Production. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem.

Transcription

1 Producer Theory

2 Producton ASSUMPTION 2.1 Propertes of the Producton Set The producton set Y satsfes the followng propertes 1. Y s non-empty If Y s empty, we have nothng to talk about 2. Y s closed A set s closed f t contans ts boundary. We need ths for the soluton exstence n the proft maxmzaton problem. 3. No free lunch You cannot produce output wthout usng any nputs. In other words, any feasble producton plan y must have at least one negatve component. 4. Free dsposal The frm can always throw away nputs for free. For any pont n Y, ponts that use less of all components are also n Y. Thus f y Y, any pont below and to the left s also n Y (n the two dmensonal model). Ths s mostly a techncal assumpton and ensures that Y s not bounded below whch has somethng to do wth the second-order condtons n the proft maxmzng problem.

3 Producton ASSUMPTION 2.2 Propertes of the producton functon n The producton functon f : + + s contnuous, strctly ncreasng and strctly n quasconcave on +, and f (0) = 0.

4 Elastcty of Substtuton DEFINITION 2.1 The Elastcty of Substtuton For a producton functon f ( x ), the elastcty of substtuton between nputs and j at the pont x s defned as dln( xj / x) d( xj / x) f( x) / f j( x) σ, j= = dln( f ( x)/ f ( x)) x / x d( f ( x)/ f ( x)) j j j where f and f j are the margnal products of nputs and j.

5 ΤΗΕΟREM 2.1 Lnear Homogeneous Producton Functons are Concave Let : n f + + be a contnuous, ncreasng and quasconcave producton functon, wth f(0) = 0. Suppose f(.) s homogeneous of degree 1. Then f(.) s a concave functon.

6 Returns to scale DEFINITION 2.2 (Global) Returns to Scale Α producton functon f(x) has the property of 1. CRS f f ( tx) = tf ( x), t > 0, x 2. IRS f f( tx) > tf( x), t > 1, x 3. DRS f f ( tx) < tf ( x), t > 1, x

7 Local Returns to scale DEFINITION 2.3 (Local) Returns to Scale The elastcty of scale at pont x s defned as [ f tx ] 1 dln ( ) = μ ( x) = lm = t 1 f ( xx ) dln( t) f( x) = 1 LCRS μ( x) > 1 LIRS < 1 LDRS The elastcty of scale and the output elastctes of the nputs are related as follows: n μ( x) = μ ( x) n = 1

8 Homothetc Functon DEFINITION 2.3 Homothetc Functon Α homothetc functon s a postve monotonc transformaton of a functon that s homogeneous of degree 1.

9 COST

10 ΤΗΕΟREM 2.2 Propertes of the Cost Functon If f(.) s contnuous and strctly ncreasng, then (, ) cwy s: 1. zero when y = 0 2. contnuous on w and y 3. Homogeneous of degree 1 n w 4. For all w >> 0 5. ncreasng n w 6. concave n w, strctly ncreasng n y Moreover, f f(.) s strctly quasconcave we have 7. Shephard s lemma: cwy (, ) s dfferentable n w and cw dw 0 0 (, y) 0 0 (, y ) x w = = 1,... n

11 ΤΗΕΟREM 2.3 Propertes of the Condtonal Input Demands Suppose the producton functon f(.) s contnuous, strctly ncreasng and strctly quasconcave on, and f (0) = 0 contnuously dfferentable. Then n + 1. x( wy, ) s homogeneous of degree zero n w 2. The substtuton matrx, defned and denoted x1( w, y) x1( w, y)... w1 w n *. σ ( wy, ) =. xn( w, y) xn( w, y)... w1 wn and that the assocated cost functon s twce partcular the negatve semdefnteness property mples that s symmetrc and negatve semdefnte. In x ( w, y) w 0,

12 ΤΗΕΟREM 2.4 Cost and Condtonal Input Demands when Producton s Homethetc 1. When the producton functon s contnuous, strctly ncreasng and strctly n quasconcave on +, and f (0) = 0 and t s homothetc (a) the cost functon s multplcatvely separable n nput prces and output and can be wrtten cwy (, ) = h( y) cw (,1), where h(y) s strctly ncreasng functon and cw (,1) s the unt cost functon (the cost of 1 unt of output) (b) the condtonal nput demands are multplcatvely separable n nput prces and output and can be wrtten xwy (, ) = h( y) xw (,1), where h(y) s strctly ncreasng functon and x( w,1) s the condtonal nput demand for 1 unt of output. 2. When the producton functon s homogeneous of degree a > 0 (a) (b) cwy y cw 1/ a (, ) = (,1) x wy y xw 1/ a (, ) = (,1)

