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 ShortRun 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 ShortRun 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 ShortRun 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 CapandTrade: On their
own, West in pollution abating technologies such as smokestack ﬁlters. As
a result, governments have increasingly turned to “capandtrade” 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: CapandTrade 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 CapandTrade 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 CobbDouglas production technologyx = f (s, v) = 50st"25 v0'25.
(a) In the absence of capandtrade 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) = = 0.5w05r05. (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 CobbDouglas 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(0.25025)(0.25025) ’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+ 0.0008p25p35x2. (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) 41225145 (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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 Fall '08
 Woroch
 Economics, Microeconomics, Supply And Demand, insurance policy, smokestack ﬁlters

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