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ECON20110 Fall 2013 - Lecture 3: Cost — Transcript

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0:00okay so um this week we're going to
0:03study how individual firms
0:06produce uh and we'll start today by
0:09talking about production itself per unit
0:13time so we'll basically Focus only on a
0:17static uh way of thinking about things
0:20where sort of time doesn't enter it's
0:22everything is thought of in terms of
0:24everything per unit
0:27time and on Thursday we'll talk about
0:30how Supply decisions are made over
0:32time um and to start things off we're
0:35going to learn two dual ways or
0:38equivalent ways of representing the way
0:41that firms produce one is their
0:43production
0:45technology and thinking of things in
0:47terms of production technology makes it
0:50easy to think about
0:52decisions uh of when there are many
0:55different inputs and you're trying to
0:56choose which inputs are most useful uh
1:00in producing the output that you want to
1:03achieve um this is an intuitive way uh
1:08to think about um information that's
1:12relevant to creating products given
1:14particular inputs but it's going to be
1:17much less intuitive and less useful for
1:20thinking about the choices of which
1:23outputs to produce when there are many
1:25different potential outputs the other
1:28way we'll talk about our costs of
1:30production and this makes it easy to
1:33represent multiple outputs and it's
1:35intuitive when you're thinking about the
1:38effect of changing the prices of
1:40different outputs but it's more
1:42difficult to think about when you have a
1:43choice of different inputs or what you
1:45would do when the price of different
1:47inputs change and we'll learn to link
1:50these two different representations
1:52together uh to one
1:54another okay so um let me use an example
1:59that's sort of close to home as
2:01motivation imagine that you're trying to
2:03get into a good
2:04college uh and that all colleges
2:06evaluate you based on some score s the
2:10higher s is the better College you get
2:12into and you can create this score s by
2:15uh two things one is extracurricular
2:18activities which for every unit of
2:20extracurricular activities uh uh it
2:24costs you a number of hours
2:26TX or by academic work which for every
2:30unit of academic work you produce costs
2:32you a number of hours
2:34ta and we'll think about um two
2:37different ways of representing this
2:39problem one is the production of the
2:42score and therefore of college
2:44admissions which will say how for um as
2:48a function of how much extracurriculars
2:50you do and how much academic work you do
2:53uh do you produce output the score for
2:55getting into
2:56college and the reason that this is a
2:58useful way to think about things things
3:00is it helps you determine how much time
3:03to spend on each of uh the two
3:06activities for a given amount of uh
3:09college that you want to get into but
3:12another way we can think of things is
3:13the cost of being accepted which is how
3:16much total time you'll have to spend in
3:19order to uh get a certain
3:22score uh and this is useful for helping
3:25you to make tradeoffs between whether
3:28you want to spend uh this amount of time
3:31on getting into college or whether you
3:33want to spend it on other aspects of
3:34your life like partying sleeping
3:37establishing ties with your family
3:42Etc okay so um let me talk about some
3:46extreme types of Technology one extreme
3:49type of technology is perfect
3:50compliments this is when the amount that
3:53you produce is the minimum of the two
3:57different uh uh score
4:00uh um aspects of your life
4:03extracurricular activities and academics
4:05so this is the case when schools only
4:08want people who are perfectly balanced
4:10so they don't care about you being great
4:13at extracurricular activities if you're
4:14not at least equally good at academics
4:16and vice
4:18versa this is an important production
4:20function in Industries with many complex
4:22products so a famous story about this
4:26was the
4:27Challenger uh that uh blew up and um
4:33apparently the reason was that there was
4:34one little like o-ring that had failed
4:37in the Challenger out of tens of
4:39thousands of pieces that went into it
4:41right and that shows how perfectly
4:44complimentary all the pieces were
4:46because they were only as strong as the
4:48weakest link
4:50right okay another technology is perfect
4:54substitutes um and uh Florence uh what's
4:59the
5:00um form of perfect substitutes uh
5:03technology uh you would have something
5:05like f of a a being X plus a or some
5:10other ratio exactly exactly so this is
5:14the case in which schools value both
5:18things according to some constant
5:20tradeoff and they really just care you
5:23know that you're good overall you could
5:25be good in academics you could be good
5:27in
5:28extracurriculars uh they you know don't
5:31care which you're good at you just
5:33overall produce things
5:36um uh and this is most relevant when
5:39different dimensions are fungible so one
5:42example of this might be if I put an
5:45advertisement in the New York Times or
5:48an advertisement in the Wall Street
5:50Journal um though each one will get me
5:53some
5:54customers right but given that not
5:57there's not that many people who overlap
5:58and read both newspap papers pretty much
6:00the amount of customers I'll get from
6:01running the ad in the two things is just
6:03additive right so they're perfectly
6:06substitutable ways of getting customers
6:07to come to my
6:09store okay then um there is uh an even
6:15more extreme form of substitutes which
6:18is called Ultra substitutes or either or
6:20production and is Mitch Zach here yeah
6:24you go by Mitch right yeah yeah uh what
6:27is what's the form of that so you you
6:30choose the maximum of one or the other
6:33exactly um so the uh production function
6:38is the maximum of what you have on X and
6:42what you have on a so this is the case
6:44where the school really only values you
6:46on your greatest strength if you're good
6:48at
6:49academics uh they don't care how good
6:51you are at extracurricular activities
6:53unless you're even better at
6:54extracurricular activities than you are
6:56at
6:56academics and this is relevant when you
6:58can only choose one mode of production
7:00so for example imagine you're starting a
7:02power plant um really only you're going
7:06to use either coal or oil being a Coal
7:12Fired power plant and also having some
7:14oil does no good for you right only like
7:17whatever you focus on uh is the uh
7:20technology you'll use right now
7:24typically this uh technology is not that
7:28commonly used I don't think it showed up
7:29in your book uh in but it's quite
7:32relevant in many cases and even when uh
7:36you don't have this very extreme form of
7:38substitution often there can be more
7:41than perfect substitutes because you
7:44specialize in one way of doing something
7:46and then it can become costly to use
7:48other ways of doing it
7:51okay so
7:54um more commonly we're going to assume
7:58that uh technology has several features
8:01first of all that it's convex sorry that
8:04it's monotonic which means that it has
8:07free disposal that is um no school holds
8:10it against you that you're better at
8:12something so it might not do you any
8:13good to be better at something in some
8:15circumstances but they're not going to
8:17actually hold it against you right you
8:19can always freely dispose of any ability
8:22you have in any area the second property
8:25is convexity and Philip could you define
8:28convexity uh
8:30a production function is convex if and
8:33only if given two points that some
8:37production level P any linear
8:40combination that lies between those two
8:43points also produces P or more exactly
8:46so this is a weak notion of pre of sort
8:49of more some amount of complimentarity
8:52or preference for diversity right so if
8:55you had perfect substitutes that's not
8:57convex because the linear combination of
8:59two points always produces the same if
9:01those two points produce the same
9:04right uh and um
9:09similarly uh Ultra substitutes is very
9:13non-convex so it's some um notion of uh
9:19of complementarity between the different
9:22inputs another way of putting it is that
9:25the isoquant curves the curves where
9:27there's the constant Quant
9:30produced by The Firm are going to point
9:32towards the
9:33origin um under perfect
9:39complement those isoquants look like
9:42this so they point to the origin in a
9:44very extreme way under perfect
9:47substitutes they're lined so they don't
9:49Point either way and under Ultra
9:52substitutes or either or they point the
9:54opposite direction and so the Assumption
9:56we're going to make is that they weakly
9:57Point towards the origin
10:02okay and
10:05um the slope of these isoquant curves is
10:10called the marginal rate of technical
10:14substitution and uh basically what
10:17convexity says is the marginal rate of
