Profit - Economics

Profit - Economics

Profit Maximization Profit Maximizing Assumptions Firm: Technical unit that produces goods or services. Entrepreneur (owner and manager) Gains the firms profits and suffers losses and has the goal of maximizing profit. Transforms inputs (aka factors) into outputs through the technology of the production function. Decides how much of each input to use and what quantity to produce. Realistic? For this class, we assume corporations with shareholders and boards and executives function like the entrepreneur. Obviously, managers and executives have other incentives besides maximizing profit. But there is a more developed theory of the firm for corporations that we will not get into. Profit maximization? Libraries and water utilities are strictly non-profit. Most hospitals claim to be non-profit, but they act like for-profit hospitals.

Why Firms? You could ask why even have firms? Why dont entrepreneurs outsource EVERYTHING? Transactions costs make that infeasible. Or not. In the 1870s-1950s vertical integration was the norm (River Rouge plant included a steel mill and processed rubber). Since then, outsourcing has been growing. New communication technology has driven this more towards the entrepreneur-only model. Profit Profit maximization could easily be 1/2 of the text as we assume firms maximize profit under a host of situations: perfectly competitive, monopoly, price discrimination, oligopoly, monopsony, etc. However, the basics are perfect competition (price taker) and monopoly (price setter). Price Taker vs Price Setter Profit is TR-TC Price Taker (competitive firm)

Treat the price they face as given when choosing quantity Price Setter (single price, no strategic behavior) Price is chosen along with quantity. Short and long run options are the same Short Run, quantity decision includes the shut down option Long Run, consideration of returns to scale and entry and exit Profit Profit = TR-TC Total costs include all implicit and explicit costs (unlike accounting cost that would only include explicit costs). In our model, we assume the firm rents capital at a rate of v. But that is exactly the same as if the firm owned the capital but could rent it out to another firm at a rate of v. Accounting profit using the firms owned resources in the next best alternative use Includes Value of the entrepreneurs time Selling off owned factors and investing elsewhere For us, SC = VC + FC = wL + vK Price Takers The rest of this lecture focuses on price takers.

Homogeneous output No barriers to entry/exit in long run Many sellers Perfect price information Revenue: Price Taker Price Taker P Market Demand However, price taker assumption is that no firm is big enough to be able to affect the market price. p = Pm 1,000 2,000 3,000 3,000,000 q If a firm charges p > Pm , they will sell q = 0

Demand for firms output is p = Pm, the firm can sell as many as it wants, until q = 3,000,000, and then need to lower the price to sell more. Revenue: Price Taker So for price taker, we assume decreasing returns to scale precludes getting large enough to have production influence price: P Demand for firm: Pm = MR = AR p = Pm 1,000 3,000 5,000 q So R = pq, where p = Pm MR

d p q dq p, p q AR p q Cost and Short Run Supply Lets for the moment assume production exhibits IRS and then DRS. Firms will, in the LR, choose a level of K commensurate with getting the lowest possible SAC. The price will be driven to the low point of AC, the break even price. At this starting point, the firms SAC, AVC, and SMC are relevant in the short run Other returns to scale options considered in the long run. C SRC SRC Exhibits

IMR, DMR C AC SAC SMC Exhibits IMR, DMR Exhibits IMR, DMR MC SMC AC SAC Ehhibits IRS, DRS q

pbe AVC q Cost and Short Run Supply But for the moment, we dont care which level of K the firm has or what the AC curve looks like. In the SR, here is what we have to work with. To determine the profit maximizing level of q = q *, and whether shut down and produce q = 0, MC and SAVC are most important. C SC SC Exhibits IMR, DMR AC SRAC SRMC Exhibits IMR, DMR

SAC AVC SMC q q Profit Max Maximize = R(q) - C(q) FOC for this yields q where slope of function = 0 SC SC R=pq =R-SC MR=P slope of R maximized at q where MR=SMC FOC, derivative of function is zero SMC = slope of SC

q q Profit Max Checking the SOC too. SC SC R =R-SC MR=P slope of R also minimized at q where MR=SMC (FOC satisfied here too) Which is why we check SOC, to make sure profit is falling where MR = SMC (i.e. MR is falling relative to SMC) SMC = slope of SC

q q Profit Max The more common graph Maximize = R(q) - C(q) FOC for this yields MR=SMC, which it does twice. SOC ensures MC is rising relative to MR R C SC SC AC SAC SMC

