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0:12
all right why don't we get started um today we're going to continue our discussion of producer theory once again to remember to put this in context we the first few lectures were working on consumer theory to help us derive a demand curve now working on producer theory to help us come up with supply curve we started last time by talking about how producers profit maximize and the profit maximization implies cost minimization uh that if you want to maximize profits
0:40
you're going to want to produce as efficiently as possible and basically to do that we need to understand how your costs vary with your output if you're going to produce at an efficient level you need to understand how what your costs are going to vary with your level of production so essentially our goal of this lecture is to develop a cost curve develop a curve which tells you how the cost of your production varies with how much you produce and that's what we're after in this lecture
1:12
okay so we're going to start in the short way the short run and then turn to the long run so start with developing a short run cost curve and then turn to long run okay and to make this lecture sort of mathematically coherent throughout the whole lecture we'll work with our favorite functional form a production form of the function q equals the square root of l times k remember firms produce goods little q using two inputs labor and capital labor is a variable input that means you
1:43
can change it in the short run and the long run capital is a fixed input which means that you can only change in the long run okay so in the short run we're going to have two kinds of costs we're going to have fixed costs which are going to come from a fixed level of capital and we're going to have variable costs we're going to come from our labor so we're going to have two kinds of costs fixed costs they're fixed because in the short run you can't change the level of capital
2:10
and variable costs which are the costs of our labor of labor and then we're going to have total costs are simply going to be fixed costs plus variable costs okay so that's how we think about costs think about any the cost of a firm's production in the short run is the sum of their fixed costs i.e their capital costs and the variable costs i either labor costs now we're going to show you how you can turn a production function into a cost function and to do so you simply need
2:44
to recognize that costs the costs of firms production are simply the amount of capital it uses k bar times the price of that capital which we'll call r plus the amount of labor it uses times the price of that laboratory we'll call w the wage okay this is the easy part think of the amount of hours of work you use times the wage per hour or the amount of workers times the salary per year in any case this is easy to understand every additional unit of labor comes with the cost that is the wage of
3:18
that unit of labor so think about an hourly model every hour you work at your convenience store they have to pay you the minimum wage for that hour okay that's the cost of that hour of labor capital is harder we call r the rental rate and the reason is because we don't think of don't think of buying capital don't think of buying a machine think of renting a machine the reason we do that is to make the periodicity work okay you don't buy a worker thank god you rent that worker
3:49
and you rent that worker at a pr at a price w so when the firm uses your time they're renting your time an hour at a time at a price w when we get machines think of a firm as renting machines for a price r per machine per time period okay so i understand firms usually don't do this they usually buy machines and we'll come back to even if you buy a machine how it effectively is like renting it but for ease of thinking about this want to think about flows not stocks
4:20
so about the firm's decision as renting a worker at the price wage at the price w or renting machine the price are yeah
4:31
all that would be in there and we'll come back to that that's right this is a this is sort of the per period cost of using a machine this is the per period cost of using a worker is the wage this is the per period cost of using machine which will include all the costs of running the machine as well as the cost of renting the machine itself okay so later we'll talk about how you own the machine and we'll we'll come back to the fact that you can actually our representation is not bad but for
4:57
now just think of renting machine or if it's a building think of this as the year as the rent you pay on that building okay not the cost of building them build the building now armed with our production function okay and let's also say to make life easy uh i did this wrong in my notes so i hope my the other teachers figure it out uh to make life easy let's say let's say that the rental rate r we're gonna say is ten dollars and the wage rate w is going to be five dollars okay now
5:33
armed with a production function if you simply have the production function and these two prices you can derive the short run cost function how do you do that well just look at the math we know that q equals square root of l times k bar okay in the short run l times k bar okay so inverting that simply means that l equals q squared over k bar l equals q q squared over k bar okay and that means that costs can be written costs can be written as 10 k bar the price the rental rate is 10