10:19technical substitution is going to get
10:22flatter As you move in this
10:25direction okay how about declining
10:28marginal product Evan what does that
10:30mean declining marginal product means
10:32it's more expensive to make the next
10:35one exactly so the first unit is cheaper
10:38than the second unit and cheaper than
10:40the third and so forth
10:43um and uh we usually go even further
10:47than this and assume that the production
10:49function is concave in both of the
10:51inputs uh convexity and declining
10:55marginal product will uh um imply that
10:58and so
10:59uh the a simple way of putting it is
11:03just that the production that you get
11:04out of each unit is
11:07concave Um this can also be justified by
11:10optimization as we'll talk about in the
11:12next
11:13lecture and finally we usually assume
11:16that the production technology is smooth
11:18so there's not going to be any of these
11:19Kinks like we had in
11:23these that's so basically so we can use
11:25calculus and you can usually uh
11:28approximate any smooth production
11:30function by one that's smooth so the
11:32most common form that satisfies these
11:34properties is constant
11:37elasticity uh sorry it's cob Douglas uh
11:40which takes the form F ofx and a is
11:44equal to um X to some power uh uh here I
11:49think it's Zeta uh and a to some power
11:54or I guess it's kai kai no it's Kai and
11:59then
11:59a to some power
12:03Alpha so um this has a number of useful
12:06properties one is constant elasticity of
12:08production with respect to the inputs in
12:12particular a p% increase in the um uh
12:17input X for a small uh X will lead to a
12:22c
12:23* uh for small P will lead to a c * p%
12:29increase in the output and that's
12:31constant regardless of how much uh of
12:34each of the goods you have so uh is
12:37Julian here uh yeah yeah Julian uh can
12:40you prove that uh let's see um so we
12:47have you can do it on the board if it's
12:49easier
13:35yeah
13:37[Music]
13:53that's right okay so that's the
13:56derivative that's for one unit increase
13:58in X but how do we make a percent
13:59increase okay right so we need to
14:03do so it's going to be larger the bigger
14:06X is right the D log of yes so that what
14:11you did is fine but then you just have
14:13to multiply the derivative by X right
14:15right because the bigger X is the bigger
14:18the increase will be
14:22right that so that drops away and we
14:26want that to cause the same percentage
14:28increase in the output so we need to
14:30divide it by the current output as well
14:33right all right okay so which is just
14:37the
14:45yeah Al okay
14:50then that cancels just epsilons then we
14:55are left with just and Epsilon exactly
14:59exctly yeah that's actually it's another
15:02crazy variable but anyways yes that's
15:04exactly right so so what he just showed
15:06is that the percentage increase is
15:08always equal to that side right
15:10regardless of um what the levels of uh
15:15the inputs are sir could you explain why
15:17do you multiply by X again okay so you
15:21want a x% increase right that is the
15:25larger X's the more you want to increase
15:27and you want it to be proportion to X
15:29right and so it's like a a change of
15:33that proportional to X leads to a change
15:37in the output proportional to the output
15:41right
15:43okay uh another property is that it's
15:47homogeneous of degree uh plus
15:51Alpha uh in particular if you double
15:55both of the inputs then you will
15:58increase the output by 2 the + uh Alpha
16:04you get that just by factoring
16:05everything out um and that's the
16:08definition of homogeneity of degree uh
16:12uh S Plus Alpha it's also linear in its
16:16logarithm so if we take the logarithm of
16:18both sides we get the logarithm of the
16:20output is equal to the logarithm of K
16:23plus * logarithm of X Plus Alpha *
16:27logarithm of a
16:30um the marginal rate of technical
16:33substitution that's how much we have to
16:36change the outputs relative to one
16:38another in order to stay along the
16:40isoquant well that's just going to be
16:42equal to the derivative of the output
16:45with respect to a divided by the
16:46derivative of the output with respect to
16:48X because that's the rate at which we
16:50can raise one and lower the other so as
16:52to keep uh the total amount
16:55constant and uh that's going to be equal
17:01by exactly this argument here to Alpha *
17:05X over * a which is proportional to the
17:09ratio of x to a so the ra uh the ratio
17:13of these two is equal to the um ratio of
17:16the quantities used that implies that as
17:19we move to higher X lower a the slope of
17:24uh is going to decline um of transform
17:29in X into
17:32a yeah sorry how did you go from that
17:34last justification like using what Julie
17:37proved like y so let me just write out
17:41what he did so he said that the
17:44derivative with respect to X is
17:49Kai x to the
17:53-1 a to the alpha if we took the
17:56derivative with respect to Alpha instead
17:58we would have gotten K Alpha to the I
18:02mean x to the a to the alpha minus 1 if
18:07we now take the ratio of this to that we
18:11get
18:13alpha x over
18:17s
18:19a right because all the other exponents
18:23cancel out and then K
18:26cancels okay so that means that the more
18:28you have of one input the more valuable
18:31the other input
18:33is
18:35um so uh these properties are all very
18:39closely related to one another because
18:42linearity and logarithms is precisely
18:44what makes you have constant elasticity
18:46because that means that one logarithmic
18:49Point change always leads to the same
18:51logarithmic Point change in the
18:53other uh similarly if you have constant
18:57percentage inre inrees then a given unit
19:00increase is going to be less valuable
19:02the larger a given uh input is and as a
19:06result you have this complimentarity
19:08property
19:11okay so another way of thinking about um
19:17uh production is in terms of costs and
19:19there are different types of costs that
19:21we discuss typically one are variable
19:24costs and these are ones that increase
19:26more or less continuously the more
19:28output that you have have so for example
19:30time spent studying or practicing Sports
19:34would be an example of a variable cost
19:37of producing uh either in the first case
19:41academic output or in the second case
19:43extracurricular activities then there
19:46are what are called lumpy or quasi fixed
19:49costs uh these are things that are not
19:53uh fixed in order to have any of the
19:54thing but which are um which as you
19:59increase output you have to
20:01discontinuously uh in increase this uh
20:05you know buy a big lump of this thing in
20:07order to increase your input now these
20:11are sometimes necessary in order to get
20:13your output Beyond some point sometimes
20:15they're merely useful to to do it so for
20:19example enrolling in a new class in
20:21school is like a lumpy input to
20:23producing academic uh output right or
20:27taking up some new activity it's not
20:29like you can just adjust that
20:30continuously you have to like do a bunch
20:33more in order uh for it to work another
20:37example would be paying a tutor or
20:39buying new equipment now those things
20:41wouldn't be strictly necessary to
20:42increase your output but they would be
20:44useful and they would certainly be
20:47lumping uh yeah philli are we assuming
20:50that you can't pay the tutor
20:52continuously by the hour you have to buy
20:54this p and Mony chunks that's the idea
20:57or you can't buy it by the second yeah
20:59you have to buite in lumpy chunks
21:02um uh fixed costs are ones that are
21:07necessary in order to produce any of the
21:10output at all uh so for example
21:12enrolling in school if you don't enroll
21:14in school you're basically going to be
21:15viewed as a zero uh academic wise right
21:18and if you don't uh have any current
21:22academic activities and the only thing
21:24you want to do is like start a new club
21:26well then starting a new club or you
21:27know have your own club and starting a
21:29new club is a big fixed cost that you
21:31need to produce any
21:32extracurricular output and then finally
21:35there are sunk costs which are fixed
21:37costs which you've un undertaken in the
21:40past and which you can't undo at this
21:41point and these shouldn't affect your
21:44future decision so for example imagine
21:46that you know you have uh enrolled in
21:49school and you've paid your yearly
21:51tuition and now you want to withdraw and
21:52they won't give you the tuition back
21:54well that's a fixed cost and you should
21:55just forget about that in making your
21:58future
22:00decisions okay so costs can come from
22:04many different sources One Source is
22:07money costs um and these are probably
22:09the most intuitive and common source
22:12like having to buy various things like
22:14equipment textbooks school supplies
22:17Etc the second is opportunity costs uh
22:21of time or some other resource and this
22:24is a classic Topic in economics so for
