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PS3+Solutions - 593 Choice and Markets in the Presence of...

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Unformatted text preview: 593 Choice and Markets in the Presence of Risk 17.3 We have illustrated in several settings the role of actuarily fair insurance contracts (b, p) ( where b is the insurance benefit in the “bad state" and p is the insurance premium that has to be paid in either state). In this problem we will discuss it in a slightly different way that we will later use in Chapter 22. A: Consider again the example, covered extensively in the chapter; of my wife and life insurance on me. The probability of me not making it is 6, and my wife’s consumption if I don’t make it will be 10 and her consumption if I do make it will be 250 in the absence of any life insurance. (a) Now suppose that my wife is offered a full set of actuarily fair insurance contracts. What does this imply for how p is related to 6 and b? Answer: Actuarial fairness implies that what my wife pays is equal to what she receives in expectation. She will receive (b — p) with probability 6, and she will pay p with probability (1 — 6). Thus, actuarial fairness implies that 6 (b — p) = (1 — 6) p or simply p = 6b. (b) On a graph with b on the horizontal axis and p on the vertical, illustrate the set ofall actuarily fair insurance contracts. Answer: This is illustrated in panel (a) of Graph 17.3. (a) Graph 17.3: Tastes over premiums p and benefits b (c) Now think of what indifference curves in this picture must look like. First, which way must they slope (given that my wife does not like to pay premiumsbutshe does like benefits)? Answer: Indifference curves must slope up. Consider any initial bundle (b, p). We know that an increase in b to b' will make my wife unambiguously better off — which means that the bundle containing b’ that is indifferent to (b, p) must have an offsetting increase in p which, by itself, would make my wife unambiguously worse off. You can thus think of this as indifference curves over two goods where one of the goods, namely the premium p, is really a “bad”. (d) In which direction within the graph does my wife have to move in order to become unam- biguously better off? Answer: She becomes unambiguously better off as p falls and b increases — thus, she be- comes better off moving to the southeast in the graph. (e) We know my wife will fully insure if she is risk averse (and her tastes are state- independent). What policy does that implyshe will buy if 6 = 0.25 .9 Answer: As was shown in the text, this would imply buying a policy (b, p) = (240, 60) which satisfies the actuarily fair relationship derived in (a). Under this policy, she would have con- sumption of only 190 in the “good” state (where she has income of 250 but needs to pay the premium of 60) but she also has consumption of 190 in the “bad” state (where she has income of 10, has to pay the premium of 60 but also gets a benefit of 240). Choice and Markets in the Presence of Risk 594 (f) Putting indifference curves into your graph from (b), what must they look like in order for my wife to choose the policy that you derived in (e). Answer: This is illustrated in panel (b) of Graph 17.3. (g) What would her indifference map look like if she were risk neutral? What if she were risk- loving? Answer: If her tastes were risk neutral, she should be indifferent between all the actuarily fair insurance policies along the budget line p = 6b. Thus, her indifference curves must be straight lines with slope 6. If she were risk loving, then she would still become better off moving to the southeast in the graph, but her indifference curves would bow in the opposite direction from those involving risk aversion. This is pictured in panel (c) of Graph 17.3. B: Suppose u(x) = 1n(x) allows us to write my wife’s tastes over gambles using the expected utility function. Suppose again that my wife’s income is 10 if I am not around and 250 if I am — and that the probability of me not being around is 6. (a) Given her incomes in the good and bad state in the absence of insurance, can you use the expected utility function to arrive at her utility function over insurance policies (b, p)? Answer: Her expected utility is U(xB,xG) — 6u(xB) + (1 — 6)u(xG) = 61an + (1 — 6)1nxG (17.21) where x3 is her consumption in the event that I am not around and xG is her consumption in the event that I am around. For any insurance policy (b, p), x3 = (10+ b — p) and x5 = (250— p). We can therefore write her expected utility of the policy (b, p) as U(b, p) = 6ln(10+ b— p) + (1 — 6)1n(250— p). (17.22) (b) Derive the expression for the slope of an indifference curve in a graph with b on the horizontal and p on the vertical axis. Answer: This is just the MRS which is MRS__6U(b,p)/6b __ 6/(10+b—p) ‘ aU(b,p)/ap ' (—6/(10+b—p)) — ((1—6)/(250—p)) 6(250— p) = 6(250— p) + (1 —6)(1o+ b— p) ' (1723) (c) Suppose 6 = 0.25 and my wife has fully insqu under policy (b, p) = (240,60). What is her MRS now? Answer: Plugging 6 = 0.25, b = 240 and p = 60 into equation (17.23) gives us 025(190) _ 025(190) +0.75(190) _ (d) How does your answer to (c) compare to the the slope of the budget formed by mapping out all actuarily fair insurance policies (as in A(b))? Explain in terms of a graph. Answer: We concluded in A(b) that the slope of the budget line is 6 which is equal to 0.25 in our case. Now we concluded that, at the actuarily fair full insurance policy, the MRS of our indifference curve is also 0.25. Thus, the indifference curve is tangent to the budget line at the full insurance policy — implying that my wife is optimizing by fully insuring in the acuarily fair insurance market. We depicted this already in panel (b) of Graph 17.3 where tastes were assumed to be risk averse (as they are when we can use the concave function u(x) = lnx to represent tastes over gambles using the expected utility function.) MRS = 0.25. (17.24) 347 One Input and One Output: A Short-Run Producer Model 1 1.4 In this exercise, we will explore how changes in output and input prices affect output supply and input demand curves. A: 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 Output ou+fv+ frontier looks like. Answer: This is illustrated in panel (a) of Graph 11.5 — an initially steep slope to the pro- duction frontier (representing initially high marginal product of lab or) which becomes shal- lower as labor input increases. (a) #91:?! A late, (Q) Ma RLD/ Graph 1 1.5: Output Supply and Labor Demand (b) On your graph, illustrate your optimal production plan fi2r a given p and w. True or False: As longas there is a production plan at which an isoprofitcurve is tangent it is profit maxi— mizing 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 One Input and One Output: A Short-Run Producer Model 348 that the 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 vertical axis and output quantity on the horizontal. This is illustrated in panel (b) of Graph 11.5. This curve mustbe 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 MP! 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 de- creases? 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 pMPg increases as p increases. The reverse happens when p decreases. B: Suppose that your production process is characterized by the production function x = f (l) = 1001n(€ + 1). For purposes of this problem, assume w > 1 and p > 0.01. (a) Set up your profit maximization problem. Answer: The profit maximization problem is max px— w! subject to x = 1001n(€+ 1) (11.26) x which can also be written as the unconstrained maximimtion problem mtax 100pln(€ + 1) — wl. (11.27) (b) Derive the labor demand function. Answer: The first order condition for the unconstrained maximization problem above is — —w=0. (11.28) Solving for l, we get the labor demand function 100p— w [(p, w) = (11.29) (c) The labor demand curve is the inverse of the labor demand function with p held fixed. Can you demonstrate what happens to this labor demand curve when p changes? Answer: The labor supply curve we graph (with l on the horizontal and w on the vertical axis) is then _ 100p w(l) — [+1. (11.30) 349 One Input and One Output: A Short-Run Producer Model When p increases, the right hand side of this equation increases — which implies that the labor supply curve shifts to the right. When p falls, the right hand side decreases — which implies that the labor supply curve shifts to the left. ((1) Derive the output supply function. Answer: To get the output supply function, we simply plug the labor demand into the pro- duction function; i.e. 100p— w w +1) =1001n(fl). (11.31) x(p. w) = f (€(p, w)) = 1001n( w (e) The supply curve is the inverse of the supply function with w held fixed. What happens to this supply curve as w changes? (Hint: Recall that lnx = y implies ey = x, where e is the base of the natural log.) Answer: To take the inverse, we first divide both sides of the supply equation by 100 and recognize that this implies 1 e(x/100) = E. (11.32) w Solving for p, we get (16/ 100) we , = —. 11.33 p(x w) 100 ( ) To see what happens as w changes, we simply take the derivative of this supply curve with respect to w — i.e. apuc’w) _ we(xl100) — — > 0. 11.34 6w 10000 ( ) Thus, as w increases, the supply curve shifts up (and as it decreases, the supply curve shifts down). We could also see this directly by simply looking at the supply function x( p, w). Just take the partial derivative of x(p, w) with respect to w to find that , —1 ‘mp w) = i <0. (11.35) 6w w Thus, as w increases, output decreases — which is equivalent to an upward shift in the supply curve. (f) Suppose p = 2 and w = 10. What is your profit maximizing production plan, and how much profit will you make? Answer: Plugging these prices into the labor demand [(p, w) and output supply x(p, w) equations, we get the production plan (I, x) = (19,299.57); i.e. you will hire 19 workers and produce approximately 300 units of output. Profit is then