6:26
we have k bar amount of capital plus 5 q squared over k bar okay that's our cost function okay i just plugged in for l and then i multiply plug that for l and multiply by the wage rate w which is five okay so for example for a fixed level of capital this is cost so for example let's imagine our short run level of capital is one let's imagine we have one unit of capital in the short run just to make the math easy then that simply says that cost equals 10 plus 5
7:03
q squared and that's our cost function 10 plus 5q squared we've just derived the short run cost function 10 is the fixed cost component that doesn't vary with the amount you produce so there's no q part of this 5q squared is the variable component that varies with how much you produce so the cost function is a fixed part which comes the fact of one fixed unit of capital and a variable part which comes from the fact that the amount of q that the amount of q drives the amount of labor we need
7:39
okay question about that math all right now armed with this cost function let's write that down again here so c equals 10 plus five q c equals 10 plus five q squared that's gonna be our short run cost function we're gonna work with and remember that short run cost function came directly from that production function to derive this equation all i needed was that production function and those two prices and i derived this okay armed with that okay we can define some of the key
8:12
concepts that will drive our entire analysis of firms and the single most important concept is marginal cost which is what it sounds like the derivative of cost with respect to quantity okay so in this it's the marginal cost is delta c delta q
8:31
okay that's marginal cost okay we'll also care about average cost which is just c over q okay it's very important in this class to keep our modules different from our averages the average is simply over the entire range of production what is the average cost of producing each unit the marginal is what's the cost of producing the next unit and since production functions are non-linear those will not be the same generally okay average will not equal marginal in general
9:01
because of nonlinear function delta c delta q is not the same as c over q okay so we can actually graph these in figure six one figure six one shows the cost curves for this cost function i just wrote down which comes from that production function okay so you can see that the marginal cost as i said is delta c delta q well that's 10q so the marginal cost the cost of producing the next unit rises with the number of units which makes sense the cost is a q squared term in it
9:39
so obviously the marginal cost can have a q term in it so basically the more you produce the higher your marginal cost the more units you produce the more little q the higher marginal cost average cost is this sort of funky shape thing where it's which is 10 over q plus 5q i just divided this by two okay where it's first declining and then increasing why is that why is average cost first declining and then increasing we'll see what just intuitively why is that why in general in the short run
10:14
would we expect that average costs first to fall and that increase you have ideas yeah
10:25
well no but it's falling first so why is it why is it falling first you're it's about that yeah really high average or first fixed cost right right the first units are paying off your fixed costs if you think about it well the first units you sell basically you start with this huge fixed cost so actually by selling two units yes you get the variable cost second unit but you get to pay off the fixed costs the first unit so look at here on this graph we show average fixed
10:54
costs and average variable costs average fixed costs are 10 over q if you only produce one unit your average your average fixed cost is 10. you produce units it's five with every unit you produce your average fixed cost is falling you're paying off that fixed cost average variable cost rises every unit you produce you're getting more and more variable cost you put those together and you get a function that first declines and then rises you first pay off your fixed costs so
11:22
your average costs are falling then your marginal cost then you start to rise because you've got marginal cost that increases quantity produced and critically the marginal cost intersects the average cost curve at the minimum of average cost and that's just mathematical okay if you have any function and you take the average then the minimum is going to be at the derivative okay that basically before you get to here 1.5 units before 1.5 units average cost is above marginal cost because you're
11:59
paying off your fixed costs once you get beyond 1.5 units average cost is below marginal cost so average cost hits marginal cost at the minimum average like cost yeah large company he eventually doesn't really have to worry about their fixed cost because like if they're having a lot more workers their entire cost is going to be considered basically in the short run it depends on this function you said large but large can be defined in two ways but absolutely true certainly if we take
12:28
