Firms and returns to scale -1- Firms and returns to scale. Increasing returns to scale and monopoly pricing 2. Constant returns to scale 19 C. The CRS economy 25 D. pplication to trade 47 E. Decreasing returns and competitive markets 56 F. Monopoly and joint costs (not covered) 59 ppendix: Technical results (not examinable)
Firms and returns to scale -2-. Increasing returns to scale and monopoly Production plan: a vector of inputs and outputs, (, ) zq Production set of a firm: The set S f of feasible plans for firm f Efficient production plan 0 0 ( z, q ) is production efficient if no plan ( zq, ) satisfying z z 0 and q 0 q is feasible. IRS The production set S exhibits increasing returns to scale on Q { q q q q} if for all q and q in Q ( z, q) S implies that ( z, q) int S. Increasing returns to scale
Firms and returns to scale -3- Proposition: If a firm exhibits IRS on Q { q q q q}, then for any 0 q 0 0 and q q in this interval C q C q 0 0 ( ) ( ) Proof: Suppose 0 z solves Min r z z q S. z 0 { (, ) } Then 0 0 C( q ) r z. Given IRS 0 0 (, ) int z q S. Then for some 0 0 0 (, ) z q S. Therefore 0 0 0 0 0 C( q ) r ( z ) r z r z C( q ). Increasing returns to scale
Firms and returns to scale -4- Corollary: If a firm exhibits IRS everywhere, then the firm cannot be a price taker. Proof: R( q) p q p q p q R( q). (*) C( q) C( q) C( q) C( q) C( q) **
Firms and returns to scale -5- Corollary: If a firm exhibits IRS everywhere, then the firm cannot be a price taker. Proof: R( q) p q p q p q R( q). (*) C( q) C( q) C( q) C( q) C( q) If 0 q is profitable so that Rq Cq 0 ( ) 0 ( ) 1 then q q with 1 is strictly profitable. 1 0 *
Firms and returns to scale -6- Corollary: If a firm exhibits IRS everywhere, then the firm cannot be a price taker. Proof: R( q) p q p q p q R( q). (*) C( q) C( q) C( q) C( q) C( q) If 0 q is profitable so that Rq Cq 0 ( ) 0 ( ) 1 then for any ˆ 1, q ˆ q is strictly profitable. 1 0 From (*) it follows that for all 1 R q R q C q C q 1 1 ( ) ( ) 1 1 ( ) ( ) 1 So it is always more profitable to expand QED
Firms and returns to scale -7- Single output firm If there is a single output, then for every z, the production efficient output is the maximum feasible output. We write this as q F() z. * Increasing returns to scale
Firms and returns to scale -8- Single output firm If there is a single output, then for every z, the production efficient output is the maximum feasible output. We write this as q F() z. IRS The production set S exhibits increasing returns to scale on Q { q q q q} if, for any 0 q 0 0 and q q in this interval, Increasing returns to scale if q F( z ) then 0 0 0 0 (, ) int z q S. Therefore for a sufficiently small 0, 0 0 (, ) z q S and so 0 0 F( z ) F( z ).
