Economics 352: Intermediate Microeconomics

Transcription

1 Economics 35: Intermediate Microeconomics Notes and Sampe Questions Chapter 8: Cost Functions This chapter inestigates the reationship beteen a production function and the cost of producing gien uantities of output, assuming that a firm minimizes its costs of production. You shoud be sure to read the first to pages of the chapter. They estabish the idea of economic costs and set out the basic assumptions of the chapter. The to simpifying assumptions are: 1. Firms use abor () and capita ().. Labor and capita are hired in perfecty competitie marets, meaning that a firm can hire as much of each input as they ant to ithout driing the price up. Aso, the age paid to abor (the price of abor) is and the renta rate of capita (the price of capita) is, so that tota costs are: Tota Costs C + Economic profit is the difference beteen a firm s reenue and its cost. If output is sod at a constant price of p, then reenue is the product of price and uantity, p, and profit is: Profit π p Profit π pf(,) Cost Minimization Economists assume that firms minimize costs (this is consistent ith profit maximization). The math behind this goes bac to the Lagrangian that e used in ooing at consumer decisions. In fact, a ot of producer theory is basicay the same as consumer theory.

2 Whereas a consumer sought to maximize utiity gien a budget constraint, a producer i minimize costs gien some uantity constraint. That is, if a producer i produce a certain uantity of output, 0, they i see to do so at the oest possibe cost. The Lagrangian for this is: Minimize + subject to f(,) 0 L + + λ( 0 f(,)) Taing deriaties ith respect to, and λ and setting these eua to zero gies us: L L L λ λf λf 0 f 0 0 (,) 0 Taing the ratio of the first to of these gies us: λf λf 0 0 λf λf λf λf f f So, just as ith the consumer s decision e had, for to goods, x and y, the reationship that the price ratio euaed the ratio of the margina utiities or the margina rate of substitution: p p x y MU MU x y MRS x,y So ith the producer s cost minimizing input decision e hae the reationship that the ratio of the costs of capita and abor is eua to the ratio of their margina products, hich is aso non as the margina rate of technica substitution (MRTS): f f MP MP MRTS, We can aso get the reationship that f f λ

3 With the interpretation being that the margina cost of increasing tota output either by adding abor or by adding capita is eua to λ, and this is the margina cost in, for exampe, doars per unit. Diagrams and hat not. The diagram for a cost-minimizing production decision is just ie the diagram for a consumer s utiity maximizing decisions. The production function gies rise to isouants, or combinations of inputs that a produce the same uantity of output. These are ie a consumer s indifference cures. So, for exampe, if e had the production function, e might dra the isouant for 10 as: A graph shoing an isouant for the production function ith 10. And if the prices of capita and abor ere 1 and 40, e coud dra the set of a input combinations that oud cost $480, an isocost ine, as:

4 A graph shoing an isocost ine ith cost of 480, a price of capita of 1 and a price of abor of 40. Putting these to together e get either the bunde of inputs that aos 10 units of output to be produced at the oest possibe cost or the bunde of inputs that aos $480 to produce as much output as possibe: A graph shoing the cost minimizing point for the production function. The math underying this, from the point of ie of minimizing the cost of producing a uantity of 10, is:

5 f MP MP 6 (,) f MP f f MP f Expansion Paths Expansion paths anser the uestion, Ho does the combination of capita and abor change as the size of a firm increases? That is, as you moe to higher and higher isouants, ho does the cost-minimizing bunde of capita and abor change, hoding and constant? Expansion paths are shon in Figures 8. and 8.3 in the textboo. If an expansion path is a straight ine, the reatie mix of capita and abor remains constant as the uantity produced increases. So, the same proportions of capita and abor oud be used. If a production function is homothetic, its expansion path i be a straight ine. Both the Cobb-Dougas and CES production functions are homothetic. In some sense this means that the optima mix of abor and capita doesn t actuay depend on the uantity of output produced. The fact that a Cobb-Dougas production function is homothetic is demonstrated by euation 8.10, hich shos that the ration of capita to abor is independent of the tota uantity of the good that is being produced and is simpy a function of the ratio of to :