approximately Profit: px — w! = 2(300) — 10(19) = 410. (11.36) 411 Production with Multiple Inputs 12.9 Business and PolicyApplication: Investing in Smokestack Filters under Cap-and-Trade: On their own, West in pollution abating technologies such as smokestack filters. As a result, governments have increasingly turned to “cap-and-trade” programs. Under these programs, dis- cussed in more detail in Chapter 21, the government puts an overall “cap” on the amount of permissible pollution and firms are permitted to pollute only to the extent to which they own sufficient numbers of pollution permits or “vouchers’I I f a firm does not need all of its vouchers, it can sell them at a market price py to firms that require more. A: Suppose a firm produces x using a technology that emits pollution through smokestacks. The firm mustensure that it has sufficient pollution vouchers v to emit the level ofpollution that escapes the smokestacks, but it can reduce the pollution by installing increasingly sophisticated smokestack filters s. (a) Suppose that the technology fi2r producingx requires capital and labor and, without consid- ering pollution, has constant returns to scale. For a given set of input prices (w, r), what does the marginal cost curve look like? Answer: The MC curve is flat when the production technology has constant returns to scale. This is depicted in panel (a) of Graph 12.16. (A) M V MC will! pa llolion Poached Graph 12.16: Cap-and-Trade and Smokestack Filters (b) Now suppose that relatively little pollution is emitted initially in the production process, but as the factory is used more intensively, pollution per unit of outputincreases— and thus more pollution vouchers have to be purchased per unit absent any pollution abating smokestack filters. What does this do to the marginal cost curve assuming some price pg per pollution voucher and assuming the firm does not install smokestack filters? Answer: It causes the MC curve to be upward sloping as depicted in panel (a) of Graph 12.16. (c) Considering carefully the meaning of “economic cost”, does your answer to (b) depend on whether the govemmentgives the firm a certain amount of vouchers or whether the firm starts out with no vouchers and has to purchase whatever quantity is necessary fi2r its production plan? Answer: It does not depend on whether the vouchers are owned by the firm or the firm has to purchase them. In both cases, the opportunity cost of using a pollution voucher to emit pollution in production is py. If the firm owns the voucher, it foregoes the opportunity to sell it at py to another firm that wishes to buy more vouchers. If the firm does not own vouchers, it must directly pay pg per voucher. Production with Multiple Inputs 412 (d) Suppose that smokestack filters are such that initial investments in filters yield high reduc— tions in pollution, but as additional filters are added, the marginal reduction in pollution declines. You can now think of the firm as using two additional inputs — pollution vouchers and smokestackfilters 7 to produce outputx legally. Does the overall production technology now have increasing, constant or decreasing returns to scale? Answer: The overall technology nowhas decreasing returns to scale. This is because, whether the firm uses pollution vouchers or smokestack filters or some combination of the two, ithas to expend increasing resources to deal with its pollution output for any marginal increase in production. (e) Next, considera graph with “smokestackfilters”s on the hor’wontal and “pollution vouchers" v on the vertical axis. Illustrate an isoquant that shows different ways of reaching a particular output level? legally — ie. without polluting illegally. Then illustrate the least cost way of reaching this output level (not counting the cost of labor and capital) given py and p3. Answer: This is illustrated in panel (b) of Graph 12.16 where A is the cost minimizing bundle of smokestack filters and pollution vouchers to produce 7 when the prices of filters and vouchers are p, and py. (f) If the government imposes additional limits on pollution by removing some of the pollu- tion vouchers from the market, py will increase. How much will this affect the number of smokestack filters used in any given firm assuming output does not change? What does your answer depend on? Answer: The increase in py will cause isocosts to become shallower. If output does not change from E, this will lead to a change in the cost minimizing bundle to B — causing the firm to use fewer vouchers and more smokestack filters. The size of the adjustment depends on the degree of substitutabilitybetween vouchers and smokestack filters in production. In other words, if it is relatively easy for the firm to install additional smokestack filters, the effect will be bigger than if it is not. (g) What happens to the overall marginal cost curve for the firm (including all costs ofproduc- tion) as py increases? Will output increase or decrease? Answer: This is