22:27example uh you know Benjamin Franklin
22:29famously said that time is money in the
22:31sense that if you don't use time
22:33studying you could use it working a job
22:36uh and you know earn money that way and
22:38you should you know never use time on
22:41something that isn't worth at least that
22:44amount of money in units of time uh but
22:46this is true for other uh scarce
22:48resources like energy or willpower if I
22:51exhaust myself on the playing field I
22:54may not be able to study as well if I uh
22:58um if I work really really hard to study
23:02I might end up eating candy and getting
23:03overweight right so like you know
23:05there's trade-offs in terms of how you
23:07use your willpower as well um Capital
23:11costs uh are another important kind of
23:13cost that's not as obvious so you can
23:16deplete a fixed resource that has some
23:18future value uh and a previous problem
23:21Set uh asked you to explore this um what
23:24it talked about was the fact that um if
23:28imagine you have a car and the more you
23:30use your car the more it gets run down
23:33then using your car uh is costly because
23:38you'll forfeit the opportunity to use
23:40your car in the future for
23:43example um and this could be true of
23:45many things uh in life so your
23:48reputation if you you know use your
23:51reputation to get a job and then don't
23:53do a good enough job there you've sort
23:56of depleted your reputation and you
23:57won't be able to use it it to get a job
23:59in the future uh relationships with
24:01people in if you you know don't attend
24:05to your relationships with people you
24:07won't be able to call on them when
24:08you're in need in the future right your
24:11physical appearance can be like this as
24:13well if you don't spend time attending
24:15to your physical appearance right you'll
24:17start to look worse and worse or this is
24:20true of your health as well right yeah
24:23how do you for like f of success with
24:25this
24:26framework like for instance Microsoft
24:34H that's an
24:35interesting uh idea
24:41um yeah I mean isn't that just an
24:43example of an opportunity cost in the
24:46like you're no yeah sure I mean look all
24:50these taxonomies are useful in some
24:53cases and not that useful in other cases
24:57uh I'm trying to think is there like a
24:59neat category into which that falls I
25:02mean you could just you can always call
25:04things just direct costs right but I'm
25:07not sure that there's any of these
25:09categories into which it particularly
25:10naturally fits yeah Christian it almost
25:12becomes like a tax like if you get to a
25:14certain level of success in the
25:16government that's right but that's not
25:17in any of these categories I mean a tax
25:19is is really just another form of cost
25:21but yes it's a cost imposed by the
25:23government
25:25um okay so uh
25:29um costs are a
25:32useful way to think about things
25:35relative to the technological way of
25:37talking about things we were discussing
25:39before in terms of production technology
25:41because it can help you think about how
25:43much to
25:44produce um so for example like when
25:47you're making the tradeoff between going
25:49to college getting into college and
25:50making other goals in your life you want
25:52to know how much time you're going to
25:54have to put in to achieve a certain
25:56amount of college so you can think about
25:58how valuable getting into certain
25:59colleges relative to other things I
26:01could do and this is easy to graph as
26:04we'll see uh shortly and this is the
26:06context in which we'll usually think
26:07about these things when firms are
26:09choosing how much to produce not what
26:11things to use to
26:12produce um it helps determine prices
26:16because it has a very natural interface
26:17with product markets because it'll say
26:19how much do firms produce as a function
26:21of price we can then use that to figure
26:23out how much the uh price has to be in
26:26order to get firms to produce what's
26:27demanded in market so for example if we
26:30want to determine what score you need to
26:32get into any given School uh we need to
26:35aggregate across students how much
26:37they're all going to be working and
26:39compare that to how many places the
26:41schools have right because that'll tell
26:43you what the threshold uh will be for
26:46getting into a given school so to find
26:48that equilibrium we need to think about
26:50things in terms of how much uh would
26:53students it cost students in order to
26:56get into a given school and then we can
26:57use how much they value getting into a
27:00given school to determine uh you know
27:03for a given set of thresholds how hard
27:05people will people work and then given
27:08the amount of places that are available
27:10what threshold will that lead uh to an
27:13equilibrium now this is difficult using
27:15technology if we had to derive that all
27:18directly uh from like how
27:21extracurricular and academics fit into
27:23all of this it would be a bunch of
27:25additional steps that usually are not
27:27useful if that's the exercise we're
27:28thinking about but if we're thinking
27:30about exactly how do you get into a
27:31school well then that will be really
27:33useful so depending on what problems
27:34we're thinking about we'll use one or
27:36the other way of analyzing them okay and
27:39with fancier math that we won't go
27:41through today it's actually pretty
27:42straightforward to think about costs of
27:45producing multiple different outputs not
27:47just a single
27:49output
27:51okay nonetheless these two concepts are
27:55equivalent or what economists called
27:57dual to one
28:00another and we can mathematically derive
28:03one from the
28:05other so this is particularly easy to
28:08see if we have only one input with a
28:11price Pi let's say well then we can just
28:14ask if we want to produce a given amount
28:16of output o and the production function
28:17is i to the gamma well then we can just
28:20invert this and say well that means that
28:22we're going to need an amount of the
28:24input equal to O the 1 over gamma in
28:27order to produce it and then just
28:29multiply that by the price of that input
28:31to determine the cost of
28:32production right now when there are
28:35multiple inputs there are two steps
28:38first you have to find the cheapest way
28:41of producing for a given output and we
28:43call this the cost of the
28:45output
28:49um
28:50uh and yeah so first we find the
28:53cheapest way of producing then we
28:55calculate at that cheapest rate uh um
28:58way of producing what is the cost of
29:00producing given output and that becomes
29:02the cost right and this is particularly
29:05easy to do for some of the Extreme
29:07Technologies that we considered above so
29:10for example um Kim uh for the uh perfect
29:15complement technology what is the cost
29:18function don't look at your
29:24slide how much does it cost to produce
29:28output
29:29o uh if we have a perfect complement
29:37technology but just think about it
29:40intuitively right remember what happens
29:42imagine you need to get score s you want
29:44to achieve score s and we know that the
29:46school is going to rate you on the worst
29:51of your two abilities you have to be
29:53equally good on the two things what is
29:55it going to cost you in terms of time uh
29:58to get a score s in that
30:05case What minimum well no so I mean just
30:09intuitively think about it if they're
30:12going to grade you on the worst of the
30:14two things what is the best strategy for
30:16you to
30:18take it's going to be to produce equal
30:20amounts of those two things right
30:22because other anything else would be a
30:24waste and how much are you going to have
30:26to produce in order to get the square if
30:28they grade you on the minimum of the two
30:30things that you
30:32do well you're just going to have to
30:34produce the amount that they say right
30:36the score s right you're going to have
30:38to do that on both of the two Dimensions
30:40because otherwise you're not going to
30:42achieve their desired score level right
30:45and so you have to sum together the time
30:47that it takes you to achieve that score
30:49on both the dimensions so it's just the
30:51sum of the two time costs multiplied by
30:53the score you need to
30:55achieve right
31:00okay now how about if the two things are
31:02independent or perfectly substitutable
31:04Phil up what what would the cost
31:05function look like then um it would be
31:09you would take the cheapest that you
31:12could produce uh and produce enough of
31:15it to get the desired score so it be S
31:19minimum txta exactly so you you choose
31:22which way which thing you were best at
31:24which thing was cheapest for you and
31:26then you would do everything
31:28that way and nothing the other way
31:30because it would be more costly to do it
31:32the other way
31:34right um and that's the same with the
31:36either or or extreme compliments
31:39technology okay so now let's consider
31:41the cob
31:43yeah
31:45yeah because that's the amount you want
