The Theory of the Firm Economic Markets

Transcription

1 The Theory of the Firm Economic Markets We ve discussed demand, from the theory of a consumer. For suppy we wi examine the firms perspective, what inputs shoud they use, what are their ong run cost functions, what form do they take? This a eads to aggregate suppy, how firms react to different prices in the market. We wi think of the firm in terms of a production function. If a firm is producing widgets, to product q widgets it s going to combine various eves of inputs. We wi keep our anaysis simpe (as we did in demand), were going to think of production as coming from two inputs. These inputs can be anything, raw materia and abor for exampe. But traditionay the way we ook at these things is to think of the two inputs as being abor and capita. But they coud be anything and we can generaize this anaysis to many inputs. Basic Concepts The Production Function The firm s production function for a particuar good (q) shows the maximum amount of the good that can be produced using aternative combinations of capita (k) and abor () q = f(k, ) Here we are describing the firm as a production function with variabes capita, k, and abor, aowing us to produce q goods. We wi be interested in understanding how firms can change the different eves of inputs and what impact that has on the units produced. We wi then think about what the firm is going to maximize. If it wants to produce so many widgets, what is the optima eve of the inputs, how much capita does it need and how much abor. Wi aso ask, if a firm wishes to maximize it s profits, how many units shoud it produce in order to do so. How much does it cost to create these widgets? Simiar to the anaysis we did ast week but wi be some differences. FNCE317 Cass 8 Page 1

2 Product Curves, cont d Margina Product To study variation in a singe input, we define margina product as the additiona output that can be produced by empoying one more unit of that input whie hoding other inputs constant For exampe, the margina product of capita is given by: q CAPITAL: MPk fk k Margina Product, if we vary one of the inputs, if we increase the amount of abor that we use, how does it change the number of units that we create? So hoding one of the inputs constant, if we change one of the other inputs how much does it change the output? We have a certain fixed eve of capita, if I increase my abor hours from 1000 to 1001 how many more units can I product. This is what we are interested in when we are taking about margina product. How much more can I produce given one input increased and the other hed constant? Exampe, if we focus on capita, the Margina Product of capita if simpy the partia derivative of the number of units produced, q, wrt the eve of capita (we denoting this as f k ). We are ooking at the rate of increase of the number of units given an increase in the eve of capita hoding abor constant. We derive the abor version of margina product by taking the partia derivative wrt abor hoding capita constant: q LABOR: MP f FNCE317 Cass 8 Page 2

3 Margina Product, cont d Diminishing Margina Productivity The margina product of an input depends on how much of that input is used In genera, we assume diminishing margina productivity Using abor as an exampe, this woud mean: MP f 2 f 0 2 We usuay assume there is diminishing margina productivity meaning as we add more and more of one input whie hoding the other input fixed we get ess bang for the buck. If I have 50 peope working in the factory and ony so much capita and I add one more person I wi increase my outputs. As I add the 52 person I m going to increase the output but it wi do so at a diminishing rate (wi not see as must as an increase as we did when we added number 51). So adding additiona inputs hoding the other input fixed is going to increase the output but the rate of change is diminishing, ess and ess benefit by adding those factors. This is what is meant by Diminishing Margina Productivity. What does this physicay mean? Its means that the second partia derivative wrt that input is negative. Diminishing Margina Productivity, cont d Because of diminishing margina productivity, 19 th century economist Thomas Mathus worried about the effect of popuation growth on abor productivity But changes in the margina productivity of abor over time aso depend on changes in other inputs such as capita We need to consider f k which is often > 0 Our mode hods the other factors constant, if the other factors can change then it may be possibe to get more productivity out of abor because the derivative wi be wrt more than one term, for instance the partia derivative wrt capita and the partia derivative wrt abor. Getting more from the abor we are empoying as we add other things incuding computer systems and management. FNCE317 Cass 8 Page 3

4 Product Curves, cont d Average Product The average product of an input, say abor, is the tota product produced divided by the quantity of abor used in its production AP q Note that AP aso depends on the amount of capita empoyed Take a particuar number of outputs, say q = 1000 widgets. And I ook at the amount of abor or capita that was necessary to produce that output, the average output is simpy the number of units divided by the inputs empoyed. We re ooking at the abor exampe, coud just as easiy be capita. We see that AP is a function of the other input, in this case capita. So dependent on the amount of capita we have empoyed, the average product can vary even with constant eves of abor. If I change the amount of capita, even with the same amount of abor I can have varying amounts of average product. f k, q f(k 0, ) Suppose we fix capita at k 0. Want to start with no abor (origin) the increase the amount of abor (combined with this fixed eve of capita) and we wi see an increase in the eve of output. Trying to q * average product map the amount of units I m producing if I increase abor hoding capita fixed, * f(k 0, ). Now we wi cacuate AP at a particuar point on the curve by drawing a ine through that point and cacuating it s sope. At this point we are producing q units with hours of abor, so the average is just q/ (since the ine is through the origin, no offset). FNCE317 Cass 8 Page 4