this very function for large enough q's average fixed cost asymptotes to zero right 10 over infinity zero so for large enough q's the average fixed cost goes away okay in the short run remember we're in the short of these fixed costs okay now so other questions good question other questions about that so that's our basic intuition the short run is that at first your costs are super high because you got to pay because you had to build the plant okay but then over time that plant cost falls
13:02
and then your only cost is based on that you got to hire more workers if you want to produce more okay now what we want to notice is that in the short run there is a really close relationship one was the key relationship between marginal cost and the marginal product of labor which we defined last time and the marginal product of labor was dqdl how much remember digging a hole and diminishing marginal product that each additional worker for a fixed level of capital
13:34
is less and less productive right we talked about that last time well the marginal cost of production is as i said equal to delta c over delta q okay well we know from last time that delta q over delta l we defined as the marginal product of labor right plugging those um and uh so we know marginal cost of delta c delta q okay and we can write this if you take the derivative of the cost function so our general cost function up there see the cost function at the top there
14:14
take the derivative of that with respect to the amount of labor well the first term drops out because it's fixed so you can rewrite delta c delta q as w times delta l delta q ah w times delta l delta q right i just rewrote delta q delta w got it's brutal sorry guys w times delta l over delta q okay a little bit better okay i read that because i just took the derivative of the cost function okay first term drops out we take the derivative second term i just took the derivative here
14:54
or the discrete derivative and i just said delta c delta q is w times delta l delta q well we know the marginal product is delta q delta over delta l so we can rewrite marginal cost as w over the marginal product of labor the marginal cost is equal to w over the marginal product of labor and that makes sense the marginal cost of the next unit will be higher the higher the wage and lower the more productive the worker making another unit with a super productive worker
15:32
is cheap okay make another unit with a very unproductive workers expensive so essentially the more you pay them for each hour of work the higher your marginal cost but the less they get done each hour of work the higher your marginal cost so that's why roughly speaking firms might want to pay a lot to people who are high skilled okay so you might say gee why are they paying my friend twice what i'm paying well maybe your friend's twice as productive as you okay that would make or two and a half
16:02
times productive as you and that makes sense so we can't just say that it's a mistake to pay someone higher wages we have to consider their wages relative to how productive they are okay and that's a key relationship uh we will um we'll come back to okay question about that all right that's the short run now let's go to the long run it gets a little more interesting actually i was riding that side i'll switch to this side today switch things up
16:34
i guess i do decide i'd block you less probably so i should do this side okay long run cost curse
16:43
okay long run cost now here what gets interesting is now k is no longer fixed now we get to choose our input mix okay and now our goal is going to be how do we choose our input mix to minimize costs that's our goal here how we're going to choose the mix of workers and machines to produce a given quantity most efficiently now that optimal mix may change with the quantity so we're going to start we're going to say two steps first we're going to say for a given
17:18
quantity picked out of a hat what's the right mix of labor and capital that minimizes the cost of producing that quantity given our production function then we're going to say as the quantity varies how does that change the optimal mix of l and k and does it okay so two steps first say for a given quantity what's the right l and k to minimize cost producing that quantity then ask well as we vary the quantity how does l and k vary optimally okay so basically we want to find the economically
17:51
efficient combination of l and k which is a combination that produces goods at minimum cost okay and to do this we are going to write down to derive this we're going to write down what we call iso cost curves iso cost curves remember last time we did isoquants which felt a lot like a difference curves costs they feel a lot like budget constraints okay iso cost curves are essentially the firm's budget constraint okay they're essentially mappings of the function c equals wl plus rk
18:26
essentially mappings for different amounts of k and l of the function c equals rk plus wl so if you look at figure six two here we see our iso cos lines okay so let's talk about this for a second so for example take the middle one the hundred dollar iso cost this is saying what combinations of labor and capital cost you one hundred dollars well with a rental rate of ten dollars and a wage of five dollars that means you could have 10 machines and no workers or 20 workers and no machines or some
19:04
