Chapter 9 Profit Maximization


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1 Chater 9 Profit Maximization Economic theory normally uses the rofit maximization assumtion in studying the firm just as it uses the utility maximization assumtion for the individual consumer. This aroach is taken to satisfy the need for a simle objective for the firm. This objective seems to be the most feasible.
2 The rofitmaximizing firm chooses both inuts and oututs so as to maximize the difference between total revenue and total cost. π = R() C() The firm will adjust variables under its control until it cannot increase rofit further. Thus, the firm looks at each additional unit of inut and outut with resect to its effect on rofit.
3 R() = () R and are functions of and =f(k,l). π() () C() R() C() To Maximize π : dπ dr dc Inverse demand function. Also, C is a function of and = f(k, L). FOC dr dc or MR MC Maximizing π is different from maximizing uantity () subject to a cost constraint (C=?) or minimizing C subject to a uantity constraint ( =?). Find that maximizes π and π =f(), so one variable and no constant.
4 MR=MC is the rofit maximization rule  Marginalism (MR is the change in R resulting from a small change in outut and MC is the change in C resulting from a small change in outut.) The SOC for rofit maximization is: 2 d π 2 At the otimal uantity ( ), marginal rofit must be declining; economic rofit [π()] must be a concave function of at..
5 $ C R $ + At, the sloe of C euals the sloe of R MC = MR and is at a maximum. But MC = MR at two oints; one is at maximum and the other is at minimum. Must check the SOC at. 2 d π 2. At the other MC = MR, the second derivative of is >! 
6 Let s examine marginal revenue only. dr MR() d () d, d MR = for a erfectly elastic D curve,. MR < for a downward sloing D curve, which haens when more outut can be sold only if the rice is reduced for all units sold. d so e,
7 d If then MR =. d If (downward sloing demand curve), then MR <. MR is a function of if REMEMBER: d e, MR (1 d (from a firm s demand curve ersective); therefore, d ) MR() (1 1 e, ). d and This formula is derived by multilying the second art of MR() by / and factoring out.
8 Given MR (1 1 e, ); with a negatively sloed demand curve, e, is negative and is greater than MR. Furthermore, if e, = , MR =. In summary: If: e, = , MR =.  < e, <, MR <. If: e, < 1 (demand is elastic), MR >. e, = 1 (demand is unit elastic), MR =. e, > 1 (demand is inelastic), MR <. See examle for a linear demand curve on the next slide.
9 If R = 12, then AR = = 1 and MR = 1 2. $ AR can be derived from chords to the R curve. =AR=$1 R R =AR=$5 R R When AR is declining, MR is below it. For a linear demand curve, the sloe of MR is twice the sloe of AR in absolute value. e, < 1 (elastic) MR > e, = 1 MR d=ar= MR> MR < =5 $ $ e, > 1 (inelastic) MR < =1 MR R dr The firm s demand curve is the firm s AR curve if the firm must sell all its outut at one rice.
10 Inverse Elasticity Rule Given that MC = MR at maximum for the firm and 1 MR (1 ) then e MC MC MC MC (1, 1 e e, e 1 e,,, ) This rule only makes sense if MR because if MR <, MC < at that also, and MC < is not ossible. Therefore, a rofitmaximizing firm will only oerate in the elastic ortion of its demand curve where MR >. This statement does not aly to industry demand curves.
11 As e, becomes more negative, becomes smaller, ie., the ga between and MC ( MC) becomes smaller. When e, = , MC = = MR at oint of Max. When the demand curve is negatively sloed, MR is below the demand curve (AR curve) and is greater than MC at the uantity where MC = MR. 1 MC 1 Given MR MC MC =MC at this. 1 e,, MC Economist can look at the inverse elasticity to tell how close is to MC. As aroaches MC, demand becomes more elastic. The inverse elasticity is a measure of market ower. At, 1 MC 1 > demand is downward sloing.
