in public schools in order to raise student achievement.
One way to model the production process for student achievement is to view the teacher/student ratio as the input. For purposes of this problem, let t be defined as the number of teachers per 1000 students; i.e. t = 20 means there are 20 teachers per 1,000 students. Class size in a school of 1000 students is then equal to 1000/t .
(a) Most education scholars believe that the increase in student achievement from reducing class size is high when class size is high but diminishes as class size falls. Illustrate how this translates into a production frontier with t on the horizontal axis and average student achievement a on the vertical.
Answer: This production frontier is pictured in panel (a) of Graph 11.13 with large slope (or marginal product) initially that falls as t increases. Note that t is not class size when t is small, there are few teachers per 1000 students, which implies a large class size. In our graph, class size therefore falls as t increases.
Graph 11.13: Setting Class Size
(b) Consider a school with 1,000 students. If the annual salary of a teacher is given by w , what is the cost of raising the input t by 1 i.e. what is the cost per unit of the input t ?
Answer: Since t is the number of teachers per 1000 students, we need to hire one more teacher to increase t by 1. Thus, the per unit cost of t is w.
(c) Suppose a is the average score on a standardized test by students in the school, and suppose that the voting public is willing to pay p for each unit increase in a. Illustrate the production plan that the local school board will choose if it behaves analogously to a profit maximizing firm. Answer: This is also illustrated in panel (a) of Graph 11.13 where A is the optimal production plan given the willingness p of voters to get a one unit increase in achievement. The input t A then results in class size of 1000/t A .
(d) What happens to class size if teacher salaries increase?
Answer: If teacher salaries increase, w goes up which implies w /p increases and the isoprof-
its become steeper. As a result, the optimal t falls which implies class size increases. This is illustrated in panel (b) of Graph 11.13.
(e) How would your graph change if the voting publics willingness to pay per unit of a decreases as a increases?
Answer: This would in effect imply that p is high when a is low but decreases as a increases. A high p implies a relatively flat slope thus, we would get isoprofit curves that, rather than being straight lines as in panels (a) and (b) of the graph, would be shaped as depicted in panel (c). (Note: The isoprofit tangent at B represents the curve when teacher salaries are higher.)
237 One Input and One Output: A Short-Run Producer Model
(f) Now suppose that you are analyzing two separate communities that fund their equally sized schools from tax contributions by voters in each school district. They face the same production technology, but the willingness to pay for marginal improvements in a is lower in community 1 than in community 2 at every production plan. How do the isoprofit maps differ for the two communities?
Answer: At every production plan the isoprofits of the community with lower willingness to pay would be steeper indicating that a greater increase in achievement would be required for every increase in t in order for the community to remain equally well off.
(g) Illustrate how this will result in different choices of class size in the two communities.
Answer: This can be illustrated in a graph identical to the one pictured in panel (c) of Graph 11.13 where the community with the lower willingness to pay optimizes at B while the other community optimizes at A.
(h) Suppose that the citizens in each of the two communities described above were identical in ev- ery way except that those in community 1 have a different average income level than those in community 2. Can you hypothesize which of the two communities has greater average income? Answer: Community 1 is poorer. Even though they care as much about education, their will- ingness to pay is lower simply because they have fewer resources. (This should be familiar from consumer theory people with identical tastes can have very different demands for goods if they have different incomes.)
(i) Higher level governments often subsidize local government contributions to public education, particularly for poorer communities. What changes in your picture of a communitys optimal class size setting when such subsidies are introduced?
Answer: You can think of these subsidies as either reducing the teacher salaries w (since the higher level government now shares in the expense) or as raising the willingness to pay p since the voters know that they get more resources if they spend more on education. Whether viewed as a decrease in w or and increase in p, the impact on w/p is the same: w/p falls which implies that isoprofit curves become shallower. This in turn has the impact of increasing the optimal choice of t (as illustrated in panels (b) and (c) of Graph 11.13) which is the same as saying that class size will decrease.
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