Firms and returns to scale -1- John Riley

Transcription

1 Firms and returns to scale -1- John Riley Firms and returns to scale. Increasing returns to scale and monopoly pricing 2. Natural monopoly 1 C. Constant returns to scale 21 D. The CRS economy 26 E. pplication to trade 39 F. Decreasing returns and competitive markets 42 G. Monopoly and joint costs 46 ppendix: Technical results (you may safely ignore!)

2 Firms and returns to scale -2- John Riley. 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 ( z, q ) is production efficient if no plan ( zq, ) satisfying z z and q q is feasible. IRS The production set on Q { q q q q} if for all q ( z, q) S f exhibits increasing returns to scale and q in Q f S implies that ( z, q) int S. Increasing returns to scale

3 Firms and returns to scale -3- John Riley Cost Function of a firm C( q, r) Min{ r z ( z, q) S f } z The cheapest way to produce q units. Proposition: If a firm exhibits IRS on Q { q q q q}, then for any q and q q in this interval C q C q ( ) ( ) Proof: Since ( z, q ) is feasible and costs It follows that Cq ( ) C q r z r z C q ( ) ( ) ( ) Increasing returns to scale Since z is not a boundary point we can lower some Input and still produce the same output. Therefore C q r z C q ( ) ( )

4 Firms and returns to scale -4- John Riley Corollary: If a firm exhibits IRS everywhere, then the firm cannot be a price taker. Proof: R( q) p q p q p q p q R( q). (*) C( q) C( q) C( q) C( q) C( q) C( q) **

5 Firms and returns to scale -5- John Riley 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 q is profitable so that Rq Cq ( ) ( ) 1 then q q with 1 is strictly profitable. 1 *

6 Firms and returns to scale -6- John Riley 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 q is profitable so that Rq Cq ( ) ( ) 1 then for any ˆ 1, q ˆ q is strictly profitable. 1 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

7 Firms and returns to scale -7- John Riley 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. This is called the firm s production function. f The production set is S {( z, q) q F( z)} * Increasing returns to scale

8 Firms and returns to scale -8- John Riley 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. This is called the firm s production function. f The production set is S {( z, q) q F( z)} IRS The production function exhibits increasing returns to scale on Q { q q q q} if, for any q and q q in this interval, Increasing returns to scale if q F( z ) then (, ) (, ( )) int z q z F z S. Therefore F( z ) is not maximized output and so F( z ) F( z ).

9 Firms and returns to scale -9- John Riley Group exercise q F() z z z 1/2 1/2 1 2 s the manager you are given a fixed budget, the input price vector is r choose the input vector z that maximizes output.. Your objective is to 1 (a) Show that z1 2 r ( ) r r 1 2 and solve for z 2. (b) Show that maximized output is q 1 1 r r 1/2 1/2 ( ). 1 2 (c) What is the lowest cost of producing q?

10 Firms and returns to scale -1- John Riley. 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

11 Firms and returns to scale -11- John Riley 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) p M M MC( q )

12 Firms and returns to scale -12- John Riley Consumer surplus The green dotted region to the left of the demand curve Is the consumer surplus (net benefit). Consumer surplus

13 Firms and returns to scale -13- John Riley Total cost Suppose that you know the fixed cost F. You also know the marginal cost at q {, qˆ, qˆ, qˆ} 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 ) ˆq to ˆq

14 Firms and returns to scale -14- John Riley y the same argument, C ( q ) C () MC () q 1 ˆ 4 C( qˆ ) C( qˆ ) MC( qˆ ) q C( qˆ ) C( qˆ ) MC( qˆ ) q C( qˆ ) C( qˆ ) MC( qˆ ) q dding these together, C( qˆ ) C() MC() q MC( qˆ) q MC( qˆ) q MC( qˆ) q C( qˆ ) C() 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)

15 Firms and returns to scale -15- John Riley Variable cost s we have seen, this can be approximated as follows: VC() qˆ MC() q MC( qˆ) q MC( qˆ) q MC( qˆ) q 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

16 Firms and returns to scale -16- John Riley 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

17 Firms and returns to scale -17- John Riley Social surplus SS() q Since d TC dq MC, M q M TC( q ) TC() MC( q) M q F 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.