5 Output Easticity Output Easticity: if I m interested in knowing if I can generate a sma change in a quantity that I produce divided by the starting eve of production, say I m at 1000 units and I make a change, this is going to give a percentage change in the number of units that I produce, and I want to compare that to a change in one of the input factors that generates this change in units. Late the second change be abor, if I change the number of abor hours by a sma number divided by the starting eve of abor I wi get the percentage change in units produced divided by the percentage change of abor empoyed HOLDING CAPITAL K FIXED. q q = q F q where F q q hoding k fixed For exampe, if I increase my abor hours by 1%, how many percentage point increase do I get in the number of units that I produce? We coud aso derive this cacuation for capita hoding abor constant. We take the deta q and ratios to their imit which gives us the starting abor over the starting quantity, /q, mutipied by the partia derivative of the number of units produced wrt abor. This is the output easticity of abor. (we can do the same for capita). We know from the previous sides that F q MP = AP This is the ratio of Margina Product Labor to Average Product Labor. Output Easticity is the Margina Product divided by the Average Product. When the two are equa a 1% increase in abor wi ead approximatey to a 1% increase in output. q MP = F AP = Notice it is possibe for MP < 0 for some eves of. This woud mean we are generating ess output for the one unit of abor added. But this is possibe, for instance a factory running at 100% production fuy staffed with workers. If I add more workers to that factory but don t actuay change the amount of capita assets they operate they actuay end up just getting in each others way. The firm becomes ess efficient. In this way we can get diminishing returns by adding more abor. FNCE317 Cass 8 Page 5

6 Basic Concepts, cont d Isoquants, this is a form of indifference curve. (a curve that represents a the possibe efficient combinations of inputs that are needed to produce a quantity of a particuar output or combination of outputs) To iustrate the possibe substitution of one input for another, we use an isoquant map An isoquant shows those combinations of k and that can produce a given eve of output (q 0 ) We are ooking at different eves of the two inputs and seeing when they give us a fixed eve of outputs. They are usuay assumed to ook something ike this. Usuay assumed to be concave to the origin (not necessary, can think of cases where adding more gives you ess utiity). In the same way as utiity was increasing for the consumer, the same is usuay true for the isoquants. The more abor or capita we have (the more we move in a northeast direction away from the origin), the higher our eve of production. Anywhere on a particuar ine we have a fixed eve of production, q, and the axis are the different combinations of abor and capita that can give us these production outputs. The more abor and capita we have the higher the overa output. Where we had RCS in the suppy case we have Rate of Technica Substitution (RTS) in this case. The idea here is in order to remain at the same eve of output, as we move down the curve how much abor has to be repaced by capita to keep the output constant. If I reduce my abor how much more capita do I need in order to produce the same amount? This is caed the RTS. It is derived as the negative sope of the Isoquant ine FNCE317 Cass 8 Page 6

7 at any particuar point. So this means it wi varies depending on the amounts of capita and abor and their sope on the isoquant at any particuar point. Rate of Technica Substitution (RTS) The sope of an isoquant shows the rate at which can be substituted for k RTS = - Sope It is usuay assumed that RTS > 0. Assume that the tangent is downward soping, and because we assume that the isoquants are concave to the origin, there is usuay a diminishing effect, the sope fattens out the further you go down the curve. How do we measure RTS? Same idea as before, the rate of change of k wrt. That wi give us the sope of that ine. The rate of technica substitution (RTS) shows the rate at which abor can be substituted for capita whie hoding output constant aong an isoquant RTS for k dk d q q 0 Where k is a function of, the sope of that ine is given by the derivative of k wrt fixing at a certain eve of output. On the next page we take this a step further (simiar to the consumer anaysis we saw ast cass) FNCE317 Cass 8 Page 7

8 Rate of Technica Substitution, cont d RTS and Margina Productivities Take the tota differentia of the production function: f f dq d dk MP d MP dk Here we have the rate of change of output wrt (considering that q=f(k, )). We aso have the rate of change of output wrt k. And each is mutipied by a sma change in their respective vaue (d and dk). We assume that we are keeping q constant, hoding on the same isoquant. So we wi set dq equa to zero. Now we can rearrange the terms k MP d MPk dk 0 MP d MP dk k And we arrive at Aong an isoquant dq = 0, so k RTS for k dk d qq 0 MP MP k Basic Concepts, Different Forms of the Production Function Shape of the Production Function The Linear Production Function Fixed Proportions Returns to Scae, if I increase both of my inputs by proportion does my output change bu the same proportion? FNCE317 Cass 8 Page 8

9 The Linear Production Function Suppose that the production function is q = f(k,) = ak + b This production function exhibits constant returns to scae f(tk,t) = atk + bt = t(ak + b) = tf(k,) A isoquants are straight ines RTS is constant The Linear Production Function, cont d Notice that this production function exhibits constant returns to scae. If I increase the amount of capita I have, say I doube it, if I increase the amount of abor I have, say I doube that as we, [increasing by a constant amount] if the Production Function is LINEAR then it wi increase the production by the same amount. For exampe, if I increase abor and capita each by 10% then overa production wi increase by 10%. Constant Returns to Scae. Ony for this type of function and ony because it is inear. In genera this wi not be true. Most industries wi see increased output but as the business becomes arger the increases wi fa off. With these types of functions the Isoquants are aso straight ines. FNCE317 Cass 8 Page 9