Exhibits IMR, DMR SMC q q Price Taker Profit Max So long as you are better off producing than not, As you increase q, the change in profit = MRSMC. Produce until MR = SMC and marginal profit (change in profit as q increases) is falling. To Maximize Profit Q MR 99 TR 12

SMC 1188 SC Change in Profit = (MR-MC) 1000 I pulled these starting values out of the air Profit 188 To Maximize Profit Q MR TR

SMC SC Change in Profit = (MR-MC) 1000 Profit 99 12 1188 100 12 1200 7

1007 +5 193 101 12 1212 8 1015 +4 197 102 12 1224

9.10 1024.10 +2.90 199.90 103 12 1236 10.40 1034.50 +1.60 201.50 104

12 1248 12 1046.50 0 201.50 105 12 1260 13.80 1060.30 -1.80 199.70

106 12 1272 16 1076.30 -3 195.70 Note, at max, AR > ATC. When AR = ATC, = 0 188 Graphically MR = P, SMC MC 16 13.80 12

MR = P 10.40 9.10 8 7 100 101 102 103 104 105 106 q MC and qs: If price was $16, then the firm would produce 106. MR = P, SMC MC 16 MR = P 13.80 12 10.40 9.10 8 7

100 101 102 103 104 105 106 q MC and qs: If price was $10.40, then the firm would produce 103. MR = P, SMC MC 16 13.80 12 MR = P 10.40 9.10 8 7 100 101 102 103 104 105 106 q MC and qs: If price was $8.00, then the firm would produce 101. MR = P, SMC MC 16 13.80

12 10.40 9.10 MR = P 8 7 100 101 102 103 104 105 106 q MC and Supply Price Takers The MC curve tells us the profit maximizing qs by the firm at any price. Since it is the MC curve that determines the relationship between p and the quantity to supply, the SMC curve IS the firms short run supply curve. Important caveat, if suffering a loss, firm might want to shut down if the loss is larger than FC. Side note: Price Setters set the price, they do not respond to it, so they have no supply curve. Shut Down Option (price takers and price setters)

Shut down: Short run situation where the firm produces a quantity of 0 while remaining in the industry. It is still considered to be in the industry as long as it cannot rid itself from its fixed inputs. The firm could start producing very easily by employing some of the variable input. Intuition A firm bearing a loss can produce qs = q*, (where MR=MC) or can shut down, produce qs = 0. If qs = q*: = R VC FC If the firm shuts down: = -FC (that is, has a loss = FC) If FC is greater than the loss from producing, qs = q*. If FC is less than the loss from producing qs = q*, better to shut down and produce qs=0. Decision Rule: Shut down if R = 10,000 R = 7,000 FC > R VC FC Profit from shut down Profit from

q=q* VC = 8,000 FC = 4,000 = -2,000 -4,000 < -2,000 qs = q* VC = 8,000 FC = 4,000 = -5,000 -4,000 > -5,000 qs =0 Decision Rule Shut down if: FC > R VC FC 0 > R VC R > VC So long as revenue covers all variable cost, the loss will be less than FC so q = q* . Side note: Which can change the firms output decision, a change in Variable Cost and/or a change in Fixed Cost?

Shut down if: FC > FC > R VC FC FC is on both sides, so a change in FC does not affect the relationship or the decision. Ok, yes, fixed costs are fixed (dont vary with output) health insurance premiums rise Tony Romo signs a $108m extension. But a change in either R or VC could change the decision. Price Taker in the Short Run Simple, just MR = MC Maximize profit w.r.t. q Maximize profit w.r.t. L Profit Max 1 Simple, supply is SMC, find q where SMC = p.

SC SC v,w,q;K as from last chapter SC w Ls w,q;K v K dSC SMC w,q;K dq Set P SMC w,q;K and solve for q qs p Check to ensure that > -FC, if not, then shut down Profit Max 2, MR = MC Optimize by choosing q.

max p q SC v,w,q;K1 , where SC* SC v,w,q;K1 q FOC q p SMC w,q;K1 0 or p SMC w,q;K1 , choose q such that the MR = SMC Solve for q to get the firm supply funct ion, q* qs p SOC

d(SMC) qq 0, which it will be if DMR dq Check to ensure that > -FC, if not, then shut down Profit Max 3, MRPL=w Optimize by choosing L. max p f K1 ,L vK1 wL L FOC L p fL w 0 p fL w, choose L such that the MRPL = w