18 Firms and returns to scale -18- John Riley 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

19 Firms and returns to scale -19- John Riley 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, ) *

20 Firms and returns to scale -2- John Riley 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

21 Firms and returns to scale -21- John Riley C. Constant returns to scale CRS The production set S exhibits constant returns to scale if for all ( z, q) S and all, (, ) z q S. Proposition: If a firm exhibits CRS then for any q and q q in Q C q C q ( ) ( ) Group Exercise: Prove this by showing that if z is cost minimizing for q then z is cost minimizing for q. Remark: If a firm exhibits CRS everywhere, then equilibrium profit must be zero for a price taker. Constant returns to scale

22 Firms and returns to scale -22- John Riley Single product firm with production function Fz () CRS The production set S exhibits constant returns to scale if, for any F( z ) F( z ). q and q q, if q F( z ) then Remark: It follows that Fz () is a homothetic function (Those interested can read the derivation of this result in the ppendix.) Convex superlevel set

23 Firms and returns to scale -23- John Riley a1 a2 Example: Cobb-Douglas production function F() z a z z where a1 a

24 Firms and returns to scale -24- John Riley Diminishing marginal product of each input

25 Firms and returns to scale -25- John Riley Exercise: CRS production function Show that q 1 a a a ( ) z z z /2 ) exhibits constant returns to scale. a1 a2 a3 Explain why S { z g( z) k } is a superlevel set z z z 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.

26 Firms and returns to scale -26- John Riley D. The two input two output constant returns to scale economy Review: CRS function is Fz () is homothetic since F( z) F( z). Therefore if z solves Max{ F( z) r1 z1 r2 z2 1} z we have shown that z solves Max{ F( z) r1 z1 r2 z2 } z F ( z ) z1 F ( z ) z 2 is called the marginal rate of technical substitution r1 r1 FOC: MRTS( z ) and MRTS( z ) r r 2 Thus the MRTS is constant along that ray through z. 2

27 Firms and returns to scale -27- John Riley D. The two input two output constant returns to scale economy The model Commodities 1 and 2 inputs. Commodities and are consumed goods. CRS productions functions: q F ( z ), q F ( z ). Total supply of inputs is fixed: z z.

28 Firms and returns to scale -28- John Riley D. The two input two output constant returns to scale economy The model Commodities 1 and 2 inputs. Commodities and are consumed goods. CRS productions functions: q F ( z ), q F ( z ). Total supply of inputs is fixed: z z. The superlevel sets are strictly convex. Hence both production functions are concave. 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.

29 Firms and returns to scale -29- John Riley Input intensity t the aggregate endowment MRTS ( ) MRTS ( ) Graphically the level set is steeper for commodity at the red marker. *

30 Firms and returns to scale -3- John Riley Input intensity t the aggregate endowment MRTS ( ) MRTS ( ) Graphically the level set is steeper for commodity at the red marker. The production of commodity is then said to be more input 1 intensive. t the aggregate input endowment, the firms producing commodity are willing to give up more of input 2 in order to obtain more of input 1.

31 Firms and returns to scale -31- John Riley Production Efficient allocations n output allocation qˆ ( qˆ, qˆ ) is production efficient if there is no feasible output vector q which is strictly larger. Then, for any qˆ qˆ Max{ F ( z ) F ( z ) qˆ, z z z }, Group Exercise: If commodity 1 is more input 1 intensive explain why the efficient input allocations must lie below the diagonal of the Edgeworth ox Efficient allocations

32 Firms and returns to scale -32- John Riley Proof: 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 ).

33 Firms and returns to scale -33- John Riley Characterization of efficient allocations t the PE allocation zˆ MRTS ( zˆ ) MRTS ( zˆ ) Step 1: long the line O MRTS ( z ) MRTS ( zˆ ) Therefore in the cross-hatched region above this line MRTS ( z ) MRTS ( zˆ ) *

34 Firms and returns to scale -34- John Riley Characterization of efficient allocations t the efficient allocation zˆ MRTS ( zˆ ) MRTS ( zˆ ) Step 1: long the line O MRTS ( z ) MRTS ( zˆ ) Therefore in the cross-hatched region above this line MRTS ( z ) MRTS ( zˆ ) Step 2: long the line O MRTS ( z ) MRTS ( zˆ ) Therefore in the dotted region above this line MRTS ( z ) MRTS ( zˆ ). It follows that in the intersection of these two sets MRTS ( z ) MRTS ( zˆ ) MRTS ( zˆ ) MRTS ( z )

