微观经济学：现代观点Ch20

Chapter TwentyCost MinimizationCost MinimizationAfirm is a cost-minimizer if it produces any given output level y  0 at smallest possible total cost.  c(y) denotes the firm’s smallest possible total cost for producing y units of output.  c(y) is the firm’s total cost function.Cost Minimization Whenthe firm faces given input prices w = (w1,w2,…,wn) the total cost function will be written as c(w1,…,wn,y).The Cost-Minimization Problem Considera firm using two inputs to make one output.  The production function is y = f(x1,x2).  Take the output level y  0 as given.  Given the input prices w1 and w2, the cost of an input bundle (x1,x2) is w1x1 + w2x2.The Cost-Minimization Problem Forgiven w1, w2 and y, the firm’s cost-minimization problem is to solve min w 1x1  w 2x 2 x1 ,x 2  0 subject to f ( x1 , x 2 )  y.The Cost-Minimization Problem Thelevels x1*(w1,w2,y) and x1*(w1,w2,y) in the least-costly input bundle are the firm’s conditional demands for inputs 1 and 2.  The (smallest possible) total cost for producing y output units is therefore * c( w 1 , w 2 , y )  w 1x1 ( w 1 , w 2 , y ) *  w 2x 2 ( w 1 , w 2 , y ).Conditional Input Demands Givenw1, w2 and y, how is the least costly input bundle located?  And how is the total cost function computed?Iso-cost LinesAcurve that contains all of the input bundles that cost the same amount is an iso-cost curve.  E.g., given w1 and w2, the $100 isocost line has the equationw1x1  w 2x 2  100.Iso-cost Lines Generally,given w1 and w2, the equation of the $c iso-cost line isi.e.w1x1  w 2x 2  cw1 c x2   x1  . w2 w2is - w1/w2. SlopeIso-cost Linesx2c”  w1x1+w2x2 c’  w1x1+w2x2c’ < c” x1Iso-cost Linesx2 Slopes = -w1/w2.c”  w1x1+w2x2 c’  w1x1+w2x2c’ < c” x1The y’-Output Unit Isoquantx2All input bundles yielding y’ units of output. Which is the cheapest?f(x1,x2)  y’x1The Cost-Minimization Problemx2All input bundles yielding y’ units of output. Which is the cheapest?f(x1,x2)  y’x1The Cost-Minimization Problemx2All input bundles yielding y’ units of output. Which is the cheapest?f(x1,x2)  y’x1The Cost-Minimization Problemx2All input bundles yielding y’ units of output. Which is the cheapest?f(x1,x2)  y’x1The Cost-Minimization Problemx2All input bundles yielding y’ units of output. Which is the cheapest?x 2*f(x1,x2)  y’x 1* x1The Cost-Minimization Problemx2At an interior cost-min input bundle: * * (a) f ( x1 , x 2 )  y x 2*f(x1,x2)  y’x 1* x1The Cost-Minimization Problemx2At an interior cost-min input bundle: * * (a) f ( x1 , x 2 )  y  and (b) slope of isocost = slope of isoquantx 2*f(x1,x2)  y’x 1* x1The Cost-Minimization Problemx2At an interior cost-min input bundle: * * (a) f ( x1 , x 2 )  y  and (b) slope of isocost = slope of isoquant; i.e. w1 MP1 *   TRS   at ( x* , x 1 2 ). w2 MP2f(x1,x2)  y’x 1* x1x 2*A Cobb-Douglas Example of Cost MinimizationAfirm’s Cobb-Douglas production function is / 3 2/ 3 y  f ( x1 , x 2 )  x1 1 x2 .  