combination in between okay this is just a budget constraint it's just saying given the amount of money you want to spend given your cost okay how many machines and workers can you have the difference is we don't start i didn't start this example by saying your parents give you x dollars that's why firm theory is harder than consumer theory i pinned down the consumer theory problem much more easily by saying your parents give you x dollars which told you which line to draft graph
19:29
i don't have that here i haven't told you that here so you have to graph the series of iso cost curves because you don't know what the optimal cost is going to be that's going to be pinned down later that's what makes supply theory harder okay you have to draw the series so you draw these series of iso cost curves different combinations that represent different amounts different totals of cost and of course the slope of that iso cost curve is delta k delta l or minus w over r
20:00
okay that's the slope or in this case minus 0.5 okay
20:08
now those you're thinking ahead i know you guys are very insightful as a class i'm sure some many of you think in your head might think gee that slope might not that slope might change as the number of workers and machines changes could you imagine the relative price of capital labor changes in different costs and it might and we'll put that aside for now for now assuming for every relevant quantity these prices five and ten dollars are fixed let's ignore them where those prices come from
20:33
they're just given now we'll come back to that later okay like i said this course is sort of like peeling an onion we raise things then we come back and go to the next layer of where'd that come from right now we're we'll tell you where w and r come from right now we're going to take them fixed and we'll assume they're always five and ten dollars regardless the amount produced okay now here's the question you're a firm that wants to produce a certain amount of units
20:58
you have a production function and a cost function how do you graphically figure out the right combination of capital labor to use to produce a certain amount of units yeah the the the tangent between the isoquant and the isocosta that's exactly right just as i asked you what is the right combination of pizza and cookies and you told me that it was the tangency of the indifference curve and the budget constraint it's the exact same logic here the optimal mix of capital and labor
21:36
comes from the tangency of the iso quant with the iso cost as we see in figure six three and ignore the like somehow that curve is sort of connected at the top it's sort of a glitch a power point just ignore that it's not actually like a square or a trapezoid it's just a curve what would you call that the curve and the two sides that uh that's not trapezoid that's not that's a it's not a polygon okay it's just a line all right um so basically uh is there a name for that a curve in
22:05
two lines i don't think so it's just a polygon right okay so it's not a polygon it's just a curve okay so the curve is the isoquant for the square root of 12.5 what do i mean by that i mean that is the combination of capital labor that delivers square root of 12.5 units of production okay so what that curve is all possible combinations of capital and labor that deliver square to 12.5 units okay just like the indifference curve is all possible combination of pizza and
22:40
cookie that leaves you equally happy this is all possible combination of capital labor that leads with that leads you to a given production level and as we said the further out the isoquant the more you can produce so you want to produce as much as you can given the prices you face in the market well those price you face in the market are delivered by the iso cost curve so the tangency is the best is the cost minimizing point that's when you're producing the most
23:07
you can given the cost you face in the market the most you can given the cost you face in the market okay and that tangency condition once again continue our parallels to consumer theory that tangency condition is going to deliver that the marginal product um is going to deliver that the marginal product of labor over the marginal product of capital which we remember called last time the marginal rate of technical substitution okay is going to be equal to w over r actually the negative of
23:47
these is going to be equal to each other but we'll just cross out the negatives so the negative of mpl over mpk which we call the marginal rate of technical substitution is equal to the negative of w over r the slope the optimal point is where the marginal rate of technical substitution equals the slope which is the wage uh which is the wage to rental rate ratio alternatively i don't know if anyone besides me likes this intuition we can rewrite this as um mpl over w equals mpk
24:21
over r my bang for the buck formulation that i like that the next dollar of wages if you ask should i spend the next dollar on wages or machines you should do it until the next dollar of wages delivers you the same return as the next dollar of machines okay this is what you get for the next dollar of wages mpl over w this will get the next dollar machines mpk over r you want to continue to trade off machines and workers until that condition is true okay so let's actually now solve for