Firms and returns to scale -9- Natural monopoly firm that exhibits IRS for all output levels. Consider the single product case. Then for all 1 C( q) C( q) C( q) C( q) q q Smaller firms have higher average costs so it Is less costly for all the output to be produced by a single firm. Moreover a larger firm with lower average costs can drive a firm out of business by lowering prices and still remain profitable. Decreasing average cost
Firms and returns to scale -10- Invert the market demand curve and consider the demand price function pq ( ), i.e. the price for which demand is q. ( q) R( q) C( q) qp( q) C( q) ( q) R ( q) C ( q) p( q) qp ( q) C ( q) p( q) C ( q). Profit is maximized at M q where Monopoly output and price Note that ( q) MR( q) MC( q) 0 p M M MC( q ) 0
Firms and returns to scale -11- Consumer surplus The green dotted region to the left of the demand curve Is the consumer surplus (net benefit). Consumer surplus
Firms and returns to scale -12- Total cost Suppose that you know the fixed cost F. You also know the marginal cost at q {0, qˆ, qˆ, qˆ} 1 1 3 4 2 4 1 i.e. in increments of q q. 4 ˆ So you know the slope of the cost curve at each of these points. Suppose output is 3 ˆq. 4 MC( q ) is the slope of the red line. 3 ˆ 4 3 ˆ 4 MC( q) q is the length of the green line It is an estimate of the increase in the firm s cost as output increases from 3 4 MC( qˆ ) q C( qˆ ) C( qˆ ) 3 3 4 4 ˆq to ˆq
Firms and returns to scale -13- y the same argument, C ( q ) C (0) MC (0) q 1 ˆ 4 C( qˆ ) C( qˆ ) MC( qˆ ) q 1 1 1 2 4 4 C( qˆ ) C( qˆ ) MC( qˆ ) q 3 1 1 4 2 2 C( qˆ ) C( qˆ ) MC( qˆ ) q 3 3 4 4 dding these together, C( qˆ ) C(0) MC(0) q MC( qˆ) q MC( qˆ) q MC( qˆ) q 1 1 3 4 2 4 C( qˆ ) C(0) is the cost of increasing output from zero to ˆq, i.e. the firm s Variable Cost VC( q ˆ) dding the fixed cost F we have an approximation of the total cost Cq ( ˆ)
Firms and returns to scale -14- Variable cost s we have seen, this can be approximated as follows: 1 1 3 VC() qˆ MC(0) q MC( qˆ) q MC( qˆ) q MC( qˆ) q 4 2 4 This is also the shaded area under the firm s Marginal cost curve The smaller the increment in quantity q the more accurate is the approximation. In the limit it is the area under the cost curve. VC ( q) MC( q) qˆ VC( qˆ ) MC( q) dq 0
Firms and returns to scale -15- Welfare analysis Consumer surplus CS The green dotted region to the left of the demand curve Is the consumer surplus (net benefit). Total enefit The total cost to the consumers is M M p q. This is the cross-hatched area. Thus the sum of all the shaded areas is a measure of the total benefit to the consumers Total benefit
Firms and returns to scale -16- Social surplus SS() q Since d TC dq MC, M q M TC( q ) TC(0) MC( q) 0 M q F 0 MC( q) where F is the fixed cost. Thus the blue horizontally lined region is the variable cost Social surplus The vertically lined region is the total benefit to the consumers less the variable cost. This is called the social surplus.
Firms and returns to scale -17- Society gains as long as the social surplus exceeds the fixed cost s long as p( q) MC( q), social surplus rises as the price is reduced. Thus a monopoly always undersupplies relative to the social optimum. Hence there is always a potential gain to the regulation of monopoly Undersupply
Firms and returns to scale -18- Markup over marginal cost d dp q dp MR qp( q) p( q) q p( q)(1 ) dq dq p dq The second term is the inverse of the price elasticity. Therefore 1 MR( q) p( q)(1 ). ( qp, ) *
Firms and returns to scale -19- Markup over marginal cost d dp q dp MR qp( q) p( q) q p( q)(1 ) dq dq p dq The second term is the inverse of the price elasticity. Therefore 1 MR( q) p( q)(1 ). ( qp, ) The monopoly chooses M p so that MR 1 ( q ) p ( q )(1 ) ( ) ( qp, ) MC q. Therefore p M MC( q) 1 1 ( q, p) Elasticity is negative. more negative denotes a more elastic demand curve. Thus the markup is lower when demand is more elastic. Class discussion: Market segmentation and price discrimination
Firms and returns to scale -20-. Constant returns to scale CRS The production set S exhibits constant returns to scale if for all ( z, q) S and all 0, (, ) z q S. Proposition: If a firm exhibits CRS then for any 0 q 0 0 and q q in Q C q C q 0 0 ( ) ( ) Group Exercise: Prove this by showing that if 0 z is cost minimizing for 0 q then 0 z is cost minimizing for 0 q. Remark: If a firm exhibits CRS everywhere, then equilibrium profit must be zero for a price taker. Constant returns to scale
Firms and returns to scale -21- Single product firm with production function Fz () CRS The production set S 0 0 F( z ) F( z ). exhibits constant returns to scale if, for any 0 q 0 0 and q q, if q F( z ) then 0 0 Remark: It follows that Fz () is a homothetic function (Those interested can read the derivation of this result in the ppendix.) Convex superlevel set
Firms and returns to scale -22- a1 a3 Example: Cobb-Douglas production function F( z) a z z where a2 a3 1 0 1 2
Firms and returns to scale -23- Diminishing marginal product of each input
Firms and returns to scale -24- Exercise: CRS production function Show that 1 2 2 2 2 1 1 2 2 3 3 q ( a z a z a z ) exhibits constant returns to scale. Explain why the super level sets are convex. Solve for the maximum output for any input price vector r. Hence show that minimized total cost C( q, r ) is a linear function of q.