12 Profit Maximization by PriceTaking Firm The firm is a rice taker in the short run. 1 = SAR 1 er unit of = SAR 1 SAC 1 = 1 SAC 1. $ SAC 1 π() SC() Economic is the area of the rectangle = ( 1 SAC 1 ). FOC: FOC: dπ dπ dr SMC dsc dsc SAC=SMR=d=SAR At >, SMC > SMR so as. At <, SMC < SMR so as. Profit u to and falls beyond. SMC at must be increasing. If: () = SC() FOC: '() = SMC() = SOC: ''() = SMC'() < because ' = for ricetaker. True only if SMC'() >.
13 Pricetaking Firm s ShortRun Suly Curve Because SMC shows how much the firm will roduce at each rice, it is the firm s shortrun suly curve. Set SMC= and solve for to get shortrun suly function. The firm will move u and down the curve so SMR = SMC, maximizing SMC SAC SAVC At rices below the firm will roduce zero outut because it cannot cover SAVC. The firm will minimize losses by shutting down comletely and only losing SAFC. If it continues to oerate, it will lose all of SAFC and art of SAVC. At rices between and 2, the firm will minimize losses (max ) by continuing to oerate to cover all SAVC and art of SAFC. It loses all SAFC if it shuts down, but only art of SAFC if it oerates. At rices above 2 the firm earns an economic rofit. Thus, the shortrun suly curve is SMC above the minimum level of SAVC curve. SMC must be ositively sloed also (SOC).
14 Profit Functions Economic rofit is defined as π f(k, L) vk wl. This is not the Profit Function. The Profit Function is π f(k, L ) vk wl, or as in the text Max Max Π(, v, w) K,L π(k, L) K,L [f(k, L) vk wl]. This is maximum rofit attainable given rices. Proerties of Profit Functions 1. Homogeneous of degree 1 Inflation does not change uantities of inuts used and outut roduced, but rofit will increase at the rate of inflation. 2. Nondecreasing in outut rice, If the firm does not change inut use and outut roduced, rofit will rise as increases. If the firm changes inut use or outut in resonse to the increase in, it must be doing so to make even more rofit. Therefore, if increases, rofit remains the same or increases; it cannot decrease for a rofitmaximizing firm.
15 3. Nonincreasing in inut rices Similar to above discussion. When rofit is maximized, a firm cannot reallocate inut use without reducing rofit. If v increases and the firm cannot reallocated resources to achieve higher rofit or it would have allocated inuts differently before. 4. Convex in outut rices Average rofits obtainable from two different outut rices will be at least as high as rofit obtained from the average of two outut rices. Π(1,v,w) Π(2,v,w) 1 2 Π[ 2 2,v,w)
16 The Enveloe Theorem allows us to calculate the firm s suly function and inut demand functions by artially differentiating the Profit Function with resect to each of the rices as follows. Π(,v,w) (, v,w); the firm's suly function. Π(,v,w) K(,v,w); the negative of the firm's derived v demand function for caital (this is not contingent demand). Π(,v,w) L(,v,w); the negative of the firm's derived w demand function for labor (this is not contingent demand).