24:53
this okay for our for our example let's actually solve for that okay if we solve for this we know that the marginal product of labor which is dq dl okay is 0.5 times k over square root of k times l and the marginal product of capital which is dq dk equals 0.5 times l over the square root 0.5 times l of the square root of k times l okay just just taking the derivative of the all i did was take the derivative of the production function okay so therefore putting these together we're going to we
25:39
know that marginal rate of technical substitution in this example is equal to minus k over l that's not a general formula okay that's just this example the marginalized substitution is equal to the negative of the ratio of capital labor we also know that the wage rate we also know we want to set this equal to the negative of the wage rate i'm sorry the wage the wage rental rate ratio okay and we know that's negative a half then we know that's a half so because that's five and that's 10. so
26:18
we want to set the margin technical solution to the wage rate rental rate ratio which means we set minus k over hell equal to minus a half or at the optimum okay that means that um your labor your capital at the optimum that means the amount of capital should be half as much as the amount of labor in this example we just solved for the efficient combination of inputs the efficient combination is the issues kate you should use capital to labor in a ratio of a half you're just half as much
26:53
capital as used labor okay so let me pause there and let's let's talk about where this is coming from yeah does that mean that at any given place of cost per
27:13
that's the graphic intuition let's come to economics intuition the economics intuition is the following the production function delivers this relationship that the marginal rate of substitution goes minus k over l in other words when you're producing goods given this production function you're indifferent between the next machine the next worker that's just the way his production function worked out that one more machine delivers you the same amount as one more worker
27:39
okay now i've just told you that one machine costs half of one more worker so which do you want more of no one machine delivers the same return as one worker you want more workers workers cost hash machines they're equally productive so you want more workers so the optimal amount of machines is going to be half as many as the number of workers you want more workers because you're indifferent look at that production function you're indifferent okay you don't give a about l versus k
28:19
they're the same to you okay you're a hard capitalist man machine or worker you don't care but the market's telling you you can get a worker for half the price of machine so you as a good cost minimizing capitalist get take twice as many workers as machines okay and that's the outcome that you get here questions about that okay so that's basically um what we do to to derive this now what i want is take this and then derive our ultimate goal which is what is the long-run cost function
29:02
that's sort of what we why we started this lecture what is the long-run cost function let's do the math we'll do the math in five steps step one q equals square root of k times l okay step two we know from up there that k over l equals w over r we derive that right leading us to the conclusion that k leaves conclusion to k equals one half l okay we just derived that therefore we can rewrite q we can rewrite q as the square root of one half times l squared just substituting in because k equals
29:48
one half l therefore we can solve for l is going to be square root of 2 over q and k is going to be square root of 2 over 2 over q
30:15
okay l the square root of 2 over q k is square root of 2 over 2 over q i'm sorry i'm sorry no not not over q i'm sorry that's my bad error error go back should always look at my notes it's square root of 2 times q my bad l is square to two times q okay and k is square root of two over two times q
30:49
okay therefore armed with this l and k we can rewrite our cost function so step five is that the cost function given this stuff the cost function equals r times square root of 2 times q i'm sorry r times square root of 2 over 2 times q plus w times square root of 2 times q i just plugged in the optimal l and k into my cost function okay now i can plug in the 10 and the 5 to get c equals 10 times square root of 2 times q and i'm done i've just derived the cost function this
31:41
is what we came here for this is what you got up this morning wanting to see okay you got this one you said i want to know how does the cost of a firm vary with the quantity it produces and i've just told you this tells you how the the costs that you pay vary the quantity you produce and i did that by by deriving the optimal mix of l and k you want to use and then simply imposing the prices of those two and i get a cost function yeah i'm sorry would you be adding the two terms which two terms
32:17
oh yeah plus i'm sorry you're right my bad that's a plus thanks this is the most math i'll do in a lecture all year you can be pleased to know that's why it's my least favorite lecture yeah generally true uh do you mean this particular functional form no this is all dependent on that production function i wrote down what's generally true is this okay or actually k wouldn't be what's generally true is just c equals in the long run what's generally true is c equals wl plus rk okay that's what's
