14.1 The Cost Minimization Prolem We ask, which is the cheapest wa to produce a given level of output for a rm that takes factor prices as given and h

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1 14 Cost Minimization Optional Reading: Varian, Chapters 20, & In principle, everthing we want to know aout competitive rms can e derived from prot maximization prolem. One can derive: Factor demands Firm suppl Adding over all rms in the econom we get market factor demands and suppl of goods and comining with the demand for goods and suppl of factors, which come from either exogenous resource constraints (land) or adding over demands and supplies from standard consumer prolems (or if a \factor" is an intermediate good from other prot maximization prolems) we get a full lown equilirium model. However, as we saw in the example, the prot maximization prolem is rather complex and sometimes ill-dened and it turns out that it is useful to separate the prolem to maximize prots into two steps: Step 1 For a given level of output ; nd the cheapest wa to produce an given level of output. Step 2 Select the est level of output. The advantages of this approach are that 1. Tractailit: It is easier to think aout the relevant trade-os when we reak up prolem in parts. 2. The rst step, the cost minimization prolem, is the same regardless of whether the market (for the output good) is competitive, the rm is a monopolist or if there is some intermediate situation with \imperfect" competition. This facilitates comparisons of dierent market forms. 137

2 14.1 The Cost Minimization Prolem We ask, which is the cheapest wa to produce a given level of output for a rm that takes factor prices as given and has access to some technolog summarized the production function f (x 1 ;x 2 ) : That is s.t. f(x 1 ;x 2 ) = : min x 1 ;x 2 w 1 x 1 + w 2 x 2 Now, ou ma think that this is something dierent from what we have done efore due to the \min" operator replacing the \max". However: If (x 1 ;x 2) is a solution to the minimization prolem that means that w 1 x 1 + w 2 x 2 w 1 x 1 + w 2 x 2 for all (x 1 ;x 2 ) that satises f(x 1 ;x 2 )=: But w 1 x 1 + w 2 x 2 w 1 x 1 + w 2 x 2,,(w 1 x 1 + w 2 x 2),(w 1 x 1 + w 2 x 2 ) ), (w 1 x 1 + w 2 x 2), (w 1 x 1 + w 2 x 2 ) for all (x 1 ;x 2 ) that satises f(x 1 ;x 2 ) = : Which means that (x 1 ;x 2) solves max x 1 ;x 2,w 1 x 1, w 2 x 2 s.t. f(x 1 ;x 2 )=: Now, in principle the constraint can e solved for x 2 as a function of x 1 and ou can then plug this into the ojective function and dervve a solution taking rst order conditions in the exact same wa as when we solved utilit maximization prolems. Alternativel, Lagrangian methods can e applied (if ou know how ou are welcome to use Lagrangians when solving constrained optimization prolems. For this particular prolem it is actuall more convenient due to the potentiall non-linear constraint). Since I want tokeep the math at a ver asic level I will make sure that all prolems ou will see can e solved without Lagrangians (that is, f (x 1 ;x 2 ) will have simple enough form so that ou can solve it out). We will return later to the (less important) prolem how to calculate solutions. Here, the more important thing is to see conceptuall how cost minimization relates to prot 138

3 maximization. Write x 1 (w 1 ;w 2 ;) x 2 (w 1 ;w 2 ;) For the solution to the cost minimization prolem (which ou can derive in exactl the same wa as the solution to the utilit maximization prolem in consumer theor). In analogue with the utilit maximization prolem the solution will depend on the parameters (the exogenous variales) of the prolem. That is, exactl as price and income changes will change the est consumer undle in consumer theor, factor price changes and how much ou are supposed to produce will change the factor inputs that produces the target output in the cheapest possile wa. Now, we can dene C (w 1 ;w 2 ;) = w 1 x 1 (w 1 ;w 2 ;)+w 2 x 2 (w 1 ;w 2 ;)= = min x 1 ;x 2 w 1 x 1 + w 2 x 2 s.t f(x 1 ;x 2 )= This function C (w 1 ;w 2 ;) is called the minimal cost function or simpl the cost function. Now the prolem to maximize prots is suj to f (x 1 ;x 2 ) max x 1 ;x 2 ; p, w 1x 1, w 2 x 2 Let e a prot maximizing output level (together with some factor inputs of course). We note that the optimal factor inputs must solve suj to f (x 1 ;x 2 ) ; max x 1 ;x 2 p, w 1 x 1, w 2 x 2 where p is just a constant. This prolem is equivalent (see discussion on max versus min aove) with the cost minimization prolem. The important consequence of this is that this means that if we have a given cost function we can look for the prot maximizing output level solving the prolem max p, C (w 1 ;w 2 ;): 139