31:47to produce right you want to produce s
31:50and in order to do that you need to take
31:52whatever the cost of producing one unit
31:54is is just the individual yeah okay now
31:58let's consider cob Douglas in the two
32:00input case we want to figure out what is
32:03the cheapest way of us producing uh s
32:07right which is the ARG
32:11minimum of the cost which is the amount
32:14of uh X we produce times the amount of
32:16time cost of that plus the amount of a
32:19we produce times the time cost of a
32:22subject to producing at least s right
32:27this is sort of the most classic
32:29constrained minimization problem and I
32:31want to review the sort of economic
32:33approach to this
32:35problem now the way that economists
32:38think about these things is we come up
32:40with a notion of an opportunity
32:42cost of failing to produce right we need
32:48to produce this amount and so we want to
32:50figure out what is the opportunity cost
32:53of failing to produce that's really
32:55going to be the cost of production right
32:58the marginal cost of production right
33:00and then we'll minimize the total cost
33:03including the opportunity cost of
33:05failing to
33:06produce right and this opportunity cost
33:09is called the lrange
33:11multiplier right but I'm going to just
33:13call it an opportunity cost from now on
33:16to emphasize the economic interpretation
33:18of what it is so then we're going to
33:21solve for the opportunity
33:23cost and this will allow us to derive
33:25the marginal cost that we were in
33:27interested in okay so let me just go
33:31through that um problem
33:33mathematically
33:38um is there an eraser I see
33:43one guess
33:47not that's
33:50strange
33:52um chalk
33:56what ch
34:00[Music]
34:01What
34:06er where's the eras you see an eraser
34:11no it's not that useful without an
34:14eraser Ste went from another
34:17room um yeah can someone go find a
34:20eraser in another
34:22room okay thanks Kim
34:30that's really strange never had it fit
34:46eraser while we're waiting for him yeah
34:49can you explain like what the
34:50opportunity cost is of like not
34:51producing I I'm having trouble so the
34:54point is like if I fail
34:57okay so
35:00um I can spend no more than 24 hours um
35:06in a day and so therefore we say there's
35:08an opportunity cost of my
35:11time what creates the opportunity cost
35:13is that I'm limited to 24 hours if I
35:15spend one more hour doing something I
35:17have to spend one less hour doing
35:18something
35:20else you can reverse that around you can
35:23say if I have to produce at least this
35:25much then there's a cost cost if I fail
35:30to meet this quota it's really just like
35:32the reverse of the exact same idea and
35:35so all that a lrange multiplier is is an
35:37opportunity cost it's nothing more and
35:40nothing less than that so when you use
35:41lrange multipliers all that is doing is
35:44mathematically formalizing the idea of
35:46opportunity costs that weal that we that
35:49you're used to talking about intuitively
35:50in economics I don't know if any has
35:52anyone made that link for you guys
35:55before I mean that's really all it is is
35:58like whenever you do constraint
35:59optimization whenever you do these
36:01lrange multiplier things the only reason
36:02in economics we use that is it's a
36:04mathematical simple that represents the
36:06idea of an opportunity cost and when you
36:08solve out for the L range multiplier all
36:10you're doing is solving for the
36:12opportunity cost so the economic way of
36:15you know all this constrainted
36:16optimization stuff it's just a way of
36:18formalizing in mathematics or really
36:21just writing down literally the notion
36:25of an opportunity cost so that you can
36:28uh solve it and then all you're doing
36:29when you solve the lran problem the
36:31reason you have that lrange multiplier
36:33up there multiplying everything is
36:34you're minimizing your total costs
36:36including the opportunity
36:38cost right and that's why it has that
36:42weird form that always seems
36:43inexplicable where you write up the cost
36:46plus some random number multiplied by
36:48something else that's because your
36:49opportunity cost
36:51multiplies uh all your uses of those
36:55scarce resources does that make sense s
36:58yeah just to be clear
37:00so maximization and so lamb would
37:04beity inome yeah um so how would How
37:07would how would you go around explaining
37:09the economic inter behind that
37:11opportunity cost so the point is that
37:14like now we're thinking of maximizing
37:16your utility right and there's going to
37:19be an opportunity cost in term in
37:21utility terms of using
37:24money right
37:29have okay well I give up um I'll go get
37:33some paper towel from the okay will that
37:36work yeah hopefully as long as you wet
37:39it yeah it should work I guess and then
37:41have some dry ones too to dry
37:45it this is so stupid I can't believe
37:49it um yeah so like you know there's an
37:52opportunity cost in units of um you
37:57utility of using your money because you
38:00only have so much money right so like I
38:02know usually think of money as being
38:04like the unit in which things are
38:05denominated but if we're thinking of
38:07things in utility terms right then money
38:10is a scarce resource and there's an
38:12opportunity cost of your money which is
38:15you could get other utility for that
38:16money right yeah so that's
38:19why last year the land the Marg utility
38:23income so it's just that's basically the
38:25same thing yeah because of course you
38:26have to set every use of money equal to
38:28its opportunity cost which is the
38:30marginal benefit that it has for you
38:33right so the marginal benefit of having
38:35an extra unit of time is the opportunity
38:36cost of time because you set all uses of
38:39time equal to the marginal benefit of
38:41having additional time right yeah can
38:45you erase that
38:50one so
38:56silly not yet
38:58you at least it's a good example of
39:02compliments yeah exactly without the dry
39:06towel the wet towel is much less
39:08valuable because you have to wait for
39:10the board to dry
39:15that okay we're going to solve this
39:21phas okay so thank you so what we're
39:25going to do is we're going to minimize
39:28total costs including opportunity costs
39:30of failing to produce right so we're
39:32going to
39:34minimize t x X Plus ta a
39:42A plus the opportunity cost of failing
39:45to produce what's that Lambda the
39:48opportunity cost
39:50multipli by the amount that we fall
39:53short of what we were supposed to be
39:55producing right so so we're supposed to
39:58be producing s s um the amount by which
40:03we fall short of s is uh K uh x to
40:10the X to the Cai a to the alpha minus s
40:18right
40:21uh is that right no I think it's the
40:24other way around um oh God
40:27but I can't
40:32erase okay just switch those in your
40:34mind guys or put a
40:36Min
40:40yeah okay so let's take the derivative
40:44of that right
40:46so the direct cost of
40:49x minus is equal
40:53to Lambda
40:57k x to the C -1 C out front a to the
41:04alpha so what is this equation saying
41:06it's saying that the cost of producing X
41:11has to be equal to the opportunity cost
41:13of failing to produce X which is the
41:16marginal product of X multiplied by the
41:20opportunity cost of failing to produce
41:22that's all that equation says you should
41:24always anytime you write down an
41:25equation you should be able to say out
41:27what the equation says in words
41:30otherwise you're letting the math run
41:33you rather than you run the math okay uh
41:37and
41:38similarly the cost of producing a is
41:41equal to Lambda * K
41:46alpha x to the C A to the alpha minus
41:511 so then we can solve out for what the
41:54opportunity cost has to be
41:57um by setting these two lambdas equal to
41:59each other moving all this stuff to the
42:01other side of the equation so we get TX
42:04over k
42:08c x to the
42:11cus1 a to the
42:14alpha is equal to TA a over k
42:20c x to
42:23the C A to the alpha - 1
42:28an alpha there okay so these Ks are
42:31going to cancel
42:33right and this stuff basically can
42:36cancels except that we
42:39get an X up
42:43there and an alpha up there and that
42:47gives
42:48us
42:49that
42:53um that gives us that
42:58um
43:00X over
43:02Alpha it should be equal
43:07to
43:10uh
43:12ta
43:16Alpha
43:18sorry
43:22ta
43:24uh s over
43:27TX
43:29Alpha so that says we use the ratio of x
43:34to Alpha we choose is equal to the cost
43:38of a over the cost of X obviously we're
43:41going to use more X if a is more
43:44costly um times the ratio of how
43:48productive C is X is C over how
43:51productive Alpha is uh yeah is this
43:54supposed to be x a
43:57X over
44:05a
44:07um okay so uh then what we know is that
44:16um we need to satisfy this constraint we
44:19need to produce at least s so then we
44:21can substitute this condition back into
44:24here so uh we get that
44:30K * a the alpha time well X is just a
44:38multiplied by this thing here so that's