6 α β (8.10) This is demonstrated in euation 8.15 for a CES production function. If the expansion path curs upard, getting steeper and steeper (as in Figure 8.3) then as a firm expands, production becomes more capita intensie. As a side note, the increasingy intensie use of capita i mae the abor that this firm empoys ery productie. That is, the aerage productiity of abor i be ery high. If the expansion path curs donard, getting fatter and fatter, then as a firm expands production i become more and more abor intensie. Cost Functions Cost functions assume that a firm is cost minimizing, that is that it i produce any uantity of output at the oest possibe cost. A cost function expresses a firm s tota cost as a function of the cost of abor,, the cost of capita,, and the tota uantity of output to be produced, : C C(,, ) A cost function is increasing (or at east non-decreasing) in a of these arguments. That is, as, or increase, the cost of production i not fa and i probaby rise. Aerage cost is defined as: (,,) AC C(,,) Margina cost, or the additiona cost incurred to produce an additiona unit, is defined as: MC (,,) (,) C, Margina cost is the partia deriatie of tota cost ith respect to uantity. It is aso the sope of the tota cost function:

7 A graph shoing an upard soping tota cost cure and shoing that the sope of the tota cost cure is the margina cost. For exampe, if C 50 +, then the aerage and margina cost functions oud be: AC C 50 + C MC Some Graphica Anaysis of Particuar Cases Here are some specific cases of cost functions and hat the associated aerage and margina cost cures oo ie. To simpify matters, e assume that and are hed constant, so a that i be arying is the uantity produced,. First, the cures as they are usuay dran inoe a firm operating ith some fixed costs and increasing margina cost. Tota cost starts out positie (tota cost is eua to fixed cost at a uantity of zero) and rises at an increasing rate. The aerage cost cure is U- shaped because aerage fixed costs start out ery high and fa as output rises. Margina cost intersects aerage cost at the minimum aerage cost.

8 To graphs, the first shoing an upard soping tota cost cure and the second shoing a margina cost and an aerage cost cure, ith the margina cost cure crossing the aerage cost cure at its minimum. As an exampe, consider the cost function C() You shoud be abe to pot out the tota cost, aerage cost and margina cost functions to confirm that they oo ie the cures shon aboe. Second, this coud be dran ith no fixed costs and constant margina cost. This oud be the case, for exampe, if there ere no fixed costs and the cost of production ere constant at $5 per unit. Then both the margina cost and the aerage cost oud be constant. This is shon beo.

9 A graph shoing an upard soping, inear tota cost cure and the associated horizonta margina cost and aerage cost cures. As an exampe of this type of cost function, consider C() 5. The margina cost function is just MC()5 and the aerage cost function is AC()5. Deriing Cost Functions from Production Functions If you start out ith a production function, you can derie the reated cost function. That is, if you start ith f(,) you can derie C C(,,). The steps in this process are to first soe the production function for optima uantities of and, to get *(,,) and *(,,) and then to input these functions into the tota cost function C + to get, finay: C *(,,) + *(,,) C(,,)

10 So that costs, C, are expressed as a function of, and. The boo (Exampe 8.) shos the resuts of this for the fixed proportions production function, min(a, b), and for the CES production function. The CES soution is apparenty a ot of fun. We go through a Cobb-Dougas exampe ith rea numbers here. The boo has the genera resut, hich is a bit compex, but a particuar numerica exampes of the Cobb- Dougas production function i satisfy the genera resut presented in the boo. Imagine the Cobb-Dougas production function f(,). Incidentay, this production function is homogenous of degree 0.9 and, thus, demonstrates decreasing returns to scae. As a resut, the margina cost of production shoud be rising. This production function, being a Cobb-Dougas production function, is aso homothetic, meaning that if e soe for the ratio of optima inputs, */*, this term shoud not depend on the uantity produced. Anyho, to derie the associated cost function, e first need to soe for optima uantities of the inputs, and, as functions of, and. We can do this either through a Langrangian or through simpy euating the ratio of the margina products ith the ratio of the input prices. Let s try the atter of these.