4 This is a simple enough univariate calculus prolem and sometimes referred to as the rm suppl prolem ( i.e., in Varian chapter 22) 14.2 Solving the Cost Minimization Prolem We will again proceed oth graphicall and actuall solving the prolem using calculus. Here these approaches complements each other to a larger extent than previousl. The picture gives a clear intuition, ut the relationship with prot maximization is not seen as easil. The calculus approach is less intuitive, ut here the wa cost minimization is used as an intermediate step to solve the prot maximization prolem is easier to see Graphical Treatment x 2 6 C w2 3 Higher costs slope, w1 w2 C w1 - x 1 Figure 1: Lines with Constant Costs The idea is to proceed as we did when \solving" the prot maximization prolem with a picture in Section??. Consider all cominations of factor inputs that corresponds to some given cost level C; that is C = w 1 x 1 + w 2 x 2, x 2 = C w 2, w 1 w 2 x 1 ; 140

5 which denes a famil of straight lines called \isocosts" in Varian. The cost minimization prolem is to produce a given output at the lowest possile cost, so in terms of a graph it is then clear that if we put in the level curve of f (x 1 ;x 2 ) that shows the cominations of inputs that gives this level of output (i.e., the isoquant corresponding to ) the solution to the cost minimization prolem occurs at the line that touches the given isoquant which is closest to the origin of the graph. Once again it is then apparent that the solution must occur where there is a tangenc etween the isoquant and the isocost since otherwise it is possile to move towards lower cost levels and still produce the same output. x 2 6 C w2 x 2 3 Higher costs s = f(x 1 ;x 2 ) - x 1 C w1 x 1 Figure 2: The Cost Minimization Prolem In our discussion of technolog we showed that the slope of the level curve is given so the optimum condition is This makes perfect 1 ;x 2 1 ;x 2 TRS(x ) 1 ;x 2 ) 1 ;x 1 1 2) 2 The technical rate of sustitution tells ou how much extra factor 2 needed if output is to e kept constant and factor 1 reduced a small unit. w 1 w 2 \The rate at which rms can sustitute factors" 141

6 The relative factor price w 1 w 2 market". gives the \rate at which factors can e exchanged in the 14.3 Some Examples Fixed Proportions C (w 1 ;w 2 ;) = min x 1 ;x 2 w 1 x 1 + w 2 x 2 s.t = min fx 1 ;x 2 g ) x 1 (w 1 ;w 2 ;) = x 2 (w 1 ;w 2 ;) = ) C (w 1 ;w 2 ;)=(w 1 + w 2 ) Co Douglas C (w 1 ;w 2 ;) = min x 1 ;x 2 w 1 x 1 + w 2 x 2 s.t = x a 1 x 2 Solving the constraint we get Plugging into the ojective we get x 2 = 1 x, a 1 Or, ou ma write this as C (w 1 ;w 2 ;) = min x 1 max x 1 w 1 x 1 + w 2 1 x, a 1,w 1 x 1, w 2 1 x, a 1 142

7 The rst order condition is,w 1, w 2 1, a x, a,1 1 =0 or or w 1 = w 2 1 a w 1 x a+ 1 = w 2 1 a a+ ) x,( 1, multipl with x a+ 1 x,( a+ ) 1 x a+ 1 = w 2 1 a x 1 (w 1 ;w 2 ;) = x a+ 1 = w 2 a w 1 1, w2 a a+ 1 a+ w 1 Smmetricall we get (either oserving the smmetr or plugging ack in constraint) that x 2 (w 1 ;w 2 ;) =! a a+ w 1 1 a+ w 2 a and plugging this into the ojective we get the cost function! a aw1 a+ C (w 1 ;w 2 ;)=w 1 1 a+ w 2 a+ + w 2 1 a+ ; w 2 aw 1 which looks reall ugl. However, competitive analsis assumes that prices and factor prices are exogenous for the rm. Hence, from the perspective of analzing the suppl prolem for the rm the reall important propert of the cost function is to sa how costs change with output. This exercise is for xed factor prices and we note that we then ma write the resulting cost function for a Co Douglas technolog as C () =K 1 a+ ; where We then see that is: aw1 K = w 1 w 2 a+ + w 2 w 2 aw 1 C 0 () =K 1 a + 1 a+,1! a a+ 143