44:42going to be a to
44:45the
44:47times um ta
44:51a over
44:53TX Alpha to the alpha and that's equal
44:56to
44:57s right and that gives us that um a is
45:04equal
45:06to
45:09uh s over
45:12K to the 1 over
45:16Alpha
45:18um times this guy to the
45:23power Alpha
45:25over oh sorry
45:29plus Alpha
45:32plus yeah is that K Alpha a to the C on
45:37the yeah okay because I substituted in
45:40the fact that X is equal to this guy
45:43times uh
45:45a and then you can substitute that in
45:47and solve back out for X okay but the
45:50most important things that we learn from
45:52this are first the classic productive
45:55efficiency condition
45:58so what we got here was that the ratio
46:02of how productive the two things is is
46:05equal to the ratio of their
46:07prices right um and that's the most
46:11fundamental condition for the efficiency
46:13of production and this is also as you
46:16guys have probably seen before in class
46:20um the tangency
46:23condition so remember we have these
46:25isoquants and then we
46:28have like the ratio of the prices is
46:31represented by this tangency point that
46:35the that the ratio of the prices is
46:38equal to the marginal rate of
46:39transformation is exactly that the
46:41prices Li tangent to the uh
46:48isoquants the lrange multipliers are
46:50just the marginal opportunity cost of
46:54failing to produce right and when we
46:56solve out
46:57for Lambda we also solve out for the
46:59marginal cost of production immediately
47:02right and finally we get the cob Douglas
47:04formula which once you plug everything
47:06in is that the cost is equal
47:10to
47:11plus Alpha times this mess
47:19here okay and the optimal Factor
47:22proportions are proportional to prod
47:25productivity and in versely proportional
47:28to the um
47:31cost
47:33okay now
47:37mostly we're going to consider these
47:39cost curves not in terms of totals but
47:42in terms of per uh unit and there are a
47:46few different relevant P per unit
47:48quantities one is the marginal cost of
47:50production which is the derivative of
47:52the cost function and which we'll denote
47:54by MC second is average cost which is
47:57cost per unit on
48:00average and um sometimes these are
48:04decomposed into two different uh Parts
48:08into average fixed cost and average
48:10variable cost so the average fixed cost
48:13is of course the fixed cost divided by
48:14the output and the average variable cost
48:16is the variable cost divided by the
48:18output yeah if there are quy fixed cost
48:21present would you classify them as part
48:23of fixed cost for this we're never going
48:25to do that in this class
48:27um so you can just to ignore it
48:31basically
48:32um these are pretty much pretty much
48:35this whole lecture is stuff that I'm
48:37like sort of have to teach you and that
48:39doesn't interest me all that much so I
48:41tried to at least I tried to at least
48:44teach you something about what a lrange
48:45multiplier is so you understand it
48:47better that's about the only thing that
48:49I think is interesting in this lecture
48:51but I'll show you but I'll show you some
48:53like empirical examples of cost curves
48:55which maybe will be slightly more
48:57interesting
48:59um I mean this stuff is like useful to
49:03like intuitively think through like you
49:07know applications but it's not an
49:09application itself so yeah Austin this
49:11kind of an odd question I know I
49:13remember reading like um in command
49:15economies they were towards the end of
49:17like the Soviet area they started using
49:19computers to try to generate like
49:20prodction functions was it this sort of
49:22MTH or it was much more complicated
49:25versions of the same idea
49:27look this the basic idea as we'll learn
49:29behind the whole market economy is the
49:32same as the basic idea behind this like
49:35basically you can view the whole economy
49:37as one big old optimization problem and
49:40all that prices are are opportunity
49:43costs so as we'll learn later basically
49:47all that an opportunity like there
49:49really is no fundamental difference
49:50between socialism and capitalism it's
49:52just how they do the computations of
49:54what's optimal right the only difference
49:57between them prices in a market economy
50:01are just how much does it cost other
50:05people that you use that resource that's
50:07what you have to pay for it because
50:09somebody else would have been willing to
50:11buy it at that price if you didn't buy
50:12it and so you should have to pay the
50:14opportunity cost of you using that
50:16resource right from the perspective of
50:19society so from a from a communist
50:23perspective uh there the these
50:25opportunity costs that a computer solves
50:27out for from a capitalist perspective
50:30they're called prices but they're
50:32exactly the same
50:36thing um oh okay this this is a good
50:39story uh okay so average cost um okay so
50:46average cost is the uh cost divided by
50:50the output so if we take the derivative
50:51of that we get the derivative of the top
50:53times the bottom minus the dtive of the
50:57bottom times the top right that's the
50:59derivative of total cost time the output
51:01minus total cost right uh which gives
51:05you this uh but we know that the
51:07derivative of total cost is just
51:09marginal cost and that the total cost
51:12divided by output is equal to the
51:13average cost so we get that the
51:16derivative of average cost is equal to
51:17marginal cost minus average cost over um
51:21over the output now that says that the
51:25uh uh total that the average cost will
51:28be rising whenever the marginal cost is
51:30above the average cost and it will be
51:32falling whenever the marginal cost is
51:34below the average cost so the way I like
51:36to think about this is in terms of test
51:37scores if you got above the average of
51:41the test you know that um you the
51:45average is higher than what it would
51:47have been had you not been
51:48included right does that make sense okay
51:52so this is related to how I met my wife
51:54so uh freshman year in Latin American
51:57politics class um uh the the professor
52:02came back at the midterm and it was
52:04mostly an upper class class there was
52:06like you know n 80% third and fourth
52:10years there were only five first years
52:13in the class out of like about 80 people
52:15or something like that and he said I was
52:16really amazed of all the classes The
52:19Freshman did the
52:21best and on the on the midterm exam now
52:25um on this exam there were um there were
52:28five freshmen I was one of them uh uh
52:32there were three other girls who were um
52:35coming to me for help on the exam so I
52:37was pretty confident that they hadn't
52:41done well enough to given that it was
52:44like a uh you know sort of Latin
52:46American politics class you can't like
52:48curve bust in a Latin American politics
52:50class right you can only get an A
52:52there's no like you know A++ Plus+ right
52:54so I knew I wasn't enough to out away
52:56the three of them and to make our
52:57average above the upper classman so I
53:00know that the fifth freshman who became
53:02my wife had to have done really really
53:05well on the exam and that was why I
53:07decided to start hitting on
53:10her so you could tell something both
53:12about my taste in women the fact that
53:14she became uh a Latin American politics
53:19uh graduate student uh and the fact that
53:22I became a mathematical Economist so it
53:24speaks to a lot of things about us
53:27um okay so uh the uh
53:35um total cost is a curve that we rarely
53:39draw uh but it can be decomposed of
53:41course into the fixed and variable
53:43components just like the average cost
53:45curves okay um
53:50so
53:54uh you guys have probably seen these
53:56curves before but you know a typical
53:59shape will be that um there's like some
54:02fixed
54:03cost uh and the average fixed cost
54:06because it's just fixed it just dies off
54:09like a you know one over uh the
54:13output so this is the average fixed cost
54:18the variable
54:20cost
54:21um well so the marginal cost
54:26often is assumed to be initially
54:29decreasing and then
54:33increasing the variable C the average
54:36variable cost is going to follow the
54:39marginal cost but more slowly right by
54:41the logic we just did so they start out
54:44together but now this is below it it
54:46sort of follows it but now this starts
54:49to come back
54:50up uh and then this starts increasing
54:54again and it sort of follows it up up
54:57and then the average total cost is the
54:59sum of these
55:07two so oops it's always above
55:20this yeah looks something like that so
55:24the key thing is that the average costs
55:27always intersect the marginal cost at
55:28their minimum right because when it's
55:30below it it's falling and when it's
55:33above it it's
55:35rising okay so um let me give you some
55:39empirical examples of cost curves so one
55:42cool one uh comes from a paper by uh
55:46Susan ay and Dennis nikov who study
55:49online search ad auctions so these are