Firms and returns to scale -25- C. The two input two output constant returns to scale economy The model Commodities 1 and 2 are inputs. Commodities and are consumed goods. Productions functions: q F ( z ), q F ( z ). ***
Firms and returns to scale -26- C. The two input two output constant returns to scale economy The model Commodities 1 and 2 inputs. Commodities and are consumed goods. Productions functions: q F ( z ), q F ( z ). oth are CRS and the superlevel sets are strictly convex. Hence both production functions are concave. **
Firms and returns to scale -27- C. The two input two output constant returns to scale economy The model Commodities 1 and 2 inputs. Commodities and are consumed goods. Productions functions: q F ( z ), q F ( z ). oth are CRS and the superlevel sets are strictly convex. Hence both production functions are concave. Total supply of inputs is fixed: z z.
Firms and returns to scale -28- C. The two input two output constant returns to scale economy The model Commodities 1 and 2 inputs. Commodities and are consumed goods. Productions functions: q F ( z ), q F ( z ). oth are CRS and the superlevel sets are strictly convex. Hence both production functions are concave. Total supply of inputs is fixed: z z. t the aggregate endowment, the marginal rate of technical substitution is greater for commodity 1. MRTS F z F z 1 1 ( ) MRTS( ) F F z z 2 2.
Firms and returns to scale -29- Input intensity t the aggregate endowment MRTS ( ) MRTS ( ) Graphically the level set is steeper for commodity at the red marker. *
Firms and returns to scale -30- Input intensity t the aggregate endowment MRTS ( ) MRTS ( ) Graphically the level set is steeper for commodity at the red marker. Suppose that each firm has a fraction of the total input endowment The production of commodity is then said to be more input 1 intensive. If the firms producing commodity are willing to give up more of input 2 in order to obtain more of input 1.
Firms and returns to scale -31- Production Efficient allocations n output allocation qˆ ( qˆ, qˆ ) is production efficient if there is no feasible output vector q strictly larger. Then, for any qˆ, which is qˆ Max{ F ( z ) F ( z ) qˆ 0, z z z 0} Group Exercise: For any interior solution show that a necessary condition for production efficiency is F F z z MRTS z MRTS z 1 1 ( ) ( ) F F z z 1 1
Firms and returns to scale -32- Consider the Edgeworth ox diagram for the two consumed goods. We assumed that MRTS ( ) MRTS ( ) Note that if zˆ then zˆ (1 ). Since CRS functions are homothetic, the MRS are constant along a ray. Therefore MRTS ( zˆ ) MRTS ( ) MRTS ( ) MRTS ( zˆ ). *
Firms and returns to scale -33- Consider the Edgeworth ox diagram for the two consumed goods. We assumed that MRTS ( ) MRTS ( ) Note that if zˆ then zˆ (1 ). Since CRS functions are homothetic, the MRS are constant along a ray. Therefore MRTS ( zˆ ) MRTS ( ) MRTS ( ) MRTS ( zˆ ). It follows that no allocation on or above the diagonal Is production efficient. In the lower diagram z ˆ is production efficient
Firms and returns to scale -34- Characterization of PE allocations t the PE allocation z ˆ MRTS ( zˆ ) MRTS ( zˆ ) Step 1: long the line O MRTS ( z ) MRTS ( zˆ ) Therefore in the shaded region above this line MRTS ( z ) MRTS ( zˆ ) *
Firms and returns to scale -35- Characterization of PE allocations t the PE allocation z ˆ MRTS ( zˆ ) MRTS ( zˆ ) Step 1: long the line O MRTS ( z ) MRTS ( zˆ ) Therefore in the shaded region above this line MRTS ( z ) MRTS ( zˆ ) Step 2: long the line O MRTS ( z ) MRTS ( zˆ ) Therefore in the shaded region above this line MRTS ( z ) MRTS ( zˆ ). It follows that in the intersection of these two sets MRTS ( z ) MRTS ( z ) MRTS ( zˆ ) MRTS ( z )