32:52
generally true i just make but what i've showed you is given three things a production all i gave you was a production function a wage rate and a rental rate given those three things you can then derive the cost function okay given those three things you can derive the cost function in fact given two things you can derive the cost function you could actually derive the cost function given one thing given just a production function to drive the cost function as a function of these input prices
33:25
okay so that is a lot of uh that's a lot of results from one function from one production function okay now other questions about this of the math i got wrong sorry about that yeah did you just repeat the three inputs that you're using all i use somebody somebody tell me what what do you need to get the magical cost function what three things do you need yeah w and r and then q no w-r-n what about q you need q but what about what do you need w-r-n the production function starting with
34:00
the production function that mathematical equation this mathematical equation w and r i'm done everything i've done this lecture comes from those three things the math is hard and annoying we will have you practice it you will not like it okay it's just it's just kind of what you got to do all right i don't like it you're not going to like it it's just what we got to do to get to the more interesting stuff okay yeah it's fixed in the short run the bar on the it shouldn't see a bar in a k over
34:34
here that's all over here when it's doing the short run yeah r w in order to get to the fifth step like yeah i needed r and w well to simplify k is one that's oh you're right that's a good point i needed r w back here you're right it's a good point it's a good point but i still could have done this whole thing as a function of r w if i wanted to if i wanted to really you know screw up my math all right okay so now armed with this let's talk about what happens when input prices change
35:05
we talked about with consumer theory what happens when the price of pizza and cookies change what happens when the price of labor and capital changes what does that do so let's talk about changes in input prices okay let's go to figure six four and let's look at with the same production function square to l times k we're not changing our production function we're gonna change the wage rental ratio so line the the the um we have our initial uh we have our initial line our initial um
35:44
wage rental ratio which is that basically um you have a wage rate of this um but that but but but the budget constraint essentially that's flatter is our original budget constraint the flatter budget constraint is original strength that's a budget screen with a price of capital of 10 and a wage of five and that intersects our isoquant at point x so we chose five units um uh we chose five units of labor now we have a new we chose five units of labor and two and a half machines
36:19
okay that was our original this is sort of a messed up graph but our original intersection was at point x the cost minimizing combination that got square root of 12.5 production was to have five workers and two and a half machines now let's say the price of workers rises to ten the wage rate rises to ten bucks an hour so now workers and machines cost the same what is now the optimal mix of workers and machines well graphically we know we still want to produce the square root of 12.5
36:56
so we want to stay tangent to the same isoquant okay so basically what we're saying is this is as if we said to consumers keeping utility changed changed the price what do we call that remember we call that keeping utility constant changing the price anyone remember we call that this who said that all right raise your hand next time be proud substitution effect that's the substitution effect it's the same idea here we want to know for a given level of production what
37:29
happens is the price of the inputs change and so we shift along the isoquant from point x to point y and you'll see we choose a mix where we use fewer workers and more machines and just as the substitution effect is always non-positive this shift as these as the wage rate rises the price to go to the x-axis rises you will unambiguously use no more and almost certainly less workers okay and you can see that graphically think about graphically you're looking for the tangency between this curve and
38:10
the line the slope of the line just got steeper therefore you must move to the left on the curve it's the same proof as we use the substitution effect why the substitution effect was always non-positive okay it's the same intuition here we use for y a rise in the wage rate will lead you to higher fewer workers and more machines guess what you just entered the debate on the minimum wage and you follow the minimum wage what do people say well if you raise the wage
38:39
you have to pay workers they'll be replaced by machines that's this this is the math this is the mathematical and graphical intuition behind the debate on which which we'll get into later in the semester but the basic idea of that debate is gee if you force firms to pay more to workers they're going to substitute towards machines that's exactly right in theory in practice there's a lot of complications but this gives you the theory of why people make that argument yeah