10 Capita and Labor are perfect substitutes. I can get the same eve of production if I reduce my abor by adding capita in a fixed proportion. What is that fixed proportion? It is given by the ratio of the constants b/a (the sope of the ine). I can produce a certain eve of output by using a abor or a capita or combinations of the two. Rate Technica Substitution is constant. It is the same where ever we are on the particuar isoquant. Fixed Proportions, the other extreme. Suppose that the production function is q = min (ak,b) a,b > 0 Capita and abor must aways be used in a fixed ratio the firm wi aways operate aong a ray where k/ is constant I can use capita or abor or ant mixture of the two. But the amount of output I get Is imited to the minimum of these two vaues, q=min(ak, b). So it s kind of an idea that in order to produce something I need a certain amount of capita to go with my abor. If I add capita over and above that I get no benefit. Fixed Proportions If I have a fixed amount of abor then I need a certain amount of capita to produce a certain number of units. If I increase my capita above that eve I get no further benefit, it s just money down the drain. If I have a certain eve of capita I need a certain eve of abor to produce a given number of units, increasing abor gives no additiona benefit. FNCE317 Cass 8 Page 10

11 The ast page is teing us that we need the mix of capita and abor right at the sweet spot, no benefit from going over. Fixed Proportions and Linear Production are two extremes, most firms are somewhere in-between. The ratio of capita to abor is fixed by the ration b/a. Returns to Scae How does output respond to increases in a inputs together? suppose that a inputs are doubed, woud output doube? Returns to scae have been of interest to economists since the days of Adam Smith We ve aready seen this from the inear production function, where we had constant returns to scae. In that case if we doube the inputs we woud doube the outputs. In genera this wi not be the case. If I add 10% to my inputs I m not necessariy going to see a 10% increase in my outputs. I may see more or ess return. What wi ead to increased or decreased returns to scae? Adam Smith focused on two things? Adam Smith focused on two things: Division of Labor Speciaization Adam Smith studies peope who make pin. He noticed that when they manufactured a sma order each aborer had to manufacture each part of the pin (the entire pin). As production got higher abor coud speciaize, divide the job amongst themseves. As the size of the production increased it aowed speciaization and division of the abor. They became more efficient as a resut. Therefore as the size of the firm increased the firm became more efficient. Adding additiona abor actuay gave more output than woud be expected by returns of scae. On the fip side however the arger you become the more difficut it becomes to manage the operation. Therefore adding additiona abor after a certain point the firm became ess efficient. It is possibe to see ower returns than returns to scae. So it is possibe for a singe business to initiay see greater output and greater returns to scae and then ater on as it becomes too arge see ower than returns to scae. This was Adam Smith s discovery. This theory hods up under examination but it depends on the type of industry you are in. If your in an industry that is very speciaized then the economies of scae ast a ot onger. A good exampe is beer manufacturers and wine manufacturers. It is much easier for beer to be produced in buk then it is for wine to be produced in buk. It has to do with economies of scae and diseconomies of scae. FNCE317 Cass 8 Page 11

12 Suppose I have the production function, it being a function of abor and capita. Then if I increase my inputs by the same amount, say 10%, if I get the same proportiona output in my output, then I describe that as CONSTANT RETURNS TO SCALE. The inear production function exhibits constant returns to scae. If I see ess output for a given percentage increase in the input, I increase a my inputs by 10% and I get ess than 10% increase in my outputs, we wou ca this DECREASING RETURNS TO SCALE. The opposite is the case where I doube my inputs and I get MORE than doube my output, this is caed INCREASING RETURNS TO SCLAE. EXAMPLES: Knob-Doper q A k 1 ( ) where 0 < < 1 increase k to tk and to t where t> q(tk, t)=a(t k )( t ) A( t t )( k )( ) 1 =tak tq( k, ). Constant return to scae. This is an exampe of a function which is not inear but aso gives constant return to scae. CES Production Function r r 1/ r q( k, ) A( k ) r r r r 1/ r q(tk, t)=a(t k t ) tq( k, ) This one is aso constant return athough it in no way ooks iner. Production Function when q(k, )= k k q(tk, t)=t 2 k t t k if t>1, t q( k, ) q( tk, t) This ast exampe is a cassic exampe of a non-constant return to scae. Is this one diminishing or increasing? This woud be an increasing return to scae if t>1. If t<1 this woud be diminishing. To consider a rea word exampe, I open up my own store, initiay I make a profit from the one store. Say I work in Hartford, I may be abe to open 4 stores and my profits grow very quicky because I can manage those 4 stores very easiy so I get increased return. Now say I open another group of stores in Ohio, I oss the abiity to manage these things hands on, I have to hire others, another ayer of management. May see the returns diminish, I don t get so much additiona profit by adding additiona stores. In manufacturing when I am sma I can speciaize, I can pick the jobs I want to run. As I become arger I have to have set mechanisms, focus on seing a certain type of output. I can t be as fexibe as I was. Therefore, I may initiay see my output increase rapidy but ater see my output decine. FNCE317 Cass 8 Page 12

13 When we speak of diminishing we are not saying the output decines, it just stops increasing at the same rate. Returns to Scae Smith identified two forces that come into operation as inputs are doubed 1. greater division of abor and speciaization of function 2. oss in efficiency because management may become more difficut given the arger scae of the firm If the production function is given by q = f(k,) and a inputs are mutipied by the same positive constant (t >1), then Effect on Output f(tk,t) = tf(k,) f(tk,t) < tf(k,) f(tk,t) > tf(k,) Returns to Scae Constant Decreasing Increasing same proportiona output ess output for given increase doube inputs give more than doube outputs Returns to Scae It is possibe for a production function to exhibit constant returns to scae for some eves of input usage and increasing or decreasing returns for other eves Economists refer to the degree of returns to scae with the impicit notion that ony a fairy narrow range of variation in input usage and the reated eve of output is being considered FNCE317 Cass 8 Page 13