Firms and returns to scale -36- Then no allocation in the shaded area above the lines O and O is a Production Efficient allocation. y an almost identical argument, no allocation in the shaded area below the lines O and O is a Production Efficient allocation either. **
Firms and returns to scale -37- Then no allocation in the shaded area above the lines O and O is a Production Efficient allocation. y an almost identical argument, no allocation in the shaded area below the lines O and O is a Production Efficient allocation either. Then any PE allocation z ˆ with higher output of commodity lies in the triangle CO. *
Firms and returns to scale -38- Then no allocation in the shaded area above the lines O and O is a Production Efficient allocation. y an almost identical argument, no allocation in the shaded area below the lines O and O is a Production Efficient allocation either. Then any PE allocation z ˆ with higher output of commodity lies in the triangle CO. Finally note that MRTS () z is constant along a ray for a CRS (i.e. a homothetic) production function Thus for PE allocations, MRTS ( z1, z 2) for commodity rises as q rises.
Firms and returns to scale -39- For a PE allocation the MRTS for both products must be the same. Therefore the equalized MRTS both increase as q increases.
Firms and returns to scale -40- Production possibility set Q (( q, q ) F ( z ) q 0, F ( z ) q 0, z z } Consider any pair of feasible vectors ( z, z, q, q ) and ( z, z, q, q ) ll convex combinations of the input vectors satisfy the input constraint. Given the concavity of the production functions, all convex combinations of the outputs are feasible. Therefore Q convex.
Firms and returns to scale -41- Since each firm is profit-maximizing, the representative firm is profit maximizing. Therefore the WE input output vector ( z, z, q ) solves Max{ p q r ( z z ) q F ( z ), q F ( z )}. z, z, q For market clearing, z z. Therefore ( z, z, q ) solves Max{ p q r ( z z ) q F ( z ), q F ( z ), z z } z, z, q We can rewrite this as follows: Max{ p q r ( q, q ) Q} q Thus the WE output vector maximizes revenue over the set of feasible outputs, Q.
Firms and returns to scale -42- Equilibrium allocation of inputs Consider industry : FOC z solves Max{ pf( z ) r1 z1 r2 z2 } z p F z 1 r 1 0 F and p r 2 z 2 0. **
Firms and returns to scale -43- Equilibrium allocation of inputs Consider industry : FOC z solves Max{ pf( z ) r1 z1 r2 z2 } z p F z 1 r 1 0 F and p r 2 z 2 0. Therefore F F p r1 z1 z1 MRTS r F F 2 p z2 z2 ( z )
Firms and returns to scale -44- Equilibrium allocation of inputs Consider industry : FOC z solves Max{ pf( z ) r1 z1 r2 z2 } z p F z 1 r 1 0 F and p r 2 z 2 0. Therefore F F p r1 z1 z1 MRTS r F F 2 p z2 z2 ( z ) y a symmetric argument for industry F F p r1 z1 z1 MRTS r F F 2 p z2 z2 ( z ) Therefore the WE input allocations are production efficient.
Firms and returns to scale -45- Conclusion Let p ( p, p ) be the initial price vector, Suppose the price of commodity rises. The New price vector is p ( p, p ). Output of commodity rises and of commodity falls. *
Firms and returns to scale -46- Conclusion Let p ( p, p ) be the initial price vector, Suppose the price of commodity rises. The New price vector is p ( p, p ). Output of commodity rises and of commodity falls. Since the production of commodity is more input 1 intensive, the increased demand for commodity raises the price of input 1 relative to input 2. r1 In the Edgeworth-ox MRTS( z ) r rises as q rises. 2 i.e. the level set at z is less steep than at z.
Firms and returns to scale -47- These conclusion are summarized in the following proposition. Proposition: If the output price ratio p p rises (falls), then the input price ratio r r 1 2 also rises (falls).