8 1. Increasing in if a + < 1: That is, the marginal cost is increasing when there is decreasing returns to scale. 2. Decreasing in if a + > 1: That is, the marginal cost is decreasing with increasing returns to scale. 3. Constant in if a + = : That is the marginal cost is constant with constant returns to scale A Remark aout Notation Once again, note that: w 1 ;w 2 ; are parameters of the cost minimization prolem x 1 ;x 2 are the choice variales. The solution to the cost minimization prolem will then in general give the choice variales as functions of the parameters. We write these as x 1 (w 1 ;w 2 ;) x 2 (w 1 ;w 2 ;) and call them conditional factor demands. Now, plugging in these in the ojective we get the cost function C (w 1 ;w 2 ;)=w 1 x 1 (w 1 ;w 2 ;)+w 2 x 2 (w 1 ;w 2 ;) : One of the more confusing aspects of economics is that sometimes we write something as a function of a long list of parameters and sometimes we write the same thing as a function of a shorter list, mae just a single parameter. This practice simpl reects that for some purposes we want tokeep a unch of parameters constant and for other purposes we want to see what happens when we change these parameters. For that reason, the list of parameters that is explicitl introduced in the notation depends on the question we want to ask. 144

9 The cost function is a perfect example. If we want to stud factor sustitution we keep w 1 ;w 2 in the notation and stud how factor shares are aected when the relative factor price changes. However, often times we will not experiment with changes in factor prices and then we write x 1 () x 2 () for the conditional factor demands and C () =w 1 x 1 ()+w 2 x 2 () For the cost function. This is purel a matter of convenience and for an particular technolog there is a particular cost function and the formula for this function tpicall involves w 1 and w 2 : However, this is now implicitl incorporated in \the functional relation". 15 Average, Marginal and Total Costs We will now take the cost function derived in the section on cost minimization as given and introduce some terminolog that is useful in order to think aout rm suppl decisions. Since we are not going to chine factor prices we are laz and write C() rather than C (w 1 ;w 2 ;); ut it is conceptuall important that ou keep in mind we are still studing exactl the same creature. Now: C() is called the average cost dc() d is called the marginal cost. Often we think aout cost functions that have a xed cost component (costs for setting up a plant or R&D etc.) and write C () =C v ()+F; where C v () isthevariale cost function and F is the xed cost. We than call 145

10 Cv() the average variale cost F the average xed cost There are lots of relations etween these curves and ou can read aout this in Varian. However, a few facts are important for understanding of graphs: Fact 1: The Area elow the Marginal Cost Curve=Total Variale Costs 6 MC() Area= C v ( 0 ) 0 - Figure 3: The Area Below MC-Curve=Total Variale Costs Idea: MC-cost of last unit Total variale costs-sum of all MCs from 1st to last unit Mathematicall this is essentiall just saing that integration is the opposite of dierentiation, so for those of ou who knows what an integral is, this should e ovious: Z 0 0 C () = C v ()+F ) dc () = dc v () d d Z dc () 0 dc v () d = d =[C v ()] 0 0 d 0 d = C v ( 0 ) If ou don't know what an integral is, ignore this and think aout this in terms of small discrete units. 146

11 Fact 2: MC and AVC curve starts at same place 6 MC() AVC() - Figure 4: MC and AVC Curves Starts at same Place Idea: Average variale cost of rst small unit and marginal cost for the rst small unit is the same thing. I.e., Marginal cost at zero is dc (0) d = dc v (0) d = lim!0 C v (), C v (0) = lim!0 C v () and Cv() is just the average cost Fact 3: AVC decreasing whenever MC curve is elow AVC curve and AVC increasing whenever MC curve is aove AVC curve. Idea: The wa to decrease an average is to add numers that are elow the average Math: d C v () d = = dc v(), C d v () = 2 dc v() d, Cv() MC (), AV C () = 147

12 6 MC()AVC() - Figure 5: MC Decreasing i AVC>MC 16 Suppl of a Competitive Firm Optional Reading: Varian, Chapter 22. A competitive rm take the price of the product(s) it sells as given, so the prot maximizing level of output is the solution to max p, C() Oserve here the analtical advantage of using the cost function rather than to write down the \complete" prolem max x 1 ;x 2 pf(x 1 ;x 2 ), w 1 x 1, w 2 x 2 : The cost function C() includes all relevant information aout the production function f (x 1 ;x 2 ) and factor prices and as ou will see this means that we can graphicall depict the rm suppl function in pictures that should e familiar from Econ 101. Now, the rst order condition to the \rm suppl" prolem is p, C 0 () =0, p = C 0 (): 148