55:52auctions where biders offer a price that
55:55they're willing to pay per click to
55:59uh to for a word like car insurance so
56:04if you want your you know uh web website
56:07to show up on Google whenever someone
56:10searches for car insurance you might
56:11enter a bid and they usually submit
56:14these bids and maybe change them once
56:16every week or two and they win slots
56:20based on a score that comes from the
56:22search engine because the search engine
56:25basically says how likely do I think
56:27this ad is going to be to get clicked on
56:29they learn that based on a machine
56:30learning algorithm you know they look at
56:32how much people have clicked on in the
56:33past
56:35Etc and you get a you get a score which
56:39says How likely is your ad to be clicked
56:40on you say how much am I willing to pay
56:43per click and then Amazon assigns you a
56:46spot based on what's going to generate
56:48the most revenue for them right um and
56:52what you have to pay is how high you
56:55would have had to have bid in order to
56:57get into the slot that you get so
56:59they're going to give the first slot to
57:00the one who's going to generate the most
57:01total revenue second to the person who
57:03gets the second Etc and then they say
57:06how much would you have to have bid in
57:07order to stay in your slot rather than
57:09get knocked down to the next slot that's
57:11what you end up
57:13paying okay
57:15so if you are in a higher slot
57:20right then you're going to get more
57:22clicks most likely right because more
57:24people click more on the top slots so
57:26you can think of yourself not as
57:27choosing a price to
57:30bid but instead choosing how many clicks
57:33I'm going to
57:34get on average right because there's
57:38like a onetoone relationship between
57:39those things the more the higher I bid
57:41the more higher higher slot I'm likely
57:43to get and therefore the more clicks I'm
57:44likely to get so you can think of me
57:46just choosing how many clicks on average
57:49do I want to get right so there's like a
57:52cost of clicks curve that's derived from
57:55how much comp competition you face in
57:56the auction what your score is all that
57:58sort of stuff right um so we can derive
58:03both the average cost of clicks and a
58:05marginal cost of clicks from that curve
58:08right if I want to have one more click
58:09in expectation how much is it going to
58:11cost me well I'm going to have to bid
58:13enough so that I would get etc etc right
58:16and you can use this to derive optimal
58:18bids and this is exactly what uh a and
58:21nikov did using the Microsoft data so
58:25first they found the average cost of
58:27clicks by just saying okay if I want to
58:30get this many clicks on average here's
58:32how much I'm going to have if I want to
58:33get this many clicks on average here's
58:35how much I'm going to have to pay for
58:37those right and then you can derive from
58:40that a marginal cost of clicks and then
58:42the company can say well here's how much
58:44I value getting a click and that gives
58:46them the optimal clicks to to get makes
58:50a lot of sense no um and the thing I
58:53liked a lot about this is if you you
58:56looked at the problem the other way if
58:57you just said how much should I bid that
59:00seems like a really complicated problem
59:01it sounds like auction Theory whatever
59:03but if you say no here's how much it's
59:06going to cost me to get a certain number
59:07of clicks
59:09right uh that's easy to figure out just
59:12using some basic data and then you can
59:14derive from that the marginal cost just
59:16through their relationship to each other
59:18and then it's easy to figure out how
59:19much you want to bit the value is the
59:22yellow line
59:23yeah okay so so another example of sort
59:28of like a complicated problem that can
59:31be um quite
59:34simply uh
59:36understood by
59:40um by uh thinking about things in terms
59:43of cost is the cost to the government of
59:46providing insurance
59:49so imagine we have just one insurance
59:52provider so it could be the government
59:53or it could be like you know I'm an
59:55employer and I'm going to figure out how
59:57many of my employees I'm going to cover
59:59by changing around the premium that it
1:00:01costs to get coverage right if I have a
1:00:03really low premium everyone's going to
1:00:05get coverage if I have a really high
1:00:06premium no one's going to get coverage
1:00:08right so by moving around that premium
1:00:10or under Obamacare moving around the
1:00:12subsidies you can figure out how many
1:00:14people will get covered
1:00:16right
1:00:17um and so the quantity there is the
1:00:20fraction of the population that gets
1:00:21covered in insurance right so as it
1:00:25lowers its price more people will sign
1:00:27up for
1:00:28coverage now if everyone had the same
1:00:30cost of being covered this would just be
1:00:33a totally linear cost constant marginal
1:00:35cost but different people typically have
1:00:37different costs of being covered right
1:00:40the typical story is that the people are
1:00:41going to most want coverage who will be
1:00:43willing to buy it even when it's very
1:00:44expensive are the really unhealthy
1:00:47people right and that people who be
1:00:50willing to buy it even when it's less
1:00:52expensive who will be willing to buy it
1:00:54only when it's less expensive are going
1:00:55to the healthier
1:00:57people right that's called adverse
1:01:00selection or the lemons problem so you
1:01:03can understand the adverse selection
1:01:05problem I don't know if you guys have
1:01:06ever heard about this but it's usually
1:01:08you know this sort of exotic thing about
1:01:10asymmetric information you can
1:01:11understand it completely in terms of the
1:01:13cost curve faced by the government for
1:01:15providing
1:01:19Insurance um the the oldest story where
1:01:24this first came up this idea of
1:01:25adversity selection was called Gram's
1:01:27law does anyone know what gresham's law
1:01:29is so Gram's law was the idea that uh
1:01:33bad money drives out good money so if
1:01:36there's like imagine that there's like
1:01:38non-old money and there's gold money uh
1:01:42and people know that the gold money is
1:01:44really worth more than the non-old money
1:01:45is but you're legally required to treat
1:01:47them the same then everyone will hoard
1:01:49the gold money and use the non-old money
1:01:53and that's very similar to the idea of
1:01:55what happens under adverse selection
1:01:57right if you're forced to treat s
1:01:59because you don't know the difference
1:02:00say sick people and healthy people the
1:02:03same all the bad money the unhealthy
1:02:06people will come and try to get
1:02:07insurance and all the uh healthy people
1:02:11will try not to get
1:02:12insurance
1:02:14right however it's possible that we
1:02:17could get the opposite thing happening
1:02:19it could be that the people who really
1:02:20want Insurance are not the sick people
1:02:23but the people who are uh hypochondrial
1:02:26right because they're really afraid
1:02:28they're going to get sick and those
1:02:29people also take really good care of
1:02:31their health right because they're
1:02:34worried about getting sick if that's the
1:02:36case you might get the opposite
1:02:37relationship where actually the people
1:02:39most eager to buy insurance are the
1:02:40least costly in that case the cost Curve
1:02:42will slope up because the people who
1:02:44come in late to the market you know who
1:02:47only come in when there's a lot of
1:02:48people in the market will be the people
1:02:49who are really healthy uh sorry really
1:02:52sick not the people who are healthier
1:02:54this is called advantageous
1:02:56selection we're going to return to the
1:02:58policy implications of this in some
1:03:00later lecture possibly lecture 12
1:03:02possibly lecture 10 actually but because
1:03:05I forgot to change this on the slide uh
1:03:08so um there was a nice paper by luron
1:03:10aov and uh Amy finlin and Mark Cullen
1:03:13that estimated this so what they
1:03:16considered is they looked at a very
1:03:17large company that was offering
1:03:19insurance and different managers and
1:03:21different divisions offered this
1:03:24insurance for different prices
1:03:27and what they looked at was and they
1:03:29assumed that this was randomly assigned
1:03:31that basically the different managers
1:03:32basically had similar workers but they
1:03:34were randomly choosing different prices
1:03:36whether that's true or not who knows but
1:03:38as a result these people were getting
1:03:40different numbers of people being
1:03:42insured um and this gave them estimates
1:03:45of the cost curve so here's what they
1:03:47got um here were the different prices
1:03:49that were
1:03:50charged uh and the white dots indicate
1:03:53how many people bought insurance at the
1:03:55different prices and as you see it sort
1:03:57of looks like a demand curve uh as you
1:04:00increase the price fewer people buy
