micvar13
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13 COST CURVES Cost function: c (w 1, w 2, y ) Take the factor prices to be fixed, cost function: c (y ). The total costs as the sum of the variable costs and the fixed costs: c (y ) = c v (y ) +F .C c (y )VC (y )FO yAverage CostsAverage cost function(平均成本):AC (y ) = c (y )/yBy the total cost:)()()()()(y AFC y AVC yF y y c y y c y AC v +=+== AVC (y ): average variable costs(平均可变成本) AFC (y ): average fixed costs (平均固定成本)C SMC SAC(y )A VC(y )AFC(y )O yMarginal Costs if we change output by some amount △y , the change in cost △c(y ) = c (y +△y ) – c (y ) Marginal cost (边际成本).)()()()(yy c y y c y y c y MC ∆-∆+=∆∆= The derivative form dy y dc y MC )()(=Relation between MC and AC.)()(y y c dy d dy y dAC ==21y (dy y dc )(y – c (y ) ) =y 1(dy y dc )( –yy c )() = y1(MC (y ) – AC (y )) When AC is decreasing, MC is less than AC .If AC is rising, MC is greater than AC.MC must intersect AC at latter’s minimum point.Marginal costs and variable costsMarginal cost in terms of the variable cost function:y F y c F y y c y MC v v ∆---∆+=])([(])([)(=yy c v ∆∆)( The derivative form.)()()(dyy dc dy y dc y MC v ==C c (y )VC (y )O yTake ∆y = 1 MC (1) = c y (1) – c v (0) MC (2) = c y (2) – c v (1)MC (3) = c y (3) – c v (2) …… MC (n ) = c y (n ) – c v (n –1) Add n equations: c y (n ) = MC (1) + MC (2) +…+ MC (n )O n yBy integration:c v (y ) = ⎰ydu u MC 0)(When AVC is decreasing, MC is less than AVC .If AVC is rising, MC is greater than AVC .MC must intersect AVC at latter’s minimum point. Long-run CostsFixed factor: k (= x 2 ) ,The firm ’s short-run cost function: c s (y,k ),For any given y , the optimal k : k (y )What would be c s (y , k (y ))?The long-run cost function of production of y : c (y ) = c s (y , k (y )).k k(y)y 3k 2y 2y 1O L 1 L 2 L 3 LPick y *, and let k * = k (y *).c (y ) ≤ c s (y , k *)c(y*) = c s(y*, k*)for all levels of y.CLTCO y y* yFor average costsAC(y) AC s(y, k*)AC(y*) = AC s(y*, k*).C SMC2LMCSMC1SAC1LACSAC2yDiscrete levels of plant sizeThe long-run average cost curve will be the lower envelope of the short-run average costs.CO yLong-Run Marginal CostsDiscrete levels of plant sizeThe long-run marginal cost curve consists of the appropriate pieces of the short-run marginal cost curves.The continuous caseThe long-run marginal cost at any output level y has to equal the short-run marginal cost associated with the optimal level of plant size to produce y.CO y14 FIRM SUPPLYMarket EnvironmentsEvery firm faces two important decisions:----choosing how much it should produce;----choosing what price it should set.Constraints facing the firm:Technological constraints: summarized by the production function.Market constraint: it can only sell as much as people are willing to buy.Demand curve facing the firm: the relationship between the price a firm sets and the amount that it sells.Pure competition.Purely competitive market: each firm assumes that the market price is independent of its own level of output.An industry composed of many firms that produce an identical product,Each firm is a small part of the market.Firm is a price taker.Firm will sell nothing if it charges a price higher than the market price.If firm sells below the market price, it will get the entire market demand at that price.If it sells at the market price, it can sell whatever amount it wants.pp*O yThe Supply Decision of a Competitive FirmFirm’s revenue: R = pyIts cost: c(y)The profit maximization problem facing a firmRcmax y)(yWhat level of output will a firm choose to produce?MR = MC (y).For a competitive firm, p doesn’t change when y changes: MR = pA competitive firm will choose a level of output y:p = MC(y)CpO y1y2yAn exceptionThere may be more than one output that satisfying p = MC(y).Some output is not the firm’s best choice.The best choices must be the points on the MC(y) with positive slope.Cc(y) RO y1y2yMR =MC is a necessary condition for profit maximization.Another ExceptionWill firm keeps producing when it is facing py– c(y) < 0?In short-tun, when AVC > p, the firm’s profitspy – c v(y) –F= y(p – AVC(y)) –FC MCACAVCpO y yThe firm is better off going out of business when–F > py –c v(y) –FC MCACAVCpO yThe Inverse Supply CurveThe supply of a competitive firm: price equals marginal cost p = MC(y) if p min AVCy = 0 if p < min AVC---- “inverse supply curve”.Profits and Producer’s SurplusForm’s profits in short-runprofits = py – c v(y) –FC MCACp AVCProfitsFCVCO y output Producer’s surplus: the area to the left of the supply curve.C MCACp AVCBAO y ' y outputPS = py – (S A + S B )S A = AVC (y ')y ' = c v (y ') = ⎰')(y du u MCS B = ⎰yy du u MC ')(S A + S B = ⎰ydu u MC 0)( = c v (y )C MCACp AVCO y ' y outputPS = py –⎰ydu u MC 0)(Prod ucer’s surplus is the revenues minus the area beneath the MC curve. PS = py – c v (y ) =y (p – AVC (y )). Producer’s surplus is equal to revenues minus variable costs. Because c (y ) = c v (y ) + F , Producer ’s surplus = py – (c (y ) – F ) = π(y ) + FThe change in producer ’s surplus When price changes from p * to p ', output changes from y * to y '. The change in producer ’s surplus p 'y ' – (c (y ') – F ) – (p *y * – (c (y *) – F ) = π(y ') – π(y *)Cp 'p *BAO y * y ' outputExample For a cost function c (y ) = y 2 + 1 Then vc (y ) = y 2 avc (y ) = y Price equals marginal costs: p = 2y . Solving for output as a function of price we have2)(p y p S == The maximum profits for each price p .)()(y c py p -=π1)2(22--=p p p 142-=pC MC ACp A VCp * 2O y * y ' outputThe producer ’s surplus4)2)(21(2p p p A ==. The Long-Run Supply Curve of a FirmThe long-run supply function for the firm measures how much the firm would optimally produce when it is allowed to adjust plant size.The long-run supply curve P = MC l (y )C MC lpO y yCompare to the short-run supply P = MC s (y, k ) and MC l (y ) = MC s (y , k (y )) The long-run supply curve will be more elastic than the short-run supply curve. The profits that the firm makes in long-run equilibrium have to be at least zero:0)(≥-y c py which meansyy c p )(≥ = vcC LMCLACpO yLong-Run Constant Average CostsOne particular case: the long-run technology of the firm exhibits constant to scale.The long-run average cost curve is constant, c min.The long-run marginal cost curve coincides with the long-run average cost curve.The long-run supply curve will be a horizontal ling at c min.Cc min LMCO y。