13 Which simpl sas that rms should produce output up to the point where the marginal cost of production is equal to the price. This should e highl intuitive since if the last produced unit would cost more than the price to produce, then the rm would e etter o reducing output. If on the other hand, the last produced unit would cost less to produce than the price the rm gets on the market, then prots would increase if output is increased How to Handle Multiple Solutions to p = MC It is important to understand what the condition p = MC is. It is a necessar condition for an interior solution. Hence there are two things to worr aout. First of all it ma ethat there are several output levels consistent with the condition. Second of all, the solution ma e a oundar solution (that is producing nothing). 6 MC() AVC() p Figure 6: Solution can't e where MC curve slope downwards In man circumstances it is reasonale to think that the marginal cost curve is U-shaped and this case is also the favorite case for undergraduate economics textooks. Then, it ma ver well e that there are two levels of output that satises p = C 0 () asin Figure 6. Here the asic insight is that whatever the solution to the prot maximization prolem is, it must e at the upwards sloping part of the marginal cost curve. Hence, in the case depicted in Figure 6 we see that 0 and 00 oth are output levels such 149

14 that price equals marginal costs. However, 00 generates a higher prot than 0 ; so if either 0 or 00 indeed is the solution, then it must e 00 : That 00 generates a higher prot can e understood from the picture directl. We estalished aove that the total variale costs equals the area elow the marginal cost curve. Hence the dierence in variale costs etween 00 and 0 is the area elow the MC curve in etween 0 and 00 : The dierence in revenues is the rectangle with ase given the line etween 0 and 00 and height p: Hence, the extra prot from increasing production from 0 to 00 is the area in etween the horizontal line at p 0 and the marginal cost curve, meaning that producing at 00 gives a higher prot than producing at 0 : Another wa to understand the same thing is that the prots can e written as () = p, C() =p, C v (), F = (p, AV C()), F Hence, conditional on p<avc( 00 ) (see discussion elow onthis) it follows that: If 00 > 0 and AV C( 00 ) <AVC( 0 ); then ( 00 ) >( 0 ) Now, if ou look at the picture ou should see that the marginal cost curve is elow p for all in etween 0 and 00 : Hence the marginal cost for each unit in etween 0 and 00 is lower than the marginal cost for the units in etween 0 and 0 : Hence, the average variale cost must e lower at 00 than the cost at 0 : Note from this discussion that if the marginal cost curve is alwas decreasing, then there CAN NOT e an interior solution to the suppl prolem. You should draw a graph and convince ourself that this is so and wh! 150

15 6 MC() AVC() p 0 - Figure 7: =0Solves Prolem if Price too Low 16.2 The Possiilit of a Boundar Solution In general we know that the rst order condition is necessar onl for interior solutions. However, if p < AV C() for all choices of ; then the est thing the rm can do is to produce nothing. Such a situation is depicted in Figure 7. It should e clear from the picture that () = (p, AV C()), F <,F = (0) for an >0: The conclusion is that: If the candidate solution on the upwards sloping part of the marginal cost curve occurs where the marginal cost curve is elow the average variale cost curve, then this is indeed the solution to the prot maximization prolem. However, if not, then the solution is to set =0: 151

16 16.3 The (Inverse) Suppl Curve Comining the \shutdown condition" with the fact that if the shutdown condition is satised we can depict the suppl curve of the rm as in Figure 8. Again, ou should note that conceptuall we think of suppl as a function of the price, i.e., the prolem max p, C() 0 has price as an exogenous parameter. Hence, it gives an optimal solution for each p (if prolem well-dened as we will assume). However, for the graphs it is more convenient to put p on the vertical axis since that corresponds to the natural wa to draw the cost curves. p, MC6 AVC MC() AVC() - Figure 8: The Firm Suppl Curve 16.4 Prots and Producer Surplus In the case when there is a xed cost, we do have to include that in the calculation of the prot for the rm. In terms of the graph this means that we have to use the average cost curve rather than the average variale cost curve. Note that the larger is output, the closer is the average cost curve to the average variale cost curve, which simpl reects that F is decreasing in with lim!1 F =0. 152