11 * *, MP MP MP MP Pugging these functions for * and * into the definition of tota costs gies us:

12 (,) C, (,) C, (,) C, * + * (,) + C, It s not pretty, but it does gie the cost function. + You shoud proe for yoursef that the margina cost function is increasing in. That is, if you cacuate the margina cost function by taing the deriatie of this cost function ith respect to and then tae the deriatie of the margina cost function ith respect to MC,, you shoud get a positie aue. This means that margina costs are increasing. Further, to sho that this production function is homothetic, you can note that the formua for the optima ratio of capita to abor, */*, does not contain the uantity of output, Properties of Cost Functions The boo ists four properties of cost functions. 1. Homogeneous of degree one in input prices That is, if the costs of a inputs doube, then the cost of any ee of output i doube. An impication of this is that if a input prices and a output prices increase by the same percentage (a so-caed neutra infation) then the firms input and output decisions on t change. Technicay, this can be ritten as C(t,t,) t C(,,).. Non-decreasing in, and If the uantity produced increases, or if the costs of capita or abor increase, production on t become ess expensie. This shoud be pretty cear.

13 3. Cost functions are concae in input prices This is iustrated in Figure 8.6. The straight ine iustrates ho costs oud change if the firm didn t substitute capita for abor as the cost of abor rose. The cured (concae) ine shos ho costs change if the firm substitutes from abor to capita as the cost of abor rises. Put somehat simpy, if firms can substitute beteen inputs, their costs i be ess than or eua to hat they oud be if they coudn t substitute beteen inputs. An impication of this is that any a reuiring firms to produce output using a certain combination of inputs i probaby resut in those firms haing higher than necessary costs. This impies that the tota aue of the resources they use in production i be higher than necessary. 4. Aerage and margina cost functions are homogeneous of degree one in input costs. This is a fairy straightforard impication of characteristic 1. aboe. Input Substitution Don t orry about this too much. The to important things to remember are: 1. Firms can and i substitute one input for another as inputs become reatiey more or ess expensie. So, if abor costs are currenty 60% of a firm s costs and there is the potentia for abor costs to doube, this does not mean that a firm s costs i rise by 60% as they i iey substitute capita for abor in the production process.. The degree to hich a firm may substitute one input for another aries from firm to firm and from time to time. Some industries hae great potentia for the substitution of one input for another hie others, by their nature, do not. The assemby of automobies or digging of hoes offers great opportunities for substitution of inputs hie surgery or massage, for the most part, do not. Shifts in production functions, shifts in cost cures, and technica progress Various things may shift cost functions oer time. Whie changes in input costs on t change cost functions because they re incuded in the cost function, other things might. The best and most important exampe of a force that i shift a cost function is technica progress, or the process through hich peope earn to mae goods using feer and feer

14 inputs. This progress may be iustrated through changed in the production function or through changes in the cost function. One ay to mode technica change is to describe it as a function of time, t, basicay taing the assumption that technica change is a disembodied process that occurs at some exogenous rate. Whie this isn t a reasonabe rea-ord story, it might be a good and simpe anaytica mode. In terms of a production function, the factor A(t) might represent a mutipier that shos ho much more output the same uantity of input can produce at time t ersus some initia time period, perhaps caed period 0. To put this in terms of a production function, e might say that the production function at time t is gien by: t α β A(t) If A(0)1.0 and A(10)1.5, this suggests that as time moes from period 0 to period 1, technoogy improes so that the same set of inputs can produce 50% more output. To put this in terms of the cost function (hich is hat the boo does) e might state this as: (,,) C ( ) 0 C t,, A(t) Where C 0 (,,) is the cost function at time zero. If, as aboe, A(0)1.0 and A(10)1.5, then costs oud hae faen by 33% oer this time period, a statement euiaent to saying that productiity rose by 50%. To thin about the structure of the process behind A(t), you might imagine that some neutra (that is, not faoring one input or another) technica progress occurs at some rate of i% annuay. This is the process described in the atter part of Exampe 8.3 here the assumed rate of technica progress is 3% and the reated production and cost functions are: t A(t) 0.03t e Ct (,,) (,,) C0 0.03t e t e The term e 0.03t is an expression for the continuous rate of change at 3% oer time. The number e is the irrationa number from natura ogarithms that is approximatey eua to