1:04:02insurance right um then you can ask okay
1:04:07these different people bought Insurance
1:04:08what was the average
1:04:11cost that was spent on covering these
1:04:13people who bought insurance right you
1:04:15can just look at that in the claims data
1:04:18for the insurance companies right you
1:04:21can just look at how much on average
1:04:22they cost when these different things
1:04:23were assigned and that gives you this
1:04:25average cost curve and from that you can
1:04:27derive a marginal cost curve now there's
1:04:31there's a notion that adverse selection
1:04:33can cause there to be too little
1:04:34Insurance in the market and the reason
1:04:36is that the company usually prices these
1:04:38plans at their average cost but it's
1:04:40socially optimal instead to price at the
1:04:42marginal cost where that equals demand
1:04:44and that creates this Distortion here um
1:04:48that we'll talk about
1:04:50later
1:04:51okay so um an important thing to note is
1:04:57that neither of these cost curves looks
1:04:59like those standard shapes of cost
1:05:00curves that I drew out there right so
1:05:03while those types of cost curves um
1:05:07correspond to a certain story in the
1:05:09real world many cost curves look
1:05:11different depending on the types of
1:05:12application you're looking at so the
1:05:15standard
1:05:17story that generates these cost curves
1:05:20is that there's some fixed cost but then
1:05:22there's decreasing
1:05:23returns um typically because the company
1:05:28has only a certain amount of
1:05:29entrepreneurial talent to run it and
1:05:31that as you get larger that Talent is
1:05:34spread thinner over uh more
1:05:38production so let's consider the cob
1:05:40Douglas example imagine that we have a
1:05:42cost which are some fixed cost plus some
1:05:45cob Douglas costs now I've gotten rid of
1:05:47all those messy Co coefficients and I've
1:05:49just collapsed into some number
1:05:51multiplied by the score you try to
1:05:53produce to some power gamma right
1:05:56right so then the average fixed cost
1:06:00right is just the fixed cost divided by
1:06:02S the average variable cost is that
1:06:04variable cost the second component
1:06:06divided by S the average uh total cost
1:06:10is adding those two things up and the
1:06:12marginal cost is the derivative uh given
1:06:15over
1:06:16there uh so we can do comparative
1:06:19Statics on this uh using
1:06:22Mathematica um
1:06:27uh am I going to try to put you guys
1:06:29through that let me see well so this is
1:06:32what you get if f is equal to one if
1:06:34Kappa is equal to one if gamma is equal
1:06:36to
1:06:381.5 um so you get the sort of
1:06:40characteristic shape the marginal cost
1:06:42is going up the average cost is uh
1:06:46average variable cost is going up but
1:06:48the average total cost has this downward
1:06:50shape and then comes back up because the
1:06:53average fixed cost is being uh
1:06:57deferred um let me see if I can figure
1:06:59this
1:07:01out yeah uh the question C function set
1:07:06up last year we always you know told
1:07:08that the uh exp the alpha and S always
1:07:12had add up to one yeah but that's only
1:07:14if there's constant returns to scale but
1:07:18there but I'm explicitly here assuming
1:07:20that there's decreasing returns to scale
1:07:22because that's how you get the um forms
1:07:26that uh you know sort of uh the standard
1:07:30ways of
1:07:32uh the standard ways of um here we
1:07:40go yeah we would have gotten like a
1:07:42linear thing there actually we can play
1:07:43around with this if you want uh here we
1:07:47go so this is
1:07:49Mathematica I'm just putting into it all
1:07:51the things that we just had right
1:07:57and here are the parameter
1:08:02values and here we get a graph so here
1:08:04I've assumed that gamma is equal to
1:08:076 right in that case we have increasing
1:08:12returns to scale so all these cost
1:08:14functions are going down if now I
1:08:16increase gamma to one what your
1:08:17suggestion right when we have constant
1:08:20returns to scale we can re now draw the
1:08:22things and you see we have now linear
1:08:24cost and so the marginal cost and
1:08:26average variable cost are completely
1:08:29flat right and that doesn't lead to the
1:08:31characteristic shape because we just get
1:08:34this like leveling
1:08:36off but if I make gamma higher so that
1:08:41um so that those things don't sum to
1:08:44less than one
1:08:46right then uh we get the characteristic
1:08:50shape and as I make gamma even higher we
1:08:52get more decreasing returns to scale
1:08:57and this gets even more extreme in fact
1:08:59if gamma gets above two then actually
1:09:03the marginal costs are
1:09:06convex if we increase the fixed cost on
1:09:09the other
1:09:10hand that's going to make the point at
1:09:12which we reach a minimum of the average
1:09:15total cost curve be further out because
1:09:18there's more fixed costs to get rid of
1:09:21you know to sort of like amortise before
1:09:23you get overwhelmed by the marginal cost
1:09:25so that's the sort of thing you can play
1:09:27around with using this
1:09:29um uh I can post this code if you guys
1:09:32want it I just yeah just send send me an
1:09:36email and I'll and I'll post
1:09:39it
1:09:45okay mathematic is very straightforward
1:09:47you literally just enter in everything
1:09:48you want and then you can play around
1:09:50with
1:09:51it let me just see where I am
1:09:55okay good we're in exactly the right
1:09:57place so the standard assumption is that
1:10:00average cost declines first and then
1:10:02increases for the reasons I just said
1:10:05now declining average cost is called uh
1:10:10uh economies of scale and why is it uh
1:10:13called what are some of the reasons uh
1:10:16Joan that are given for there being
1:10:18economies of scale usually um for
1:10:20example if you
1:10:23have you might
1:10:26some
1:10:37and yeah that's a good good idea it's
1:10:40like um you there might be certain
1:10:43things that you need that are fixed but
1:10:46that once you have those you can produce
1:10:48a lot right
1:10:50um
1:10:52so uh an example I uh like to give of
1:10:57this is there's a restaurant in the
1:10:59outskirts of Bogota actually they now
1:11:01have one in downtown Bogota Columbia
1:11:03called Andre Carnes day race Andre
1:11:06carese race is sort of a giant show
1:11:09it's like uh this crazy Place restaurant
1:11:13for like 10,000 people basically uh and
1:11:16they have like giant daycare center
1:11:20there they've got like you know dancing
1:11:23for young people they've got you know
1:11:26like a playground for like you know kids
1:11:28all this whole thing right why are they
1:11:30able to do that well if a normal
1:11:32restaurant can't do that because they um
1:11:36you know they only have a small number
1:11:38of people and it costs a lot to build
1:11:40each one of those facilities but if
1:11:41you've got a huge number of people
1:11:43coming to your restaurant right you can
1:11:45amortise those fixed costs over a large
1:11:47number of people and it makes sense to
1:11:49do it right so that's sort of Janes
1:11:50example right you can attract these
1:11:52people but you have to have enough
1:11:54people to make it worth it having all
1:11:55those facilities Yeah
1:11:57Austin um another reason is that you can
1:12:02specialize uh when you have a larger
1:12:04facility so rather than having one
1:12:07person do lots of different things
1:12:08people can focus on something and get
1:12:10very good at that and that's Smith's pin
1:12:13Factory example I don't know if you've
1:12:14ever read The Wealth of Nations but
1:12:16there he talks about how in the pin
1:12:18Factory each person does one thing and
1:12:20they get very good at that one thing and
1:12:22in particular they're not constantly
1:12:23changing between different things so
1:12:24they don't waste time in the transition
1:12:26and that's like the principle on which
1:12:28assembly lines are basically built
1:12:30right another reason can be economies a
1:12:33scale over time where by doing something
1:12:35for a while you can get better and
1:12:37better at doing it um and I'll show you
1:12:39an empirical example of that in just a
1:12:42moment uh a third case is learning by uh
1:12:46sorry a final case is Network effects so
1:12:49it may be that um just because uh the
1:12:53consumers like interacting with of other
1:12:55consumers that it's cheaper to supply a
1:12:57service to many consumers than to few so
1:13:00Facebook's a classic example of that
1:13:01right like having a million different
1:13:04Facebooks uh providing equal Quality
1:13:07Service would be very very hard because
1:13:08consumers like being able to interact
1:13:10with everybody right so if you slice
1:13:12Facebook up it's much more expensive to
1:13:14provide a service of the equal quality
1:13:17and a final example is adverse selection
1:13:19as we just talked about right it's
1:13:21actually cheaper to provide Insurance to
1:13:23the whole population than to provide it
1:13:24to a small small part of the population