Solve for L* to get the labor demand function, L * L w,p,K 1 Plug into q f K1 ,L to get profit maximizing q* qs p Plug into p f K1 ,L vK1 wL to get maximal profit function w,p SOC LL p fLL 0, which it will be if DMR Check to ensure that > -FC, if not, then shut down Producer Surplus Producer surplus is the amount by which a firm is better off than shut down (q=0) If shut down, loss is = FC

By producing, the firm covers this potential loss, plus gains profit. If profit = 0, then better off by the amount of FC PS = + FC PS = R-VC-FC+FC PS = R-VC Producer Surplus If = 0, then producer surplus = FC If < 0, but - < FC, producer surplus > 0 If < 0, and - = FC, producer surplus = 0 Shut down point If < 0, and - > FC, producer surplus < 0 Will shut down, so PS = 0 and profit = -FC. Profit Maximization Price Taker, Long Run Returns to Scale Matter IRS, LMC falling CRS, LMC constant DRS, LMC rising Increasing Returns to Scale IRS only: incompatible with competition as the biggest firm has the lowest average cost natural monopoly results C AC MC C AC MC q q Decreasing Returns to Scale DRS only: an infinite number of infinitely small firms. C AC MC C

MC AC q q Constant Returns to Scale CRS only: any size firm can produce at the same AC. AC = AVC = MC (Firm LRS is horizontal at MC). C AC MC C AC=MC q q

IRS, DRS IRS, DRS: MC rising. Firm LRS = MC above AC (exit otherwise. C C AC MC MC AC Exhibits IRS, DRS q q IRS, CRS, DRS IRS, CRS, DRS: MC rising, but flat spot while CRS. Perhaps most realistic, but not easy to solve or find a production function that creates this. C

C AC MC DRS Firm LRS = MC for p pBE MC CRS AC IRS q q Profit Max. vs. Perfect Comp. We will eventually assume that in the long run K will be fixed to yield this SAC and the market price will be pbe (so in the LR q* will be at the low point of SAC) But in this chapter, we want to explore the possibility that

price will exceed pbe for a while. So we need a firm LRS curve. AC MC SAC SMC MC SMC SAC AC Firm exit if p < pBE q Production and Exit Essentially, a firms long run supply curve will be its long run MC curve While shut down is a viable option in the short run, in the long run all costs can be avoided by exiting the market. If p < pbe, (minimum value of AC curve), the firm should exit the industry. So firm long run supply is MC above pbe. Price Taker in the Long Run Simple, just MR = MC Maximize profit w.r.t. q Maximize profit w.r.t. K, L Profit Max 1 Simple, set MC = P, find q. C* C v,w,q as from last chapter C* w L v,w,q v K v,w,q dC* MC v,w,q dq Revenue: p q dR MR q dq Set MR=MC and solve for q* qs p d MC SOC : 0 dq Check to ensure that > -FC, if not, then shut down Profit Max , MR=MC

Optimize by choosing q max p q C v,w,q , where C* C v,w,q comes from q cost minimization. FOC q p MC v,w,q 0 or p MC v,w,q , choose q such that the MR = MC Solve for q to get the firm supply funct ion, q* q p SOC d2 d(MC) d(MC) qq 2 0, or 0 which it will be if DRS dq dq dq Check to ensure that > 0, if not, then exit Profit Max, MRPL=w; MRPK=v Optimize by choosing inputs

max p f K,L vK wL L FOC L p fL w 0 p fL w, MRPL = w K p fK v 0 p fK v, MRPK = v Solve for L* , K* to get the factor demand functions L* =L(w,v,p) K* =K(w,v,p) Plug into q f K* ,L* to get profit maximizing supply function q* q K(v,w,p),L(v,w,p) q(p) Profit Max, choose K and L Ratio of FOC L : p fL w K : p fK v fL w fK v So the profit maximizing input choice al so minimizes the cost of producing that level of q. SOC Profit Max, choose K and L

The function is strictly concave at L *, K* H LL LK KL KK 0, negative definite LL p fLL , LK p fLK KL p fKL , KK p fKK 2 H p ffLL fKK 2 LK0