Firms and returns to scale -48- D. pplication: Distributional effects of reducing trade barriers Suppose that a country opens its borders to trade. s a result the input 1 intensive commodity becomes the export good and commodity becomes the import good. This pushes up the price of commodity and pushes down the price of commodity. *
Firms and returns to scale -49- D. pplication: Distributional effects of reducing trade barriers Suppose that a country opens its borders to trade. s a result the input 1 intensive commodity becomes the export good and commodity becomes the import good. This pushes up the price of commodity and pushes down the price of commodity. Since it is only relative prices that matter we normalize and fix the price of commodity.
Firms and returns to scale -50- Lemma: In a CRS economy, r r p C ( r, r ) r C (1, ) r C (,1) 2 1 j j 1 2 1 j 2 j r1 r2 Proof: In a CRS economy C is independent of output and profit is zero so p j C j( r1, r2). Increasing input prices in proportion does not change the cost minimizing input vector. Therefore total cost (and hence average cost) increase by the same proportion, i.e. 1 p j C( r1, r2) and so p j C( r1, r2) 1 1 To complete the proof choose and r r 1 2
Firms and returns to scale -51- Heckscher-Ohlin Theorem: If constant, then *** p, the price of the input 1 intensive commodity, rises and r 1, the price of input 1 rises and r 2, the price of input 2 falls. p is
Firms and returns to scale -52- Heckscher-Ohlin Theorem: If constant, then p, the price of the input 1 intensive commodity, rises and, the price of input 2 falls. r 1, the price of input 1 rises and r 2 p is Proof: ppealing to the Lemma, ** r p C r r r C. (*) r 1 ( 1, 2) 2 (,1) 2
Firms and returns to scale -53- Heckscher-Ohlin Theorem: If constant, then Proof: ppealing to the Lemma, p, the price of the input 1 intensive commodity, rises and r 1, the price of input 1 rises and r p C r r r C. (*) r 1 ( 1, 2) 2 (,1) 2 r 2, the price of input 2 falls. p is For the CRS model an increase in the price of commodity results in an increase in 1 an increase in ( r C,1). Since r 2 p is constant it follows from (*) that r 2 must fall. r r 1 2 and therefore QED
Firms and returns to scale -54- Heckscher-Ohlin Theorem: If constant, then p, the price of the input 1 intensive commodity, rises and r 1, the price of input 1 rises and r 2, the price of input 2 falls. p is Proof: ppealing to the Lemma, r p C r r r C. (*) r 1 ( 1, 2) 2 (,1) 2 For the CRS model an increase in the price of commodity results in an increase in 1 an increase in ( r C,1). Since r 2 p is constant it follows from (*) that r 2 must fall. Remark: If input 2 is labor, workers have a lower wage and face higher prices. The gains are only to those who have significant endowments of the other input. r r 1 2 and therefore QED
Firms and returns to scale -55- Free trade and input price equalization Suppose that there is free trade between two countries. oth have the same technology (production sets). Let C be the input endowment of country C and let U be the input endowment of country U. Proposition: Input price equalization Suppose that at the aggregate endowment, commodity is more input 1 intensive in both countries. i.e. MRTS C C U U ( ) MRTS ( ) and MRTS ( ) MRTS ( ). Then WE input prices must be the same
Firms and returns to scale -56- Proof: Under the assumption that industry is the input 1 intensive industry in both countries, we have shown that for each country the input price ratio rises as the output price ratio rises. r / r Therefore in each country there is also an inverse mapping from the input price ratio to the output price ratio. 1 2 p / p
Firms and returns to scale -57- Under CRS, p C ( r), p C ( r). ppealing to the Lemma r1 (*) p C( r1, r2 ) r2 C(,1) r and r1 (**) p C ( r1, r2 ) r2 C (,1) r Therefore 1 1 2 (,1) C(,1) p r2 r2 p r r C r r C 2 1 1 (,1) C(,1) r2 r2 Thus the mapping must be the same in the two countries. 2 2 r r ppealing to (*) and (**), the input prices must be equal.