17 2 1 ShortRun Producer Surlus SMC = shortrun suly curve (Set SMC= and solve for to get B A SAVC 1 2 Shortrun roducer surlus at the revailing market rice is shortrun suly function.) The gain in shortrun roducer surlus from an increase in rice from 1 to 2 is the area above the shortrun suly curve between 1 and 2 ; the area 2 AB 1. With the rice increase, roducers gain 21 /unit of original roduction and they gain 2 ? (? is between 2 and 1 ) on increased roduction between 1 and 2. This change in roducer surlus ends u being Welfare gain (2, v, w)  (1, v, w). (1, v, w)  (, v, where is the shutdown rice at minimum SAVC. Producer surlus is the extra return the roducer makes from market transactions at the market rice over and above what he/she would earn if nothing were roduced. Finally, shortrun roducer surlus is: Π(1, v, w)  Π(, v, w) Π(1, v, w)  (vk1) 1 1 vk1 wl1 vk1 1 1 wl1, because Π(1, v, w) 1 1 vk1 wl1 and Π(, v, w) (vk1). A firm s shortrun roducer surlus is its total revenue minus its variable cost, which is what the firm gains at the market rice ( 1 ) by roducing rather than shutting down. w),
18 Profit Maximization and Inut Use Earlier we showed: π() () C() but f(k,l) and C() vk wl So, the rofitmaximizing decision is a matter of choosing otimal amounts of the inuts K and L. Max π(k, L) f(k, L) (vk wl) (Assuming a ricetaking firm in outut and inut markets. is not a function of and v and w are not functions of K and L.) π f FOC : v, K K MP v MRP K K v MP w MRP L w L π L Two variables and no constraint. f L w Define Marginal Revenue Product (MRP) as the marginal change in R for a small change in inut use. Sloe of as K and L change is. These FOCs mean that any inut should be emloyed u to the oint where its marginal contribution to revenue euals its marginal inut cost (v or w).
19 Further, divide the second FOC by the first FOC to get: f L w MP w w The sloe of isouant L or or RTSLK euals the sloe of f v MPK v v isocost line. K We get the same solution as the two constrained otimization roblems. This is the costminimizing combination of K and L for the given (otimal) outut. It is also the outut maximizing combination of K and L for the given (otimal) cost. The SOC identify this otimal combination of K and L as giving maximum rofit rather than a minimum or sadle oint. The SOC are:,, and KKLL KL. KK LL 2 If SOC are met, MC is increasing at. Diminishing MP K (f KK < ) and MP L (f LL < ) mean that KK and LL < (because KK = f KK and LL = f LL ). But diminishing MP does not ensure increasing MC with two or more inuts. The cross effects, KL and LK, which are eual, must be small enough to be dominated by the own effects to ensure that MC is increasing.
20 The FOCs for rofit maximization can be solved for the otimal combinations of K and L (K and L ) for any inut and outut rices (,v,w). Then K and L would be exressed as functions of, v, and w (for a given roduction function) to give unconditional derived inut demand functions ( is not constant). K L K(, v, w) L(, v, w) Substitute K and L into the roduction function, = f(k,l), to get otimal outut. (K,L ) (K(, v,w), L(, v,w,)) (,v, w) This is the otimized roduction function, which is the firm s suly function. This suly function shows how much will be sulied at different outut and inut rices. Is it a shortrun suly function as shown in revious grahs?
21 Inut Demand Functions K K(, v, w) L/ w always, and L L(, v, w) K/ v always. Single Inut Case: Otimality (FOC) reuires that w = (MP L ). If is fixed and w increases, MP L must increase. Because MP L is diminishing as L increases, less L must be used to cause an increase in (MP L ) to maintain euality. Thus, if w increases, L must decline for FOC to continue to hold. Mathematically: Totally differentiate L or 1 fll or w the FOC L 1 w f LL to get dw fl L L dw w The final ineuality holds because f LL is assumed to be, i.e., MP L = f L diminishes, or remains unchanged, when L increases and vice versa.
22 Two Inut Case This situation is more comlex than the single inut case because the firm would need to adjust the amount of K as well as L in resonse to a change in w. The entire MP K function moves when L changes. L MP K. When w changes, the effect on L can be decomosed into 1) the Substitution Effect and 2) the Outut Effect. Substitution Effect K K 1 A If is held constant while w decreases, there will be a substitution of L for K in the otimum inut mix. Because the minimum cost use of K and L reuires that RTS LK = w/v, a decrease in w will cause a new otimal oint at a lower RTS (less K and more L). The substitution effect will be negative. K 2 B L 1 L 2 S 1 L An increase in w will decrease L (and increase K) and a decrease in w will increase L (and decrease K). The change in L in going from A to B is the substitution effect = L 2 L 1.