So long as fLL 0 and fKK 0, and fLK2 is small. Profit Max, choose K and L Profit function, maximal profits for a given w, v, and p. Plug K* =K(v,w,p) and L* L(v,w,p) into p f K,L vK wL to get the profit optimizing profit func tion: p f K(v,w,p),L(v,w,p) vK(v,w,p) wL(v,w,p) Properties of the Profit Function Homogeneous of degree one in all prices with inflation, K*, L*, and q* are the same profit will keep up with that inflation Nondecreasing in output price profit 0 with p > 0 Nonincreasing in input prices profit 0 with w or v > 0 Convex in output prices profits from averaging those from two different output prices

will be at least as large as those obtainable from the average of the two prices (p1 ,v,w) (p2 ,v,w) p1 p 2 ,v,w 2 2 Envelope Results Long run supply d f K*,L * dp p f K(v,w,p),L(v,w,p) vK(v,w,p) wL(v,w,p) d f K(v,w,p),L(v,w,p) p fKKp fLL p dp vKpdp wL pdp dp d f K(v,w,p),L(v,w,p) pfKKpdp pfLL pdp vKpdp wL pdp dp

d f K(v,w,p),L(v,w,p) pfKKpdp vKpdp pfLL pdp wL pdp dp d f K(v,w,p),L(v,w,p) pfK v Kpdp pfL w L pdp dp By FOC d f K(v,w,p),L(v,w,p) 0 Kpdp 0 L pdp dp d f K(v,w,p),L(v,w,p) which is f(K*,L*) q* dp d gives us the long run supply equation: q* q(v,w,p) dp To maximize profit when there is a change in price , q=q*= f(K*, L*), continue producing such that L* = L(w, v, p) K* = K(w, v, p)

Envelope Results Profit maximizing factor demand functions p f K(v,w,p),L(v,w,p) vK(v,w,p) wL(v,w,p) d p fKKw fLL w dw vKwdw L* wL wdw dw d pfKKwdw pfLL wdw vK wdw L* wL wdw dw d pfKKwdw vK wdw pfLL wdw wL wdw L* dw d pfK v Kwdw pfL w L wdw L* dw By FOC d 0 Kwdw 0 L wdw L* dw By FOC Similarly, d K(v,w,p) dv

To maximize profit when K* there is a change in v, K K* K(w, v, p) To maximize profit when there is a change in w, choose L such that L = L*=L(w, v, p) d L* dw d gives us the demand equation, L* L(v,w,p ) dw Comparative Statics Price taker Long run Comparative Statics (L/w, K/w) K* and L * back into the FOC to create the

following identities p fL L(w,v,p),K(w,v,p) w 0 p fK L(w,v,p),K(w,v,p) v 0 differentiate w.r.t. w L* K* p fLL p fLK 1 0 w w L* K* p fKL p fKK 0 w w Comparative Statics (L/w, K/w) L* pfLL pfLK w 1 * pfKL pfKK K 0

w * L pfKK 2 0 2 w p (ffLL fKK ) LK pfKL K* 2 w p (ffLL fKK ) 2 LK 0 Comparative Statics (L/w, K/w) Increase in wage, increases MC, q* falls K L falls, K rises

L Comparative Statics (L/w, K/w) Increase in wage, increases MC, q* falls K L falls, K falls L Comparative Statics (L/v, K/v) * L pfLL pfLK v 0 * pfKL pfKK K 1 v * pfLK L 0 2

2 v p (ffLL fKK ) LK * pfLL K 2 v p (ffLL fKK ) 2 LK 0 Note that: L* K* v w because ffLK KL Comparative Statics (L/p, K/p) K* and L * back into the FOC to create the following identities

P f L (w,v,P),K (w,v,P) v 0 P fL L* (w,v,P),K* (w,v,P) w 0 * * K differentiate w.r.t. P L* K* fL P fLL P fLK 0 p p L* K* fK P fKL P fKK 0

p p Comparative Statics (L/p , K/p) L* pfLL pffLK w L * pfKL pffKK K K w * ffff L L KK K LK 0 2 p p(ffLL fKK ) LK K* ffff K LL p p(ffLL fKK )

L LK 2 LK 0 So long as fKL is positive or small, these will both be > 0. Since an increase in P causes MR to rise, at least one of these must be > 0 Comparative Statics (L/p, K/p) Increase in p, increases MR, q* rises K L rises, K rises L Comparative Statics (L/p, K/p) Increase in p, increases MR, q* rises K

L rises, K falls K inferior Obviously, L could be inferior instead Expansion path L Comparative Statics (q/p) q = f(L,K) q*=f(L*=L(w,v,p),K*=K(w,v,p)) How does this respond to a change in p? q* L* K* ffL K p p p and we now know L* ffff L KK

p p(ffLL fKK ) K LK 2 LK K* ffff K LL p p(ffLL fKK ) L LK 2 LK Comparative Statics (q/p) And so we can substitute to get: q* ffffffff ffL L KK p p(ffLL fKK ) and finally