14 Basic Concepts, cont d The easticity of substitution () measures the proportionate change in k/ reative to the proportionate change in the RTS aong an isoquant % ( k / ) d( k / ) RTS n( k / ) % RTS drts k / n RTS The vaue of wi aways be positive because k/ and RTS move in the same direction What we are ooking at here is if we change the ratio of two inputs then how does our ate of Technica Substitution change? If I change the ratio how does it change the substitution? This is going to give us a measure of the curvature of the isoquant. The vaue of sigma is aways going to be positive. The Rate of Technica Substitution and the ratio pair are aways going to move in the same direction as ong as we assume that the isoquants are concave wrt the origin. If they are not than a bets are off. Anaysis: % ( k / ) d( k / ) RTS n( k / ) % RTS drts k / n RTS We start off with a ratio of k/, we decrease it, how does our rate of technica substitution change? This point is greater, the rate of technica substitution is greater, as we decine isoquant smoothes out. That s what easticity of substitution measures. It measures the curvature of that isoquant ine. So is sigma is high RTS is not going to change much reative to a change in the ratio of the inputs. This means the isoquant is reativey fat. We woud be in the high eves of abor area. If your in the high eves of capita then the isoquant is going to be sharpy curved and the sigma is going to be ower in those points. We are saying that it is certainy possibe for sigma to change on an isoquant, in fact this wi amost aways be the case. (basicay saying that k/ is a function of RTS, can be soved that way then the derivative taken) FNCE317 Cass 8 Page 14

15 Easticity of Substitution, cont d Easticity of Substitution, cont d If is high, the RTS wi not change much reative to k/ the isoquant wi be reativey fat If is ow, the RTS wi change by a substantia amount as k/ changes the isoquant wi be sharpy curved It is possibe for to change aong an isoquant or as the scae of production changes Optimizing Behavior Constrained Cost Minimization FNCE317 Cass 8 Page 15

16 The economic cost of any input is the payment required to keep that input in its present empoyment The remuneration the input woud receive in its best aternative empoyment Tota costs for the firm are given by tota costs = C = w + vk Different behaviors the firm wi use to optimize, minimize cost for exampe. Many different ways. This exampe is a constrained cost minimization. If I want to produce a certain eve of output, say 1000 widgets, what is the optima mix of capita and abor I shoud put forth? What is the owest cost to produce a fixed number of widgets. But this is not the ony question we can answer. We can answer the question Constrained Cost Minimization Cost-Minimizing Input Choices To minimize the cost of producing a given eve of output, a firm shoud choose a point on the isoquant at which the RTS is equa to the ratio w/v It shoud equate the rate at which k can be traded for in the productive process to the rate at which they can be traded in the marketpace Constrained output maximization. Suppose we have a cost budget, say $10,000 to spend any way we ike on a mixture of capita and abor. Given that fixed eve of cost, the question becomes, given that fixed budget how do I maximize output? We wi first ook at constrained cost minimization. If I want to produce a certain eve of widgets what is the owest cost required. Then we wi ook at a maximization probem that says how many widgets do I need to produce and how do I need to manufacture them in order to maximize my profits. So we see there are 3 or 4 maximization probems we can ook at in this area. Tota Cost of a Firm: tota costs = C = w + vk w: abor rate, cost per something such as empoyee v: cost of capita k: capita, cost I need to put into the business. L: abor Tota cost is the number of units of each of the cost that I empoyee mutipied by the cost per unit. (w and v are ike the cost of and k). FNCE317 Cass 8 Page 16

17 So to minimize the cost of producing a given eve of output how do I get to that optima point. We wi change the mix of abor and capita so that the rate of technica substitution, RTS, is equa to the ratio of the cost of either input. Simiar to what we did in ast cass. We wi see that the optima point is the mix of v and w which is tangentia to the isoquant. Cost-Minimizing Input Choices, cont d Mathematicay, we seek to minimize tota costs given q = f(k,) = q 0 Setting up the Lagrangian: L = w + vk + [q 0 - f(k,)] First order conditions are L/ = w - (f/) = 0 L/k = v - (f/k) = 0 L/ = q 0 - f(k,) = 0 We sove this probem using Lagrangian maximization for fixed q (output to manufacture). Cost function, price of abor times abor units pus price of capita times capita units. Budget constraint in this case is the number of units I need to produce minus the production function. Take the partias isted above, set them equa to 0. Sove each of the above partias for amda, then set the equations equa (via amda) and sove for W/v (ratio of the input prices): f f Notation: f then... f k k w v w f RTS= f f v f k k If I can give abor and take on more capita, given the prices I have to pay for both to make that economicay sensibe decision, then I wi carry on doing that unti the ratio of prices in the market pace are equa to the ratio of the utiization in my production equation. FNCE317 Cass 8 Page 17