15 .7188 and for hich the natura og is 1. That is, n e 1. So, hoding inputs constant, output coud increase by 3% annuay. Hoding output constant, costs coud fa by 3% annuay. Getting Input Demand Functions from Cost Functions Shephard s Lemma If you hae cost functions based on cost-minimizing or profit maximizing behaior that are of the form: CC(,,) Then Shephard s emma says that the uantity of an input demanded can be expressed as the partia deriatie of the cost function (based on cost minimizing behaior) ith respect to the input price. In terms of the function aboe this is: c c (,,) (,,) C C The underying assumption is that the firm is minimizing its costs. The demand functions are contingent (as indicated by the superscript c ) on the uantity of output produced. You shoud be sure you can or through this cacuation for the Cobb-Dougas cost function as gien in Exampe 8.4 Short Run and Long Run Stuff From the point of ie of one firm, the distinction beteen the short run and the ong run is that at east one input (usuay capita) is fixed in the short run, but a inputs are ariabe in the ong run. In the rea ord, the time horizon to the ong run reay depends on the nature of the business. A company that operates espresso carts can probaby get more carts up and running in ess than a month. A auto company, on the other hand, may reuire seera years to buid a ne pant and get it up and running. For purposes of our anaysis here, in the short run, capita is fixed at some ee, hich the boo cas 1, and short run costs are:

16 SC 1 + In the short run, the firm is constrained in its choice of capita and can ony ary its output by arying the amount of abor it hires. As a resut, a firm on t, in genera. Be minimizing the cost of producing the uantity of output that it is producing. This is shon in Figure 8.7 here the firm is constrained to hae capita of ee 1. In this figure, if they produce 0 or, they i produce at a cost higher than the minimum possibe cost for those uantities. Ony 1 is consistent ith cost minimization ith capita ee 1. Margina cost, in the short run, is the margina cost of additiona output that is produced through the hiring of additiona abor. The boo gies this as: Cost SC SMC (8.54) Output Another ay to thin about short run margina cost is that it is eua to the age paid to an extra unit of abor diided by the additiona output that abor proides: Cost SMC Output age MP abor Short Run and Long Run Cost Cures Here are some fairy simpe diagrams of the reationship beteen short run and ong run aerage cost cures. First, a short run aerage and margina cost cure, hoding capita constant and atering ony the abor input to change output, oud oo ie this:

17 A graph shoing that the margina cost cures intersects the short run aerage cost cure at the minimum of the short run aerage cost cure. Note that aerage cost is U-shaped and the margina cost cure intersects the aerage cost cure at minimum aerage cost. Second, you coud diagram to short run aerage cost cures, each associated ith a different uantity of capita, on the same graph: To short run aerage cost cures, based on different ees of capita, ith minima at different uantities. SRAC 1 is associated ith a oer ee of capita and has a minimum at a oer uantity. SRAC is associated ith a higher ee of capita and has a minimum at a higher uantity. No, e coud dra seera short run aerage cost cures on the same graph. Again, each oud be associated ith a different ee of capita. To point out a typo, the cures shoud be abeed SRAC and not SRAS, hich is a bit of a sip bac to teaching macroeconomics for me. 1 1 This is, in fact, an economics joe. If you thought it as funny, you may consider yoursef an economist ith a decent sense of humor.

18 A series of four short run aerage cost cures. Finay, the ong run aerage cost cure is a smooth cure that runs aong the bottom of a of these aerage cost cures, sort of ie it as a hammoc in hich a of the SRAS cures ere resting on a nice summer day. The art or is a bit imperfect, but the fooing diagram (a simpified ersion of the boo s Figure 8.9) shoud gie you sense of this. A series of four short run aerage cost cures and a U-shaped ong run aerage cost cure that runs aong the bottom of the short run cures and is tangent to each of them. The ong run aerage cost cure is the aerage cost cure that is reeant if a inputs can be atered. This i ao any particuar uantity of output to be produced at the oest aerage cost possibe for that uantity.

19 Practice Probems 1. For the production function 0., use a Lagrangian to soe for the costminimizing input bunde to produce 100 units of output if 10 and 15. What is the margina cost of production at the soution you find?. Dra an expansion path in hich a. both capita and abor are norma inputs b. abor is an inferior input c. capita is an inferior input Find the aerage and margina cost functions for each of the fooing cost functions. 3. C (,,) + a b 1 α β γ γ γ 4. C(,,) B 5. C() Do the fooing uestions from the textboo: 8.3, 8.5, 8.9a, 8.10a