39:04
so in this example only the like only the wage for workers in this house not the machine why would the iso cost like not have the same uh y intercept ah great point because here i'm drawing a new iso i am drawing the iso cost that i would use while still producing square to 12.5 so that's just the substitution effect i'm just i i'm not drawing the full set of iso costs at the new price i'm just saying to produce the same amount what's my new if i want to produce the same amount
39:39
what combination i now have to use okay yeah the total cost of production let's see it use yeah it has to be uh no uh let's see no total cost doesn't have to be the same the total cost used to be five workers at five bucks an hour that's 25. it used to be 50. now what is it it's now it's 70. the total cost has gone up exactly it's not like the budget income is fixed that's what's hard about producer theory because basically the budget constraint was sort of asking
40:14
you know keeping your budget fixed this is like asking keeping your total production fixed and so now you have to pay more to get to get that level of production okay it's good questions okay now hopefully what does this lead to this so so that gives us our change in input prices okay other questions about that now remember it's at the beginning of the lecture we're first going to solve for what is the cost minimizing combination of inputs for a given quantity
40:42
we derive that up there it's half as much going back to our old prices it's half as much capital as labor now we want to ask how does your cost change as the quantity changes and we call that the long run expansion path
41:02
the long run expansion path which is how do your costs expand as you produce more and we see that in figure six five in figure six five we show the particular case of a linear long run expansion path that's what you get in this example it's a particular case what this case says is at any given level of production the optimal mix of labor and capital is the same in other words you always want to have given essentially given the price of labor is half the price of capital you always
41:40
want to have twice as many workers as machines so if you want to produce square to 12.5 you want five workers and two and a half machines if you want to produce square to 50 you want 10 workers and five machines if you want to produce square to 112.5 you want 15 workers and seven and a half machines so the long run given this production function the long run expansion path is linear you always want the same ratio of workers to machines yeah
42:08
the reason why we don't necessarily have a strict budget per se we can yeah a loan out this is what i tried to say it's sort of i always say every year but it's hard we have to come back to it the reason producer theory is harder is because we're not given a fixed constant we are with consumers consumers we're saying look you've got an amount you've got a resource you've got a constrained maximization we haven't constrained their maximization yet there's another constraint we need
42:36
they have an extra degree of freedom relative to consumers now in fact consumers have degree of freedom too when you grow up your parents don't give you money you decide how much to make so in reality consumers you could do this will have the same degrees of freedom but we started with the easy consumer theory case where you constrict we we we took away a degree of freedom now we're adding it back which you can choose how much to produce like you being able to choose your
42:57
income as a consumer okay and that leaves the wrong long run expansion path let me go on because i want to make sure i get through this stuff okay now the long run expansion path does not have to be linear so think about look at figures 6 6 five b and six five c so six five b is a long run expansion path for a production function such that capital becomes less productive the more you produce i don't have the example the production function but you write down production functions
43:27
which have the feature that the more you produce the less productive capital becomes the less each edition you have capital helps we'll teach this unit of workers okay so you know we could think of this roughly as sort of like a fast food restaurant that kind of you know each addition that basically there's so much stuff to do where workers can efficiently share tasks and things these distal worker that the marginal product of labor essentially diminishes less quickly than
43:54
the marginal product of capital okay on the other hand in figure 65c we can have a long run expansion path where where labor becomes less productive relative to capital i think it's like heavy machinery where like basically all workers can do is run the machine so that second worker workers don't really do much but sit there and flip a switch you need the worker to flip the switch that's all they do so the second worker that you're already flipping the switch
44:21