18 Cost-Minimizing Input Choices, cont d Dividing the first two conditions we get w f / v f / k RTS ( for k) The cost-minimizing firm shoud equate the RTS for the two inputs to the ratio of their prices To minimize cost set the RTS equa to the ratio of market prices for the two units (capita and abor). That wi optimize capita. We can aso think about it in these terms Cost-Minimizing Input Choices, cont d Cross-mutipying, we get fk f v w For costs to be minimized, the margina productivity per doar spent shoud be the same for a inputs f k v is the rate of change of my production by increasing capita, cost of capita. f w is the rate of change of my production by increasing abor, cost of abor. fk f, margina productivity per doar spent is going to be the same. Why? We if I v w have a choice between spending a doar on abor or a doar on capita, and there is a trade off between the two, I m going to spend it on the input that gives me the biggest return on doar spent. I m going to keep making that exchange unti the two ratios descend (?). This is teing me that the return I m getting per doar of capita is the same as the return I m getting on abor per doar of abor. If this ration equaity is out of baance then I m going to change the mixture of abor and capita to bring it back into baance. This is teing me if I increase my spending on capita this is the increase in output I get given a change in capita. This is the increase in productivity per doar of capita set FNCE317 Cass 8 Page 18

19 equa to the increase in productivity per doar of abor per doar of abor. Therefore I m going to change the mix of capita and abor aong an isoquant unti these two ratios are set equa. If one is higher than the other I m going to spend more on the higher one and ess on the ower productivity one. I m going to change my position on the isoquant unti these two ratios are set equa. Cost-Minimizing Input Choices, cont d Note that this equation s inverse is aso of interest w v f f k The Lagrangian mutipier shows how much in extra costs woud be incurred by increasing the output constraint sighty. In this case the Lagrangian Mutipier is teing us how much extra cost I wi incur by increasing the number of units. My constraint in this probem is the number of units I have to produce are fixed. In the maximization of utiity my budget is fixed. So what this is teing me is if I can increase the number of units produced by a fixed amount (say one more unit) the shadow price tes us how much the costs need to increase to produce that one extra unit. Very usefu, it avoids our having to sove two maximization probems. This wi give us an approximate idea of how much additiona costs I need to produce one additiona unit (it s ony APPROXIMATE). FNCE317 Cass 8 Page 19

20 Cost-Minimizing Input Choices, cont d We want to produce a certain number of widgets, q 0. The most efficient way of my producing the certain number of widgets is given by the eve of abor and capita where the rate of technica substitution, RTS, is equa to the ratio of prices of abor and capita. In this case the tangent of q 0 at the intersection of isoquant q 0 and c 2 (at the ). So the minimum way of producing q 0 is to manufacture at the c 2 at the mix of and k indicated by the star, this wi give optima production, minimum cost. (see next page) FNCE317 Cass 8 Page 20

21 Cost-Minimizing Input Choices, cont d FNCE317 Cass 8 Page 21

22 Input Demands Input Demand Functions Cost minimization eads to a demand for capita and abor that is contingent on the eve of output being produced The demand for an input is a derived demand It is based on the eve of the firm s output In the same way that we can construct demand functions in a utiity sense, we can construct input demand functions. Given some eve of output, I can find the minimized cost by mixing the appropriate units of capita and abor. If I want to produce so many widgets I can do so at the owest cost by finding the optima eve of capita and abor to use, where the rate of technica substitution, RTS, is equa to the market ratio of prices. The Firm s Expansion Path The firm can determine the cost-minimizing combinations of k and for every eve of output If input costs remain constant for a amounts of k and the firm may demand, we can trace the ocus of cost-minimizing choices Caed the firm s expansion path If we do this, given a different eve of output I can sove the probem over and over. So I can actuay construct, for a given eve of output, how much abor or capita I use. I can construct, for the firm, input demand functions. Given a eve of output I can te you the optima eve of capita needed to generate that output at the owest cost to me and the firm. For a given eve of output we are tracing the point which minimizes the cost. So given a eve of output, such as q 3, we can te you the eve of abor and capita needed to sove the minimization probem. (see graph beow) FNCE317 Cass 8 Page 22

23 The Firm s Expansion Path Expansion Path The expansion path does not have to be a straight ine The use of some inputs may increase faster than others as output expands Depends on the shape of the isoquants The expansion path does not have to be upward soping If the use of an input fas as output expands, that input is an inferior input Two Step Process: 1) Given output eve 2) Find minimum cost by finding the ratio of capita and abor I need to minimize my cost given the eve of output. The ine which shows me the optima mix of capita and abor in the above case, we ca that he firms Expansion Path. As the firm ramps up production, the expansion path te me the mix of capita and abor needed at any eve of production. It s the optima mix of capita and abor. How do we find it? We trace out the cost minimization choice of capita and abor. Given a eve of production I find a minimized cost which wi te me the mixture of capita and abor, that point of capita and abor is going to be the expansion path point for that isoquant / capita/abor pair. Connecting each expansion path point gives the expansion path. The points progressing outward represent the progressivey arger isoquant / capita-abor pairs, the northeast direction shows the firm is expanding aong these points, increasing it s output. And this doesn t necessariy have to be a straight ine FNCE317 Cass 8 Page 23