35 Firms and returns to scale -35- John Riley Then no allocation in the vertically lined area above the lines O and is an efficient allocation. O y an almost identical argument, no allocation in the shaded area horizontally lined region below the lines is an efficient allocation either. ** O and O

36 Firms and returns to scale -36- John Riley Then no allocation in the dotted area above the lines O and is an efficient allocation. O y an almost identical argument, no allocation in the dotted area below the lines O and O is an efficient allocation either. Then any efficient z ˆ with higher output of commodity (the point Ĉ in the figure ) lies in the triangle ˆ CO. *

37 Firms and returns to scale -37- John Riley Then no allocation in the dotted area above the lines O and is an efficient allocation. O y an almost identical argument, no allocation in the dotted area below the lines O and O is an efficient allocation either. Then any efficient z ˆ with higher output of commodity (the point Ĉ in the figure ) lies in the triangle ˆ CO. Since MRTS () z is constant along a ray, zˆ2 zˆ2 Two implications: (i) z ˆ zˆ (ii) MRTS ( zˆ) MRTS( zˆ) 1 1 Key result: For efficient allocations, MRTS MRTS rises as q rises.

38 Firms and returns to scale -38- John Riley Proposition: If output of the input 1 intensive input rises then the input rice ratio r r 1 2 must rise r1 Proof: In the equilibrium MRTS MRTS so this follows directly from the previous result. r (Thus the owners of the input 1 intensive input benefit more.) 2

39 Firms and returns to scale -39- John Riley E. Opening an economy to trade Suppose the country opens its borders to trade and as a result the price of commodity (now exported) rises relative to the price of commodity (facing import competition). Proposition: If the price of commodity rises then the equilibrium output of commodity must rise Closed economy Let ( z, q, q ) (, q, q ) be the aggregate equilibrium demand for inputs and equilibrium supply of the two outputs when prices are p Open economy and r. Let ( z, q, q ) (, q, q ) be the aggregate equilibrium demand for inputs and equilibrium supply of the two outputs when prices are p and r Since both are technologically feasible, profit-maximization implies that (i) p q r p q r, hence p ( q q) (ii) p q r p q r, hence p ( q q) Therefore p ( q q) p ( q q) and so ( p p) ( q q) p q.

40 Firms and returns to scale -4- John Riley Since it is only relative prices that matter we normalize and choose commodity 2 as the numeraire commodity so p p 1 and so.it follows from the above result that p q p It follows from the previous result that relatively more. r r 1 2 must rise. The owners of the input 1 intensive input gain

41 Firms and returns to scale -41- John Riley Proposition: The price of input 1 rises and the price of input 2 falls. Proof: Under CRS: C( q, r) qc(1, r) and C( q, r1, r2 ) qc(1, r1, r2 ). Therefore MC ( q, r) C (1, r) and MC ( q, r) C (1, r) In equilibrium price = MC. Therefore C(1, r1, r2) p and (ii) C(1, r1, r2) p. If p (a) rises it follows from (i) that at least one input price must rise. It follows from (ii) that (b) the other input price must fall. r1 Suppose r 2 rises. We showed that the ratio r 2 must rise therefore r 1 must rise as well. ut this contradicts (b). Thus opening the economy to exports of commodity raises the price of input 1 and lowers the price of input 2.

42 Firms and returns to scale -42- John Riley F. Decreasing returns to scale The production set S exhibits decreasing returns to scale on Q { q q q q} if, for any q and q q in Q if ( z, q ) is in S then (, ) 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 q and q q in this interval, if q F( z ) then 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).

43 Firms and returns to scale -43- John Riley 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.

44 Firms and returns to scale -44- John Riley 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. *

45 Firms and returns to scale -45- John Riley 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

46 Firms and returns to scale -46- John Riley G. Joint Costs n electricity company has an interest cost co 2 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 1 in the day time and 1 at night. p 2 q, p 1 q Formulating the problem Solving the problem Understanding the solution (developing the economic insight)

47 Firms and returns to scale -47- John Riley Try a simple approach. Each unit of turbine capacity costs 1+1 to run each day plus has an interest cost of 2 so MC=4 With q units sold the sum of the demand prices is 3 2q so revenue is TR q(3 2 q) hence * MR 3 4q. Equate MR and MC. q 65.