1:13:26because you don't get everybody into the
1:13:29market uh because you the healthy people
1:13:31stay out of the market unless you're
1:13:32covering
1:13:34everybody right
1:13:37increasing uh decre increasing marginal
1:13:39cost is called dis economies of scale
1:13:42and Christian what might be some reasons
1:13:44for diseconomies of scale well not all
1:13:47of your inputs will scale up at the same
1:13:50rate so for example if uh I think in the
1:13:53book it talked about the Farmers on the
1:13:54same amount of land so you can increase
1:13:57the amount of farmers but if you have
1:13:58the same amount of land eventually not
1:13:59going be able to each farmer is not
1:14:02going
1:14:02to yeah that's right there's also one
1:14:06entrepreneur so entrepreneurial Talent
1:14:09yeah um will stay constant even as you
1:14:11increase operation that's right so a
1:14:14classic example of this is restaurants I
1:14:16don't know if you've noticed but like
1:14:18most restaurants they start they could
1:14:19start out being really good and then
1:14:21they try to like build another location
1:14:24and suddenly the quality starts going
1:14:26down because like they you know what was
1:14:28making them so good was that they had a
1:14:29chef that was like really really good
1:14:31and keeping things going but if he's now
1:14:33getting spread thin over all these
1:14:34restaurants the quality starts going
1:14:36down another reason yeah EV should it be
1:14:39increasing average
1:14:42cost uh yes that's true that that that's
1:14:47right that's right that's
1:14:52right um I now of course you could try
1:14:55to hire other people right to run the
1:14:58company but if you don't trust the
1:15:00people or they're not acting in your
1:15:01interest for the reasons that we were
1:15:03talking about last lecture that becomes
1:15:04hard so in particular in developing
1:15:07countries where it's much harder to
1:15:10enforce contracts it's much harder to
1:15:12trust people it's much harder to keep um
1:15:15you know people from stealing uh
1:15:17companies tend to be quite small because
1:15:20uh they can't find people to delegate to
1:15:24who they Trust
1:15:27um another reason might be Market power
1:15:30so you remember that internet auction
1:15:32thing we saw question is why does why
1:15:35doesn't it just cost a constant amount
1:15:37more to get additional clicks and the
1:15:39reason is as you try to buy more clicks
1:15:42now you're the one who's really setting
1:15:43the price right because you're having to
1:15:47bid higher in order to get it whereas
1:15:49before like you know as you were bidding
1:15:50higher for clicks it was basically other
1:15:53people's demand that was setting the
1:15:54price and that's one reason why you had
1:15:56that increasing marginal cost in that
1:15:58graph and average cost yeah it doesn't
1:16:01Market power lot times also cause
1:16:04economies of scale you to you know
1:16:06increase so no actually I mean Market
1:16:08power might be caused by economies of
1:16:10scale but the reason it requires
1:16:12economies of scale to cause cause it is
1:16:14that it itself leads to this economies
1:16:17of scale you see what I mean yeah this
1:16:20Market power is also equivalent let's
1:16:23say to a dis econom is a scale for
1:16:26education in an area because if you're
1:16:29you make too many schools you start to
1:16:31tap out the reserve of available yeah
1:16:33absolutely
1:16:36absolutely nonetheless I think the most
1:16:38common thing is constant returns to
1:16:40scale because it's often possible to
1:16:42replicate anything you can do in a small
1:16:44area you can sort of do the same thing
1:16:47in some sense more broadly
1:16:51um in particular if you really want to
1:16:53expand something Broad
1:16:55you usually have to replicate the fixed
1:16:57cost that you had so it might be true
1:16:59that you know for Andre car St race that
1:17:02you know in order to get efficient scale
1:17:04they needed to have that but if they
1:17:05wanted to build another Andre cares to
1:17:06race it's not like they could uh just
1:17:09share the same uh dance floor with that
1:17:12other you know place and even if they
1:17:13wanted to make it much bigger at some
1:17:15point the Dance Floor would get crowded
1:17:16and they'd have to build a second dance
1:17:17floor right okay so here's an example
1:17:21from data of economies of scale over
1:17:24time in particular learning by doing so
1:17:28this uh is from a paper by lanir Benard
1:17:31and this shows the um cost in manh hours
1:17:36to produce uh an
1:17:38airplane um after they had produced so
1:17:42many units right and this shows and and
1:17:45this here shows how much they were
1:17:47producing any given year so what's
1:17:50happening is they're going down this
1:17:52what's called learning curve because
1:17:54they've produced produce more and more
1:17:55and they're getting better at better at
1:17:56producing but then they get a reduction
1:17:59in their orders and their costs go back
1:18:01up because they actually forget a little
1:18:04bit of what they learned and then once
1:18:06their orders go back up again then it
1:18:08their cost come comes down
1:18:11again okay so
1:18:15um uh yeah Evan I was just wondering
1:18:17could that cost also be explain rather
1:18:19than foret just by it becomes so
1:18:22expensive to create so many additional
1:18:23units they have to Le methods to bring
1:18:26down well I think it's particularly
1:18:28indicative that you see that when their
1:18:30volume goes down then their costs go
1:18:33back up you see what I mean it's not
1:18:35just in general that it goes up but it's
1:18:38precisely at the moment when they their
1:18:41their volume goes down that their costs
1:18:43go back up again yeah is that because
1:18:44people leave because uh they they don't
1:18:47work they're not getting yeah they
1:18:49probably have to fire a lot of the
1:18:51people uh or you know there's just turn
1:18:54over in the business and they are not
1:18:57the new people are not getting trained
1:18:59as
1:19:00much okay now you can measure economies
1:19:03of scale by something called the
1:19:05elasticity of scale which is the ratio
1:19:07of average to marginal
1:19:09cost um if this is greater than one you
1:19:12have economies of scale and if it's less
1:19:14than one you have dis economies of scale
1:19:16and this is a generalization of the
1:19:18notion of homogeneity of degree K
1:19:21because the uh elasticity of scale when
1:19:23you have homogeneity is equal to
1:19:26k um okay so when this is basically
1:19:31comes from the fact that the
1:19:33elasticity of
1:19:36um because something which is
1:19:38homogeneous has constant elasticity uh
1:19:41by the exact same argument we were
1:19:42giving
1:19:44earlier uh when there are multiple
1:19:46outputs we often talk about economies of
1:19:48scope so this is things that are cheaper
1:19:51to produce together than to produce
1:19:53separately so one reason for economies
1:19:55of scope is that um there can be common
1:19:58inputs so for example Walmart finds it
1:20:01cheaper to have uh all of the goods sold
1:20:06together rather than to have you know
1:20:08other companies that would sell them
1:20:09separately because they can have one
1:20:11truck bring all the stuff in right and
1:20:13so that makes it cheaper for them to
1:20:15sell the things together than to sell
1:20:16them separately another reason can be
1:20:19byproducts so for example crop rotations
1:20:22are very popular it's cheaper to produce
1:20:24two different crops on alternating years
1:20:27than to produce the same crop over and
1:20:28over again because what one crop takes
1:20:30out of the soil another crop can put
1:20:32back into
1:20:33it uh another example is agglomeration
1:20:36in cities so cities bring together lots
1:20:38of different types of production because
1:20:41it can benefit um someone in the tech
1:20:44industry to have people uh from Finance
1:20:47around because they can provide Finance
1:20:48for the tech industry you see what I
1:20:50mean so having those things together can
1:20:52be valuable because they feed off of one
1:20:54another
1:20:55but there can also be diseconomies of
1:20:57scope and I think a good example of that
1:20:59is when there could be clashing cultures
1:21:01between the production of different
1:21:02things so in Peru IND copi uh is their
1:21:07um antitrust and intellectual property
1:21:09regulator but the cultures of antitrust
1:21:12and intellectual property are very
1:21:13different from one another intellectual
1:21:15property is like a service to the
1:21:17business whereas uh competition policy
1:21:20tends to try to serve consumers and
1:21:23crack down on businesses that are being
1:21:24abusive and those two having those two
1:21:26people around can reduce each of their
1:21:28productivity because it can undermine
1:21:30their morale and confuse them about what
1:21:31they're trying to
1:21:32accomplish okay thanks guys sorry about
1:21:35the mess ups

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