23 K K 1 K 2 K3 Outut Effect for a Normal Inut The Outut Effect is negative. That is, a reduction in w will reduce MC of outut, which will cause an increase in the rofitmaximizing. A B C L 1 L 2 L 3 S 2 1 L $ 1 2 MC 1MC2 P For a normal inut, a decrease (increase) in w reduces (increases) MC, causing to increase (decrease). Looking at the isouants above, the firm moves to 2 and increases L use from L 2 to L 3, so a decrease in w causes an increase in L use. In the above case, the outut effect is negative, L/w <. If the MC curves for all firms in the industry decrease with a decrease in w, the industry suly curve (S=MC) would shift outward. This would cause P to fall and industry outut would also be higher as shown in the grah. $ P 1 P 2 MC 1 MC 2 $ S 1 Q 1 Q 2 S 2 =MC D In the firm and industry case, L/w <. Both substitution and outut effects are negative. Summary: Change in L from a decrease in w in going from A to C = total effect = L 3 L 1 euals the change in L in going from A to B = substitution effect = L 2 L 1 lus the change in L in going from B to C = outut effect = L 3 L 2.
24 K 1 K K 2 K 3 Outut Effect for an Inferior Inut A The Outut Effect is also negative. That is, a reduction in w will increase MC of outut, which will cause an decrease in the rofitmaximizing. B C L 1 L 2 L 3 Sub utut 1 L $ MC 1 MC 1 P For an inferior inut, a decrease (increase) in w increases (reduces) MC, causing to decrease (increase). Looking at the isouants above, the firm moves from 1 to and increases L use from L 2 to L 3, so a decrease in w causes an increase in L use. In the above case, the outut effect is negative, L/w <. Summary: The change in L from a decrease in w in going from A to C = total effect = L 3 L 1 euals the change in L in going from A to B = substitution effect = L 2 L 1 lus the change in L in going from B to C = outut effect = L 3 L 2. For an inferior inut, a decrease in w causes L to increase because of the substitution effect and to increase further because of the outut effect; thus, the total effect is negative.
25 CrossPrice Effects (Effect of w on K?) In the grah, the cross effect of a decrease in w is: K K 1 K 2 K 3 A B C 2 1 1) The crosssubstitution effect caused a decline in K: K Point A to oint B. K1 to K 3 w 1 2) The outut effect caused an increase in K due to : L 1 L 2 L 3 S L Point B to oint C. K to K 2 K w 3 Thus, the total effect on K of a decrease in w could be either ositive or negative deending on shae of isouants and the amount increases when w decreases ). K ( w
26 Mathematical Derivation of Substitution and Outut Effects. Begin with the FOC s for choosing K and L to maximize : v = (MP K ) and w = (MP L ). Solving these simultaneously shows that the rofitmaximizing amounts of K and L are functions of, v, and w for a given roduction function. Thus, K = K (,w,v) and L = L (,w,v). These are derived demand functions for the inuts. and either w or v are shifters of the K = f(v,w) or L = f(w,v) demand curves. We will use L as an examle and show how changes in w affect L. Remember that, at the rofitmaximizing choice of L, derived demand for L euals contingent demand for L: L(,v,w) = L c (v,w,). c c L(, v, w) L (v, w, ) L (v, w, ) Differentiate both sides to get:. w w w This says that the total effect of a change in w on demand for L has two arts: 1)the change in contingent labor demand holding constant (substitution effect), and 2) the change in contingent labor demand from a change in the level of outut (outut effect). The first term on the righthand side is negative because of strictly convex isouants.
27 The second term on the righthand side (the outut effect) is: c c L L MC ; where (MC) at P MC. w MC w MC For a normal inut, L c L MC For an inferior inut, and w so the outut effect is always negative. c MC and, w, $ 1 2 MC 1MC2 In any case, the outut effect is negative. This result along with the negative substitution effect combine to give a negative total effect on inut demand resulting from an inut rice change. L(, v,w) w always. The Giffen aradox cannot occur in the demand for inuts. P
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