17 6 MC() AVC() AC() p AC( ) - Figure 9: Firm Prot Given price p In Varian and other textooks it is common to call the prot net of the xed cost F the producer surplus. The reason for this name (I think) is that diagrammaticall it is the analogue of the consumer surplus in utilit theor (don't panic-i haven't even mentioned that in the class, ut ou ma recall this name from Econ 101). Now, the distinction etween prots and producer surplus is kind of trivial and unimportant when solving prolems using calculus. Whether one maximizes p, C() =p, C v (), F or p, C v () doesn't matter at all since the prolems onl dier a constant. However, for drawing pictures it is actuall good not to have the xed cost included. The reason is that we can measure total variale costs as either the rectangle with ase and height AV C( ) or as the area etween the marginal cost curve and the horizontal axis (Fact 1 in our discussion of cost curves). Hence, we can measure the producer surplus in an of the two was depicted in Figure

18 p 6 p 6 MC() AVC() MC() AVC() p 0 p Figure 10: Two Equivalent Was to Measure Producer Surplus 16.5 Example Let the cost curve (i.e., the minimized cost function from some prolem of cost minimization) e given C() = and we have that: AC() = C() = = AV C() = C V () = = AF C() = C(0) = 1 MC() =C 0 () = +1 We can easil solve the rm suppl prolem explicitl in this example. The prolem is or, with the particular cost function max p, C() 0 max 0 p, 1 2 2,, 1 154

19 6 MC(),,,,,,, AC(),,, AVC(),,, - Figure 11: Cost Curves for C() = The rst order condition is p,, 1 = 0, Solving this we get the candidate solution p = +1 {z } MC (p) =p, 1 Since the marginal costs are alwas increasing in we don't need to worr aout multiple solutions and this is reected in the algera aove the fact that solving the price equals marginal cost condition we get a unique solution for : However, we do need to worr aout the \shutting down constraint". If (p) 0; the candidate solution is p, 1 and the corresponding prot is (p) = p(p), c((p)) = = p(p, 1), 1 2 (p, 1)2, (p, 1), 1 = (p, 1) p, 1 2 (p, 1), 1, 1 = (p, 1)2 2, 1 155

20 to e compared with (0) =,1 Under the assumption that p, 1 > 0wehave that (p) = (p,1)2 2, 1 >,1 =(0); so in this case the solution to the interior rst order condition is indeed the solution to the prolem. Now, if p<1; the rst order condition p = +1isn't satised for an 0: Intuitivel it is rather clear that in this case (p) =0is the optimal solution since the marginal cost for the rst unit is 1 and the revenue for the rst unit is p<1: To see this formall, note that if p<1 the prot satises p, 1 2 2,, 1 <, 1 2 2, 1=, 1 2 2, (0) <(0) for all >0: Hence, the suppl curve is (p) = 8 >< >: 0 for p<1 p, 1 for p 1 and the inverse suppl curve can e plotted as in Figure 12 6 MC(),,,,,,, AC(),,, AVC(),,, )) - Figure 12: Firm Suppl Given Cost Function for C() =

21 16.6 Suppl with Constant Returns to Scale An important special case is when there are constant returns to scale. We saw in two examples (xed proportions and Co-Douglas with a + = 1) that this leads to a cost function of the form C() =K and this is true in general for constant returns technologies. Now: AC() =AV C() = K = K MC() =C 0 () =K So the rm suppl is (p) = 8 >< 0 if p<k anthing if p = K >: not dened if p>k That is, the suppl curve is at the vertical axis for prices elow K and then a horizontal line. You ma look ack at what we said aout wh maximized prots must e zero in equilirium with constant returns to scale. The ottom line here is that the equilirium price can e determined from the cost side onl (which includes stu from the technolog and factor prices) Remark on a Figure in Varian You ma e confused when ou read section 22.2 and look at Figure 22.1 in Varian. The discussion is actuall sort of OK, ut it ma not e clear to ou what the issue is and what the fuzz is aout. The Figure and the section is meant to explain wh the assumption of a price taking rm makes some sense. To do this Varian loosel descries a \game" where rms actuall sets prices. We will look at this particular game towards the end of the course, ut the point here is that if rms set prices and customers are free to choose 157

22 whatever rm the want (and know what prices are posted everwhere), the'll u at the lowest price. If ou don't get it now, ignore it and go ack to the section after we've discussed \Bertrand Competition". 158