Firms and returns to scale -58- E. Decreasing returns to scale The production set S exhibits decreasing returns to scale on Q { q q q q} if, for any 0 q 0 0 and q q in Q if 0 0 ( z, q ) is in S then 0 0 (, ) int z q S. Single product firm DRS The production set S exhibits decreasing returns to scale on Decreasing returns to scale the interval q q q if, for any 0 q 0 0 and q q in this interval, if q F( z ) then 0 0 0 0 F( z ) F( z ). Proposition: C( q ) increases under DRS Proof: Check on your own that you can prove this (See the discussion of IRS).
Firms and returns to scale -59- In many industries, technology exhibits IRS at low output levels and DRS at high output levels. Thus at low output levels C falls and at high output levels C rises. Let q be the output where C is minimized. Then MC( q) C( q) for q q and MC( q) C( q) for q q Group exercise: Prove this claim.
Firms and returns to scale -60- Equilibrium with free entry of identical firms t any price ˆp p the profit maximizing output is ˆq q. ll the firms are profitable Thus entry continues until the price is pushed down very close to p. *
Firms and returns to scale -61- Equilibrium with free entry of identical firms t any price ˆp p the profit maximizing output is ˆq q. ll the firms are profitable Thus entry continues until the price is pushed down very close to p. If q is small relative to market demand then to a first approximation the equilibrium price is p. Thus demand has no effect on the equilibrium output price. t the industry level expansion is by entry so there are constant returns to scale
Firms and returns to scale -62- D. Joint Costs n electricity company has an interest cost co 20 per day for each unit of turbine capacity. For simplicity we define a unit of capacity as megawatt. It faces day-time and night-time demand price functions as given below. The operating cost of running a each unit of turbine capacity is 10 in the day time and 10 at night. p 200 q, p 100 q 1 1 2 2 Formulating the problem Solving the problem Understanding the solution (developing the economic insight)
Firms and returns to scale -63- Try a simple approach. Each unit of turbine capacity costs 10+10 to run each day plus has an interest cost of 20 so MC=40 With q units sold the sum of the demand prices is 300 2q so revenue is TR q(300 2 q) hence * MR 300 4q. Equate MR and MC. q 65.
Firms and returns to scale -64- Try a simple approach. Each unit of turbine capacity costs 10+10 to run each day plus has an interest cost of 20 so MC=40 With q units sold the sum of the demand prices is 300 2q so revenue is TR q(300 2 q) hence MR 300 4q. Equate MR and MC. * q 65. Lets take a look on the margin in the day and night time TR1 q1 (200 q1 ) then MR1 200 2q1. TR2 q2(200 q2) then MR2 100 2q2. MR2 0!!!!!!!!!! Now what?
Firms and returns to scale -65- Key is to realize that there are really three variables. Production in each of the two periods q1 and q 2 and the plant capacity q 0. C( q) 20q 10q 10q 0 1 2 R( q) R ( q ) R ( q ) (200 q ) q (100 q ) q 1 1 2 2 1 1 2 2 Constraints h1 ( q) q0 q1 0 h2 ( q) q0 q2 0. **
Firms and returns to scale -66- Key is to realize that there are really three variables. Production in each of the two periods q1 and q 2 and the plant capacity C( q) q 0. c q where c ( c0, c1, c2) (20,10,10) R( q) R ( q ) R ( q ) (200 q ) q (100 q ) q 1 1 2 2 1 1 2 2 Constraints: h1 ( q) q0 q1 0 h2 ( q) q0 q2 0. L R ( q ) R ( q ) c q c q c q ( q q ) ( q q ) * 1 1 2 2 0 0 1 1 2 2 1 0 1 2 0 2
Firms and returns to scale -67- Key is to realize that there are really three variables. Production in each of the two periods q1 and q 2 and the plant capacity C( q) q 0. c q where c ( c0, c1, c2) (20,10,10) R( q) R ( q ) R ( q ) (200 q ) q (100 q ) q 1 1 2 2 1 1 2 2 Constraints: h1 ( q) q0 q1 0 h2 ( q) q0 q2 0. L R ( q ) R ( q ) c q c q c q ( q q ) ( q q ) 1 1 2 2 0 0 1 1 2 2 1 0 1 2 0 2 Kuhn-Tucker conditions L MR j ( q j ) c j j 0, q j with equality if q j 0, j=1,2 L c0 1 2 0, q 0 with equality if q0 0 L q0 qi 0, i with equality if i 0, i=1,2
Firms and returns to scale -68- Kuhn-Tucker conditions L 200 2q1 10 1 0, q 1 L 100 2q2 10 2 0, q 2 with equality if q1 0 with equality if q1 0 L 20 1 2 0, q 0 with equality if q0 0 L q0 q1 0, 1 L q0 q2 0, 2 with equality if 1 0. with equality if 2 0. *
Firms and returns to scale -69- Kuhn-Tucker conditions L 200 2q1 10 1 0, q 1 L 100 2q2 10 2 0, q 2 with equality if q1 0 with equality if q1 0 L 20 1 2 0, q 0 with equality if q0 0 L q0 q1 0, 1 L q0 q2 0, 2 with equality if 1 0. with equality if 2 0. Solve by trial and error. Class exercise: Why will this work? (i) Suppose q0 0and both shadow prices are positive. Then q0 q1 q2. (ii) Solve and you will find that one of the shadow prices is negative. ut this is impossible. (iii) Inspired guess. That shadow price must be zero. Then the positive shadow price must be 20.