48 Firms and returns to scale -48- John Riley Try a simple approach. Each unit of turbine capacity costs 1+1 to run each day plus has an interest cost of 2 so MC=4 With q units sold the sum of the demand prices is 3 2q so revenue is TR q(3 2 q) hence MR 3 4q. Equate MR and MC. * q 65. Lets take a look on the margin in the day and night time TR1 q1 (2 q1 ) then MR1 2 2q1. TR2 q2(2 q2) then MR2 1 2q2. MR2!!!!!!!!!! Now what?

49 Firms and returns to scale -49- John Riley 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. C( q) 2q 1q 1q 1 2 R( q) R ( q ) R ( q ) (2 q ) q (1 q ) q Constraints h ( q) q q h2 ( q) q q2. **

50 Firms and returns to scale -5- John Riley 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. C( q) c q where c ( c, c1, c2) (2,1,1) R( q) R ( q ) R ( q ) (2 q ) q (1 q ) q Constraints: h1 ( q) q q1 h2 ( q) q q2. L R ( q ) R ( q ) c q c q c q ( q q ) ( q q ) *

51 Firms and returns to scale -51- John Riley 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. C( q) c q where c ( c, c1, c2) (2,1,1) R( q) R ( q ) R ( q ) (2 q ) q (1 q ) q Constraints: h1 ( q) q q1 h2 ( q) q q2. L R ( q ) R ( q ) c q c q c q ( q q ) ( q q ) Kuhn-Tucker conditions L MR j ( q j ) c j j, q j with equality if q j, j=1,2 L c 1 2, q with equality if q L q qi, i with equality if i, i=1,2

52 Firms and returns to scale -52- John Riley Kuhn-Tucker conditions L 2 2q1 1 1, q 1 L 1 2q2 1 2, q 2 with equality if q1 with equality if q1 L 2 1 2, q with equality if q L q q1, 1 L q q2, 2 with equality if 1. with equality if 2. *

53 Firms and returns to scale -53- John Riley Kuhn-Tucker conditions L 2 2q1 1 1, q 1 L 1 2q2 1 2, q 2 with equality if q1 with equality if q1 L 2 1 2, q with equality if q L q q1, 1 L q q2, 2 with equality if 1. with equality if 2. Solve by trial and error. Class exercise: Why will this work? (i) Suppose q and both shadow prices are positive. Then q 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 2.

54 Firms and returns to scale -54- John Riley ppendix: Proposition: Convex production sets and decreasing returns to scale Let S be a production set. If S {( z, q) S ( z, q) } is strictly convex, then the production set exhibits decreasing returns to scale. Proof: Consider ( z, q ) (,) and 1 1 ( z, q ). Since S is strictly convex, ( 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

55 Firms and returns to scale -55- John Riley 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 q and 1 q C q C q C q 1 ( ) (1 ) ( ) ( ) Let z be cost minimizing for q and let 1 z be cost minimizing for 1 q. Then C( q ) r z and 1 1 C( q ) r z. Therefore (1 ) ( ) ( ) (1 ) [(1 ) ] C q C q r z r z r q q r z (*) Since S 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 1 ( ) (1 ) ( ) ( ) QED

56 Firms and returns to scale -56- John Riley 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.

57 Firms and returns to scale -57- John Riley

58 Firms and returns to scale -58- John Riley Technical Lemma: Super-additivity 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 for any input vectors F( x y) F( x) F( y) xy, For some, 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) is in Z.

59 Firms and returns to scale -59- John Riley Therefore F( ( x y)) F( x) 1. ppealing to CRS Therefore Therefore F( ( x y)) F( x y) 1 1 F( x y) F( x) F( x y) (1 ) F( x) F( x) F( x) From the previous slide, Therefore 1 F( y) F( x) (*) F( x y) F( x) F( y). QED

60 Firms and returns to scale -6- John Riley 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 (1 ) z and z 1. Then from the super-additivity proposition, z 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 1 1 ( ) ((1 ) ) ( ) (1 ) ( ) ( )