24 and it amost certainy won t be a straight ine. As I get more and more production, I may need essor quantities of capita reative to abor. I may need to increase the abor wrt the point (?). So there is no reason way the path of expansion has to be a straight ine. The reative quantities of the two inputs may change. What wi determine that is the shape of the isoquants. So these wi have to be very speciaized curves in order for the expansion path to be a straight ine. It depends on the production and technoogy of the firm. Tota Cost Function The tota cost function shows that for any set of input costs and for any output eve, the minimum cost incurred by the firm is C = C(v,w,q) As output (q) increases, tota costs increase We can aso tak about the tota cost function for a particuar company. What wi it ook ike? If I have an output eve, I can sove for the best mix of abor and capita I need to minimize my cost. Therefore I can construct cost function that tes me what my cost is going to be to produce any eve of output. This wi be a function of my production function (the shape of my isoquants) and aso a function of the reative prices of the input functions. So it s going to be dependent on the shape of the isoquants, and aso dependent on the sope of the costs ines (k/). So the prices of the factors, the cost of abor and the cost of capita) are going to be important. If one factor increases it s cost, then I am going to have to rebaance my inputs in order to equate the RTS to the new ratio of prices. I m going to have to resove this probem. If this ine suddeny shifts because one of the input cost goes up, then I m going to have to resove the minimization of those costs, I m going to have to reset the tangent of that ine equa to the new ratio of prices. It is a dynamic process. As output increases tota cost is certainy going to increase, assuming we don t have any negative return to any of our factors. As we focus on the cost function there are a coupe of ratios we are going to be interested in. The first one is the Average Cost Function FNCE317 Cass 8 Page 24

25 Average Cost Function The average cost function (AC) is found by computing tota costs per unit of output C( v, w, q) average cost AC( v, w, q) q We ve moved on now from ooking at abor and capita in terms of units of input. We are now ooking at cost in tota doar terms. Given the output that we need we can find the optima mix of abor and capita. If I know the prices I can te you what the tota cost is going to be to manufacture that eve output. Now when we are ooking at the cost function we are truy ooking at doars worth of cost. Average cost: if I ook at the tota costs of the optima production eve, of the optima mix of capita and abor, then I can compare tota cost to produce so many units to how many units are being produced. Can cacuate the average cost in doar terms per unit widgets produced. Average cost = tota costs / number of units Margina Cost Function The margina cost function (MC) is found by computing the change in tota costs for a change in output produced margina cost MC( v, w, q) C( v, w, q) q Cost is a function of input prices and units of production. Therefore to find the rate of change of cost wrt units of output it is simpy the partia derivative of cost wrt units produced. Taking the derivative of cost with respect to quantity hoding a other variabes constant and assuming the price is going to change. Consider what this wi ook ike graphicay FNCE317 Cass 8 Page 25

26 Graphica Anaysis of Tota Costs Suppose that k 1 units of capita and 1 units of abor input are required to produce one unit of output C(q = 1) = vk 1 + w 1 To produce m units of output (assuming constant returns to scae) C(q = m) = vmk 1 + wm 1 = m(vk 1 + w 1 ) C(q = m) = m C(q = 1) Suppose I am going to produce 1 unit of output. To do this I need k 1 units of capita and 1 units of abor, those are the optima mix that I need to produce 1 widget. Therefore the cost of producing 1 widget is going to be given to me by cost of capita times the number of units of capita pus the cost of abor times the number of units of abor. Be cear, this eve of capita and this eve of abor comes from the maximization soving probem (a few pages back). I coud use different ratios of k and to produce the same number of outputs, but I m interested here in producing to the optima ratio. Now say I need to produce m units. Assuming constant returns to scae, in order to produce m units I m going to use m units of abor and m units of capita, if I have constant returns to scae. If I do have constant returns to scae my tota costs is a straight ine. Graphica Anaysis of Tota Costs If I do have constant returns to scae my tota costs is a straight ine. As output goes up cost goes up, there is a inear reationship between the two. What s more, my average cost is equa to my margina cost. Why? We average cost is simpy the sope of this ine (since it s straight) and margina cost is aso the sope of this ine because it s the rate of change. In this case both AC and MC wi be constant. IF I HAVE CONSTANT RETURN TO SCALE AVERAGE COST IS EQUAL TO MARGINAL COST FNCE317 Cass 8 Page 26

27 Graphica Anaysis of Tota Costs, cont d Suppose instead that tota costs start out as concave and then becomes convex as output increases One possibe expanation for this is that there is a third factor of production that is fixed as capita and abor usage expands Tota costs begin rising rapidy after diminishing returns set in Initiay it takes a ot of cost to produce a sma number of units. There are a ot of fixed costs say. Costs jumps at the one or two units eve (sma q in genera). At this stage there is not much efficiency. After that I hit a sweet spot where I add additiona cost and produce additiona units. After that inefficiencies start to creep in, just become too big. Think of it in terms of a factory, if I start off producing widgets I need a certain eve of additiona factory foor, weather I m producing one or 10 widgets. Eventuay that same abor force is abe to produce 100 or even 1000 units. But if I want to produce 10,000 units I have to put an extra shift on, I start to see inefficiencies. So initiay it is very costy per unit, then sweet spot, then after that very costy again because inefficiencies start to creep into the system. So our curve has kind of an S shape. Another component of this behavior may be a third factor which we are ignoring or pushing into the background but which may be producing different effects on the cost structure. This is pausibe. Now what s this going to ook ike? Pick a point (above). Average Cost, it takes me so many doars to produce that many units, the average cost is the sope of the ine joining the origin and the production point. Margina Costs is the tangentia point to the sope of the curve at the point of production. FNCE317 Cass 8 Page 27