Firms and returns to scale -70- ppendix: Proposition: Convex production sets and decreasing returns to scale Let S be a production set. If S {( z, q) S ( z, q) 0} is strictly convex, then the production set exhibits decreasing returns to scale. Proof: Consider 0 0 ( z, q ) (0,0) and Since S is strictly convex, 1 1 ( z, q ) 0. 0 1 0 1 1 1 ( z, q ) ((1 ) z z,(1 ) q q ) ( z, q ) is in the interior of S and hence in the interior of S Local decreasing returns
Firms and returns to scale -71- Cost functions with convex production sets Proposition: If a production set S is a convex set then the cost function C( q, r) is a convex function of the output vector q. Proof: We need to show that for any 0 q and 1 q C q C q C q 0 1 ( ) (1 ) ( ) ( ) Let 0 z be cost minimizing for 0 q and let 1 z be cost minimizing for 1 q. Then 0 0 C( q ) r z and 1 1 C( q ) r z. Therefore Since S 0 1 0 1 0 1 (1 ) ( ) ( ) (1 ) [(1 ) ] C q C q r z r z r q q r z (*) is convex, ( z, q ) S. Then it is feasible to produce q at a cost of r z. It follows that the minimized cost is (weakly smaller) i.e. C( q ) r z ppealing to (*) it follows that C q C q C q 0 1 ( ) (1 ) ( ) ( ) QED
Firms and returns to scale -72- Profit maximization by a price taking firm Since C( q, r) is a convex function of q, C( q, r) is concave and so the firm s profit ( q, p, r) p q C( q, r) is a concave function of output. It follows that the FOC are both necessary and sufficient for a maximum.
Firms and returns to scale -73-
Firms and returns to scale -74- Technical Lemma: Super-additivity of a CRS production function For a single product firm, if the production function convex superlevel sets, then for any input vectors Fz () xy, exhibits constant returns to scale and has F( x y) F( x) F( y) For some 0, F( x) F( y) F( y). First note that 1 F( y) F( x) (*). convex combination Consider the superlevel set Z { z F( z) F( x)} Since y is in Z, all convex combinations of x and y are in Z. In particular, the convex combination 1 ( ) x ( ) y ( x y) 1 1 1 is in Z.
Firms and returns to scale -75- Therefore F ( ( )) ( ) 1 x y F x. ppealing to CRS Therefore F( x y) F( x). 1 1 1 F( x y) (1 ) F( x) F( x) F( x) ppealing to (*) F( x y) F( x) F( y). QED
Firms and returns to scale -76- Proposition: Concavity of a CRS production function For a single product firm, if the production function Fz () exhibits constant returns to scale and has convex superlevel sets, then Fz () is a concave function. Proof: Define y 0 (1 ) z and z 1. Then from this proposition, z 0 1 F( z ) F((1 ) z ) F( z ). Since F exhibit CRS it follows that F z F z F z F z F z 0 1 0 1 ( ) ((1 ) ) ( ) (1 ) ( ) ( )