K LK 2 LK K K LL p(ff f )LL KK 2 ff q L 2ffff KK L fK LK p p(ffLL fKK ) LK2 * where ffL 2 2ffff

KK 2 K LL L LK 2 LK 0 2 f 0 if are convex L K LK K isoquants LL to origin (required for cost minimization)

where ffLL fKK LK02ifproduction function is con cave at K* and L* (required for rising marginal cost at q* , profit decreasing in q) The Short Run and the Long Run Le Chteliar Principle How does the short run demand for L differ from the long run demand? K Increase in the wage rate K2 K0 q*(w1) q*(w2) LL* Ls * L* L Short Run Profit Max

FOC: p f L,K w 0 SOC: p f L,K 0 Demand: L L w,p,K , once K is fixed, v does not affect L decision. max p f L,K wL vK L L LL LL * S LS To get: , substitute demand (L*) into F OC w

*L p fL LS w,p,K ,K w 0 differen tiate w.r.t. w LS p fLL 1 0 w LS 1 0, as fLL 0 w p fLL Long Run vs. Short Run LS 1 0 w p fLL Remember the comparative statics result: LL fKK w p(ffLL fKK ) 2

0 LK LL LS fKK 2 w w p(ffLL fKK ) LKp f 1 LL 2 ffKK (ff f ) LL LS LL LL KK LK

2 w w p(ffLL fKK )f LK pLL f (fffLL )LL KK LK2 2 LL LS ffKK LL ffLL ffKK LK2 LK 2 w w p(ffLL fKK )f LK LLpf (ff fLL )LL KK And we can deduce 2 0 LK > 0, by SOC Short Run Profit Max Input demand in the long run is more elastic.

fLK2 LL LS w w pfLL (ffLL fKK ) 2 0 LK LL LS 0 w w LL LS , but both negative, so mult by -1 w w LL LS , LR change in L is > the short run change. w w

These types of relations are referred to as Le Chtelier effects, after the similar tendency of thermodynamic systems to exhibit the same types of responses. Silberberg, 3rd ed. (p. 85) Cobb-Douglas Examples Three cases: .25 .25 q K L .5 .5 q K L q KL .25 .25 q K L Cost Min minL wL vK (qo L.25K.25 ) FOC

K.25 L.25 LL w .75 0; LK v .75 0; L qo L.25K.25 0 L K wL Expansion path: K v .25 .25 q K L Cost Min SOC LLL 3K.25 7 4L L.25 4K.75

32L 7 4L 4 4L.75K.75 5 4K K.25 4L.75 3K.25 0 K.25 4L.75 4 0; LKK

5 4K 4 3L.25 7 0 4 L.25 4K.75 3 3 5 5 .75 .75 5 5 5 5

5 5 4L K 64L 4K 4 64L 4K 4 64L 4K 4 64L 4K 4 3L.25 7 4K 4 3 5 5 0 5 5 32L 4K 4 8L 4K 4 .25 .25 q K L L *, K * .5 2 v 2 w

L* q ; K* q w v .5 C* .5 * 2 v 2 w C wq vq w v C* 2q 2 wv .5 .5 *

.5 .5 * MC 4q wv ; P 4q wv ; q AC* 2q wv .5 p 4 wv .5 .25 .25 q K L MC, AC MC,AC MC* 4q wv *

AC 2q wv q .5 .5 .25 .25 q K L Profit Max max p L.25K.25 wL vK FOC pK.25 pL.25 L .75 w 0; K .75 v 0 4L 4K wL

Expansion path: K v .25 .25 q K L Profit Max SOC LL 3pK.25 7 16L 4 0; KK 16K 3pK.25 7 16L

p 3 16L 7 0 4 p 3 4 3 4K 4 8p2 32 LK 3pL.25

1.5 3 2 .25 2 .25 16L 4K 4 9p KL p KL 7 7 4 3pL.25 256 LK 256 LK 4 16K 0 7 4

.25 .25 q K L L *, K * p2 p2 L* 1.5 .5 ; K* 16v w 16w1.5 v.5 * 2 p q 1.5 .5 16w v * .25