28 Consider the sope, starts steep, fattens out, becomes steep again. Initiay the average cost is going to be ower than the margina costs, the ine starts off steep. But eventuay the margina costs is ess than the average cost. It reverses. The margina costs is going to be greater than the average costs. WHAT WILL THIS LOOK LIKE IF I MAP A GRAPH OF AVERAGE COSTS AND MARGINAL COSTS? Graphica Anaysis of Tota Costs, cont d Margina costs starts high, diminishes, and then increases. Average costs, starts off at a certain eve, diminishes, eventuay the inefficiencies give a knee up to average cost (?) and we start to see average cost increasing. Notice if average costs are greater than margina costs then average costs must be faing. Visa-versa, see graph above. In fact, if you ook at the point of intersection of these two ines, that s where margina costs overtake average costs. When we tak about monopoies this point of intersection wi have a huge impact on monopoies behavior. As ong as markets are not competey competitive there is going to be an impact of margina price versus margina cost. That wi give us the answer. FNCE317 Cass 8 Page 28

29 Input Substitution A change in the price of an input wi cause the firm to ater its input mix We wish to see how k/ changes in response to a change in w/v, whie hoding q constant. k w v Here we are taking about prices of the inputs, the prices w and v of capita and abor. Suppose the ratio of the prices (w/v) change. On my maximization ine the sope of the straight ine, the trade off between the cost of capita and abor, changes because the price reative to capita and abor changes. What we are interested in is seeing how our optima mix of capita and abor changes because the reative price of those two goods changes. We can see in the graph if the ratio of the prices in capita and abor shift then it s going to ead us to a different mix of capita and abor. If we are trying to minimize cost for a certain eve of output the way we do this is set the ratio of technica substitution (RTS) equa to the ratio of prices. If the ratio of prices changes then it s going to ead to a different mix of capita and abor. Wi ead to a different outcome when we do the maximization probem. So what we are interested in is seeing how much our mix changes given a change in prices. So a change in the ratio of prices, how that impacts the change in the reative portions of capita and abor. This is what is meant by RATIO OF INPUT SUBSTITUTION. FNCE317 Cass 8 Page 29

30 Input Substitution Putting this in proportiona terms as s ( k / ) w/ v n( k / ) ( w/ v) k / n( w/ v) gives an aternative definition of the easticity of substitution In the two-input case, s must be nonnegative (due to idea that ratio of one thing goes up you wi start using the other input more). Large vaues of s indicate that firms change their input mix significanty if input prices change (a change in prices is going to ead to a dramatic change in the eve of inputs) w w + ( w ) v v v k k + ( k ) The two ratios undergo a sma percentage change. The ratio of our input capita to abor given our initia capita to abor. This ratio is the percentage change. So we are adding the percentage change of each respective ratio to the initia proportion. What we are interested in is this, if the ratio of abor cost to capita cost increases by a sma positive amount, that means the abor costs go up reative to capita costs, by some sma amount. That ratio changes by a very sma amount. Now what is going to happen to the mix of capita and abor? We this wi ead to a change in capita and abor. Notice how these ratios work, if the cost of abor goes up reative to the coast of capita then we wi have a sma positive increase in this ratio. This is the amount of capita compared to the amount of abor. So if this increases by a sma amount you woud expect capita to be substituted for abor. Labor is becoming more expensive reative to capita. Therefore you woud think that capita shoud increase reative to abor (to offset the need for more expensive abor). A sma increase in this ratio shoud ead to a sma increase in this ratio. So we are interested in the percentage change, so the sma change in capita over abor divided by the amount of capita over abor (the percentage change in that ratio). Divided by the percentage change in the reative prices. If that ratio changes by 1%, what percentage increase do we see in the mix of capita and abor that we actuay use? In the extreme this is what we get when we take it to the imit. FNCE317 Cass 8 Page 30

31 Partia Easticity of Substitution The partia easticity of substitution between two inputs (x i and x j ) with prices w i and w j is given by s ij ( xi / x j ) wj / wi n( xi / x j ) ( w / w ) x / x n( w / w ) j i i j j i S ij is a more fexibe concept than because it aows the firm to ater the usage of inputs other than x i and x j when input prices change Here we are extending this mode to ook at more genera setups where we have more than two inputs. In this case the s does not necessariy have to be positive. Short-Run, Long-Run Distinction In the short run, economic actors have ony imited fexibiity in their actions Assume that the capita input is hed constant at k 1 and the firm is free to vary ony its abor input The production function becomes q = f(k 1, ) We ve been taking about how businesses can change the mix of inputs given a certain eve of output that they need. But we have to think about the distinction between short run and ong run. What we mean is If we te a company to produce a certain eve of units, if I suddeny change their target number of units, they may not be abe to increase their factor inputs in an optima way. Some of their factor inputs may be reativey fixed over the short term. Exampe, is a factory is manufacturing widgets and they have a budget to manufacture 100,000 of these widgets, if I suddeny ask them to manufacture 120,000 widgets it may take them a ong time to increase pant equipment and capita, they may not be abe to do so immediatey. They may sove the probem by using more abor. They may do that even though that new mix of abor and capita is not efficient. The reason they do this is because they have a certain target to hit and because they cannot vary one of their inputs they do a second best. They move away from the isoquant that they woud ike to be on and over the short term they are wiing not to produce at the optima eve. FNCE317 Cass 8 Page 31