so really adding more machines is a more productive way to expand expand output none of these is right or wrong we're just saying that this shape of this expansion path can basically vary with how much uh with different production functions okay but they're all the same idea now so basically that tells you um and the bot but here's the bottom line that we want to come to that long run expansion path is our long run cost curve so ultimately if you want to ask how do my costs vary with how much i
45:03
produce this curve tells you because what it does it says for every level of production i'll tell you the optimal combination of l and k given the price of l k that will tell you the cost and so you trace out the cost with every level of production this is your cost curve this long run expansion path tells you what the costs are for every level of production and it tells you that because you've made you're doing the efficient level of production okay that's what the long expansion pass
45:30
telling you okay this is hard i'm about to make it harder which is we're now going to talk about the relationship between short run costs and long run costs okay and the key insight is that long run costs are everywhere lower than short run costs without looking to figure because the figure doesn't help with this why does that make sense why are long run costs why if you can optimize over the long run will you always have costs that are no higher and in general lower
46:06
than optimizing the short run yeah you've already had the right capital because you have an extra degree of freedom i think this le chatelier's principle is that right for the chemist among us that basically like essentially an extra degree of freedom means the more you can optimize over the better you can do in optimizing in the short run you could strain by the size of the building in the long run you can choose so let's to see that let's go to figure six six
46:34
this is a confusing figure so bear with me as i walk you through it okay consider a firm with three possible sizes of plants they're going to build a plant so the capital here is the building and there's three possible sizes small medium and large okay the small plant has the curve srac1 what does that curve mean that means that the small plant i'm sorry small plant is srec1 the median planet is srac2 and the large planet has srac3 compare srac1 to srac3 what this is saying is for small
47:13
quantities of production srac1 lies below srac3 right for small quantities of production if you extend srac3 out you see it levels of production like q1 or even q2 srac3 is way above srac1 when you go to level of production like q3 srac1 if you extend that dash line out it's going to be much much higher than srac3 so the right an srac2 is in between so essentially for different levels of production these give the different optimal short run cost curves in the long run you get to choose
47:57
so the long run average cost curve is the lower envelope of the short run average cost curves because in the long run you say well here's my production level i know in the long run so if i know i'm going to build a lot of things i choose srac3 i choose i choose the biggest plant if my own production is going to be low i choose the smallest plant but i can optimize in the long run by choosing the right size plant for my production level this is hard and i'm almost out of time
48:25
so let me end with an example that perfectly illustrates this tesla elon musk everybody's favorite guy these days okay tesla when they came out had to decide how big a plant to build how many batteries to make batteries are the key and put it teslas okay and they expected to make to have demand for 20 000 cars by the year 2017. so they built a plant like srac1 they built a plant that was the efficient plant to produce 20 000 cars the problem is the man was for 200 000 cars
49:01
as a result there's a three year waiting list to get teslas it turned out that was not the right size to produce they lost money relative to the author they made money musk is incredibly rich but they didn't do what was most efficient given the underlying demand but now musk can reoptimize now he's saying wait a second people want way more cars well producing them at the tiny plane was exorbitantly expensive i had to run it over and over time pay workers overtime to produce 200k
49:27
cars that tiny plant just was exorbitantly expensive that's if you take that dashed line and said it way the hell up the srac1 line extended way to hell up but incredibly expensive so what is musk doing now building the largest battery plant in the world in nevada he is building a battery plant that can produce batteries for 500 000 cars so we shifted from srac1 to srac3 he's now saying in the long run i can more if i'm going to produce 200 000 cars i can do that more efficiently
49:56
with a giant battery plant and that's what he's doing so he's reoptimizing now what if musk is wrong what if it turns out teslas suck and people like i don't want them anymore someone else or you know chevy finally figures it out makes a good electric car okay then what's gonna happen is he's gonna have a mistake in the long run then the third period he'll go back to a smaller plant again but he always can do what's efficient in the long run given the underlying demand
50:24
so tesla is an example of this sort of long-run short run dichotomy anyway it's a lot of stuff for one lecture we'll come back next time talk more about costs and then we'll start getting into competition you
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