p * p .5 4 wv * p2 8 wv .5 2 p 1.5 .5

16v w .25 p 4 wv .5 p2 p2 v w 1.5 .5 1.5 .5 16w v 16v w .25 .25

q K L Envelope Results * p * q p 4 wv .5 * 2 p L 1.5 .5 w 16v w * 2 p K*

v 16w1.5 v.5 * .25 .25 q K L Profit Max MC,AC * MC 4q wv AC* 2q wv MR q* p2 p2 L* 1.5 .5 ; K* 16v w 16w1.5v.5

p2 q 1.5 .5 16w v * .25 p2 1.5 .5 16v w .25 p 4 wv .5 q

.5 .5 .5 .5 q K L Cost Min minL wL vK (qo L.5K.5 ) FOC K.5 L.5 LL w .5 0; LK v .5 0; L qo L.5K.5 0 2L 2K wL Expansion path: K v .5 .5 q K L Cost Min

SOC K.5 L.5 LLL 1.5 0; LKK 1.5 0 4L 4K 0 K.5 2L.5 L.5 2K.5 K.5 2L.5 K.5 4L1.5 4L.5K.5 L.5 2K.5

4L.5K.5 16L.5K.5 16L.5K.5 16L.5K.5 16L.5K.5 L.5 4K1.5 0 .5 .5 .5 .5 .5 .5 8L K 8L K 4L K .5 .5

q K L L *, K * .5 C* v w L* q ; K* q w v .5 v w C wq vq w v .5 .5 *

* C 2q wv .5 .5 .5 MC* 4 wv ; P 4 wv ; q* any * AC 2 wv .5 .5 .5 q K L MC, AC MC,AC MC* 2 wv * AC 2 wv

q .5 .5 .5 .5 q K L Profit Max max p L.5K.5 wL vK FOC pK.5 pL.5 L .5 w 0; K .5 v 0 2L 2K wL Expansion path: K

v .5 .5 q K L Profit Max SOC pK.5 pL.5 LL 1.5 0; KK 1.5 0 4L 4K pK.5 4L1.5 p 4L.5K.5 p 2 p2 4L.5K.5 p 0 .5

16 LK 16 LK pL 4K1.5 .5 .5 q K L L *, K * L* undefined; K* undefined * q* any * undefined Envelope Results? No. .5 .5 q K L MC, AC MC,AC At MR1 > MC, -max at q = At MR2 < MC, -max at q = 0 At MR = MC, -max at any q (=0) MR1 MC* AC* 2 wv MR2 q .5 q KL Cost Min minL wL vK (qo LK) FOC LL w K 0; LK v L 0; L qo LK 0 wL Expansion path: K v q KL Cost Min SOC LLL 0; LKK 0 0

K L K 0 KL KL 2KL 0 L 0 q KL L *, K * C* .5 v w

L* q ; K* q w v .5 v w * C w q v q w v C* 2 qwv .5 .5 .5 .5 .5 wv wv wv

* MC ; p ; q 2 p q q * wv AC 2 q * .5 q KL MC, AC MC,AC

wv AC* 2 q wv MC q .5 .5 * q q KL Profit Max max p LK wL vK FOC

L pK w 0; K pL v 0 wL Expansion path: K v Cost minimizing tangency q KL Profit Max SOC LL 0; KK 0 0 p 0 p2 0 p 0 SOC indicate we have a profit min, not a max. q KL L *, K * * v w L* ; K* p

p q* SOC not satisfied * SOC not satisfied q KL MC, AC MC,AC wv AC 2 q .5 * wv MC* q .5 Maximizing profit means q=, where

MR MC MR Only produce units where MR < MC! q* q Appendix Short Run vs. Long Run Labor demand Demand for labor can be written: LL (w,v,p) LS w,p,K KL (w,v,p) Differentiate w.r.t. w: L LS LS KL w w K w

But lets try to sign LS KL K w The slopes of the SR and LR factor demand functions differ by a term that is the product of two effects: change in K from a change in w and the change in L that WOULD be caused by a change in the fixed amount of K. Differentiate original equation w.r.t. v LL LS KL v K v Now what? From the differential w.r.t. v: LL LS KL v K v

sign-wise, all we know is this is < 0 Which tells us these two must have opposite signs. Rearrange for this: LL LS v L K K v Because of the opposite signs, this is < 0 And substitute into: L LS LS KL w w K w Now what? Yields: LL L LS v KL L w w K w v And from reciprocity LL KL v w Yielding: KL L LS w KL w w v 2 Finally

And we can say: L K L LS w KL w w v 2 >0 <0 Since all terms are < 0, it is clear that the short run effect is smaller than the long run effect.

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