32 Over the ong run we can change our eves of capita and abor to the point where we need them. So notice the distinction between short run and ong run. Short run, economic actors, firms have ony imited fexibiity in their actions. They may find it easier to change one input where the other may be ess fexibe. It may be easier to add one shift of workers than it is to add new capita to the business (such as increase the shop foor). So ought to assume in the short run that one of the variabes is hed constant. We usuay assume capita is the one hed constant in short run cases and the firm is free to vary it s abor ony. It can hire new empoyees and pay them overtime. In this case the production function becomes simpy a function of abor, capita remains fixed. It becomes a singe variabe function. Does this change what the firm does? YES. Because short run tota costs to the firm are going to be equa to a fixed part and a variabe part. Over the short run the cost of my capita times the units of capita are reativey fixed. Therefore my cost function becomes a fixed variabe of abor and a fixed cost of capita. Short-Run Tota Costs Short-run tota cost for the firm is SC = vk 1 + w There are two types of short-run costs: short-run fixed costs are costs associated with fixed inputs (vk 1 ) short-run variabe costs are costs associated with variabe inputs (w) In the short run we usuay have fixed costs and variabe costs. For exampe, in accounting we think of costs being fixed and variabe. Most costs, over the ong run, are variabe. You can aways buid that extra factory. Whereas over the short run, some of the costs are constrained, they cannot be increased very easiy. And other costs are reativey variabe. Short-Run Tota Costs Short-run costs are not minima costs that we need for producing the various output eves The firm does not have the fexibiity of input choice To vary its output in the short run, the firm must use non-optima input combinations The RTS wi not be equa to the ratio of input prices If we are constrained on one of our costs we are going to make a second best attempt to produce the number of units that we need. Ideay we woud vary both inputs. If one input is fixed we woud have to be very ucky to end up at the optima eve of output. FNCE317 Cass 8 Page 32

33 It is more than ikey that we are going to be spending more costs than we need we woud need to if we were given free ong run choices to make. So we wi use non-optima input combinations. When we do so our RTS is very unikey to be equa to the ratio of input prices. Physicay what does this mean? Lets suppose our capita is fixed. We can t change the eve of capita. Given a different eve of output and the fixed eve of capita, there is ony one cost ine which wi be tangentia. And that wi be at that particuar point. If by some fuke we are asked to output at this eve, then this is optima (but this is unikey). Otherwise we are forced to use too much capita or too itte capita. Fexibiity comes into the system because we can change the amount of abor that we need. Short-Run Tota Costs Because capita is fixed at k 1 we are not abe to get to the optima point on the isoquant. We woud ike to be at the optima point but if we are forced to change our production but we cannot change our capita, then we have to make a second best choice. And we can see that in either case we are not at the best production eve. The optima points woud have ower costs given the same eve of production. In the above graph 1 uses too much capita (k) and 3 uses too itte capita. 2 is about right but that is the point we are being driven off of due to the changed production demand. [End of Lecture] He wi post a practice exam. Monday he wi answer any practice exam questions. Everything after midterm. Litte on rationa/efficient markets, and the ast three casses. The homework is a itte more invoved than the exam wi be, ess math more interpretation. FNCE317 Cass 8 Page 33

34 Optimizing Behavior Profit Maximization A profit-maximizing firm chooses both its inputs and its outputs with the soe goa of achieving maximum economic profits Seeks to maximize the difference between tota revenue and tota economic costs Profit Maximization, cont d If firms are stricty profit maximizers, they wi make decisions in a margina way Examine the margina profit obtainabe from producing one more unit of hiring one additiona aborer Output Choice Tota revenue for a firm is given by R(q) = p(q)q In the production of q, certain economic costs are incurred [C(q)] Economic profits () are the difference between tota revenue and tota costs (q) = R(q) C(q) = p(q)q C(q) Output Choice The necessary condition for choosing the eve of q that maximizes profits can be found by setting the derivative of the function with respect to q equa to zero d dr dc '( q) 0 dq dq dq dr dq dc dq Output Choice To maximize economic profits, the firm shoud choose the output for which margina revenue is equa to margina cost dr dc MR MC dq dq FNCE317 Cass 8 Page 34

35 Second-Order Conditions MR = MC is ony a necessary condition for profit maximization For sufficiency, it is aso required that 2 d d q dq 2 qq* '( ) dq qq* 0 margina profit must be decreasing at the optima eve of q Profit Maximization Margina Revenue If a firm can se a it wishes without having any effect on market price, margina revenue wi be equa to price If a firm faces a downward-soping demand curve, more output can ony be sod if the firm reduces the good s price dr d[ p( q) q] dp margina revenue MR( q) p q dq dq dq FNCE317 Cass 8 Page 35

36 Margina Revenue If a firm faces a downward-soping demand curve, margina revenue wi be a function of output If price fas as a firm increases output, margina revenue wi be ess than price Margina Revenue Suppose that the demand curve for a sub sandwich is q = p Soving for price, we get p = -q/ This means that tota revenue is R = pq = -q 2 / q Margina revenue wi be given by MR = dr/dq = -q/ Profit Maximization To determine the profit-maximizing output, we must know the firm s costs If subs can be produced at a constant average and margina cost of $4, then MR = MC -q/ = 4 q = 30 Profit Maximization and Input Demand A firm s output is determined by the amount of inputs it chooses to empoy the reationship between inputs and outputs is summarized by the production function A firm s economic profit can aso be expressed as a function of inputs (k,) = pq C(q) = pf(k,) (vk + w) FNCE317 Cass 8 Page 36