13 DEFINITION 2.4 The Short-Run Cost Functon Let the producton functon be f( x, x ). Suppose that x s a subvector of varable nputs and x s a subvector of fxed nputs. Let w and w be the assocated nput prces for the varable and fxed nputs, respectvely. The short-run total cost functon s defned as sc( w, w, y; x) mn wx + wx s. t. f ( x, x) y x If x( wwyx,, ; ) solves ths mnmzaton problem, then sc( w, w, y; x) wx( w, w, y; x) + wx total varable cost total fxed cost

14 THE COMPETITIVE FIRM

15 PROFIT MAXIMIZATION wth producton sets max py s.t. y Y y m max py s.t. F(y)=0 y m y( p ) frm s net supply functon proft functon π ( p) = py( p)

16 PROFIT MAXIMIZATION wth producton sets

17 ΤΗΕΟREM 2.5 Propertes of the Net Supply Functon If the producton set Y satsfes assumpton 2.1, then the net supply functon y( p ) has the followng propertes: (a) y( p ) s homogeneous of degree 0 n p The producton set Y s not affected by re-scalng all prces by the same amount, hence the optmal soluton of the PMP does not change. y( p) y( λ p), λ > 0. (b) If Y s a convex set, y( p ) s a convex set. If Y s strctly convex, then y( p ) s a sngle pont (a vector). The reason for ths s the same as n the UMP. Recall that a strctly convex set Y corresponds to decreasng returns technology.

18 PROFIT MAXIMIZATION wth a sngle output Profts = Revenues Cost Revenues = R( y) = py Cost = wx max py wx s.t. f ( x) y x n +, y 0

19 ΤΗΕΟREM 2.6 Propertes of the Proft Functon A. If the producton set Y satsfes assumpton 2.1, then the proft functon π ( p) followng propertes: has the 1. ncreasng n p (whch means ncreasng n output prces and decreasng n nput prces, snce n π ( p) = py = p1y py... p jy j... pmym ) 2. homogeneous of degree 1 n p 3. convex n p 4. (Hotellng s lemma) If y( p ) s a sngle pont (mpled when Y s a strctly convex set), then π (.) s dfferentable at p and π ( p) p = y ( p), = 1,... m

20 B. If the producton functon f(.) satsfes assumpton 2.2, then the proft functon π ( p, w), where well-defned, s contnuous and 1. ncreasng n p and decreasng n w 2. homogeneous of degree 1 n (p,w) 3. convex n (p,w) 4. dfferentable n (p,w). Moreover, π ( pw, ) p π ( pw, ) w = y( p, w) = x ( pw, ) = 1,..., n

21 ΤΗΕΟREM 2.7 Propertes of Output Supply and Input Demand Functons Let π ( p, w) be a twce contnuously dfferentable proft functon for some compettve frm. Then 1. Homogenety of degree 0: y( tp, tw) = y( pw, ), t> 0 x ( tp, tw) = x ( p, w), t > 0, = 1,2,... n The substtuton matrx y( p, w) y( p, w) y( p, w)... p w1 w n x1( pw, ) x1( pw, ) x1( pw, )... p w1 w n xn( pw, ) xn( pw, ) xn( pw, )... p w1 w n y( p, w) 0 2. Own-prce effects: p x ( p, w) w s symmetrc and postve semdefnte 0 = 1, 2,.., n

22 ΤΗΕΟREM 2.8 The Short-Run Proft Functon Let the producton functon be f ( xx, ), where x s a subvector of varable nputs and x s a subvector of fxed nputs. Let w and w be the assocated nput prces for varable and fxed nputs, respectvely. The short run, proft functon s defned as π ( pwwx,,, ) max py wx wx s.t. f( xx, ) y yx, The solutons y( p, w, w, x ) and x( pwwx,,, ) are called the short-run output supply and varable nput demand functons, respectvely. For all p > 0 and 0 w >>, ( p, wwx,, ) π where well-defned s contnuous n p and w, ncreasng n p, decreasng n w, and convex n (p,w). If π ( p, wwx,, ) s twce contnuously dfferentable, y( p, w, w, x ) and xpwwx (,,, ) possess all three propertes lsted n Theorem 2.7 wth respect to output and varable nput prces.