illustrated in panel (c) of Graph 12.16. The marginal cost of production increases as py increases, rotating the MC curve from MC to M C' . For a given output price p, this implies that the profit maximizing output falls from 36" to x3 . (h) Can you tell whether the firm will buy more or fewer smokestack filters as py increases? Do you think it will produce more or less pollution? Answer: It is not clear whether the firm will buy more or fewer smokestack filters — because it is not clear by how much the firm will reduce its output. We know from panel (c) that the firm will produce less, and we know from panel (b) that, for the same level of output, it will buy more filters. But if the firm decreases production sufficiently much, it may end up buying fewer filters. No matter what, however, it will produce less pollution — because it produces less output with more filters for that level of output than it would have used before. (i) True or False: The Cap-and-Trade system reduces overall pollution by getting firms to use smokestack filters more intensively and by causing firms to reduce how much output they produce. Answer: This is true. As we have shown, the firm uses more smokestack filters for any given output level (panel (b) of the graph) but also produces less output (panel (c)). B: Suppose the cost function (not considering pollution) is given by C(w, r, x) = 0.5w0'5 r0'5x, and suppose that the tradeofi" between using smokestack filters s and pollution vouchers v to achieve legal production is given by the Cobb-Douglas production technologyx = f (s, v) = 50st"25 v0'25. (a) In the absence of cap-and-trade policies, does the production process have increasing, de- creasing or constant returns to scale? Answer: The marginal cost function derived from C(w, r, x) is MC(w,r,x) = = 0.5w0-5r0-5. (12.108) 413 Production with Multiple Inputs This function is independent of x — i.e. the marginal cost is constant, which implies con- stant returns to scale. (b) Ignoring for now the cost of capital and labor, derive the cost function for producing different output levels as afitnction of p3 and [2,, 7 the price of a smokestack filter and a pollution voucher: ( You can derive this directly or use the fact that we know the general form of cost functions for Cobb-Douglas production functions from what isgiven in problem 12.4). Answer: Plugging in A = 50 and a = fl = 0.25 into the cost function given in problem 12.4, we get 0.25 0.25 2 xps pv — =0.0008 05 0'5x2. 12.109 50(0.250-25)(0.250-25) ’73 p” ( ) C(ps. Pva) = 0.5 (c) What is the full cost function C(w, r, p3, py) .9 What is the marginal cost function? Answer: The cost of producing output level x is then simply the cost of labor and capital plus the cost of complying with the requirement that pollution is produced legally; i.e. C(w, r, p3, py) = 0.5w°'5 r°-5x+ 0.0008p2-5p3-5x2. (12.110) The marginal cost function is then MC(w, r, p3, py) = 0.5w0'5 r"'5 + 0.0016pg'5 pg'sx. (12.111) ((1) For a given output price p, derive the supply function. Answer: We set p equal to MC and solve for x to get x(w r p p p) _ p_0'5w0.5r0.5 J J SJ UJ (12.112) (e) Using Shephard’s lemma, can you derive the conditional smokestack filter demand function? Answer: Shephard’s lemma tells us that the partial derivative of the cost function with re- spect to an input price is equal to the conditional input demand function for that input; i.e. 6C J J J 0'5x2 8(w,r,p3,py,x) = M = 0.0004 p” ops p25 ' (f) Using your answers, can you derive the (unconditional) smokestack filter demand function? Answer: If we plug the supply function x(w, r, p3, py, p) into the conditional smokestack filter demand function s(w, r, p3, pmx), we will get the unconditional smokestack filter de- mand function. We then get (12.113) 4122-5145 (g) Use your answers to illustrate the effect of an increase in py on the demand for smokesth filters holding output fixed as well as the effect of an increase in py on the profit maximizing demand for smokestackfilters. Answer: The derivative of the conditional demand function s(w, r, p3, py,x) with respect to py is positive — indicating that we will buy more smokestack filters conditional on produc- ing the same quantity of output as before. The derivative of the unconditional filter demand s(w, r, py , p3, p) with respect to py, however, is negative — indicating that we will buy fewer pollution filters when we arrive at our new profit maximizing production plan. This is not because we pollute more — but rather because our supply function x(w, r, p; , py , p) tells us that we will produce sufficiently less such that we will need fewer overall filters even through we use more filters for the quantity that we do produce than we would have before. s(w.r.py.ps.p) = (12.114) ...
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This note was uploaded on 11/17/2010 for the course ECON 100A taught by Professor Woroch during the Fall '08 term at Berkeley.

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