Input prices are w1 and w2.  What are the firm’s conditional input demand functions?A Cobb-Douglas Example of Cost MinimizationAt the input bundle (x1*,x2*) which minimizes the cost of producing y output units: (a) y  ( x* )1/ 3 ( x* ) 2/ 3 and1 2(b)w1  y /  x1   w2  y /  x2* 2 / 3 * 2 / 3 (1 / 3)( x1 ) (x2 )  1/ 3 * 1/ 3 ( 2 / 3)( x* ) (x2 ) 1 * x2  . * 2x1A Cobb-Douglas Example of Cost Minimization* 1/ 3 * 2/ 3 (a) y  ( x1 ) ( x 2 )w 1 x* 2 .  (b) w 2 2x* 1A Cobb-Douglas Example of Cost Minimization* 1/ 3 * 2/ 3 (a) y  ( x1 ) ( x 2 )2w 1 * * x1 . From (b), x 2  w2w 1 x* 2 .  (b) w 2 2x* 1A Cobb-Douglas Example of Cost Minimization* 1/ 3 * 2/ 3 (a) y  ( x1 ) ( x 2 )2w 1 * * x1 . From (b), x 2  w2w 1 x* 2 .  (b) w 2 2x* 1Now substitute into (a) to get 2/ 3 * 1/ 3  2w 1 *  y  ( x1 )  x1   w2 A Cobb-Douglas Example of Cost Minimization* 1/ 3 * 2/ 3 (a) y  ( x1 ) ( x 2 )2w 1 * * x1 . From (b), x 2  w2w 1 x* 2 .  (b) w 2 2x* 1Now substitute into (a) to get 2/ 3 2/ 3  2w 1  * 1/ 3  2w 1 *  * y  ( x1 )  x1   x1 .   w2   w2 A Cobb-Douglas Example of Cost Minimization* 1/ 3 * 2/ 3 (a) y  ( x1 ) ( x 2 )2w 1 * * x1 . From (b), x 2  w2w 1 x* 2 .  (b) w 2 2x* 1Now substitute into (a) to get 2/ 3 2/ 3  2w 1  * 1/ 3  2w 1 *  * y  ( x1 )  x1   x1 .   w2   w2 *  w2  So x1     2w 1  2/ 3y is the firm’s conditional demand for input 1.A Cobb-Douglas Example of Cost Minimization2w 1 * *  w2  * x1 and x1   Since x 2    2w 1  w2 2/ 3 1/ 3  2w 1  2w 1  w 2  * x2  y    y  w2  w 2  2w 1 2/ 3yis the firm’s conditional demand for input 2.A Cobb-Douglas Example of Cost MinimizationSo the cheapest input bundle yielding y output units is* * x1 ( w 1 , w 2 , y ), x 2 ( w 1 , w 2 , y )  w  2/ 3  2w  1/ 3  1   2  y,  y .    2w 1     w 2  Conditional Input Demand Curvesx2Fixed w1 and w2.y yyx1Conditional Input Demand Curvesx2Fixed w1 and w2.yyy x* 2 ( y )y yx* 2x* 2 ( y )yx1yx* 1 ( y )x* 1 ( y )x* 1Conditional Input Demand Curvesx2Fixed w1 and w2.yy yy x* 2 ( y )x* 2 ( y )x* 2x* 2 ( y ) x* 2 ( y )y yyx1y yx* 1 ( y ) x* 1 ( y )x* 1 ( y ) x* 1 ( y )x* 1Conditional Input Demand Curvesx2Fixed w1 and w2.yyy yx* 2 ( y ) x* 2 ( y ) x* 2 ( y )* x* ( y ) 1  x1 ( y ) x* 1 ( y )* * ( y )  x ( y )  x y x* 2 2 2 *y y yx 2 ( y )yx1y y* * x* ( y )  x ( y )  x 1 1 1 * x1 ( y )Conditional Input Demand Curvesx2Fixed w1 and w2.yyoutput expansion pathx* 2 ( y ) x* 2 ( y ) x* 2 ( y )* x* ( y ) 1  x1 ( y ) x* 1 ( y )y y* * ( y )  x ( y )  x y x* 2 2 2 *x 2 ( y )y y yyx1y y* * x* ( y )  x ( y )  x 1 1 1 * x1 ( y )Conditional Input Demand Curvesx2Fixed w1 and w2.yyCond. demand for input 2output expansion pathx* 2 ( y ) x* 2 ( y ) x* 2 ( y )* x* ( y ) 1  x1 ( y ) x* 1 ( y )y y* * ( y )  x ( y )  x y x* 2 2 2 *x 2 ( y )y y yyx1Cond. demand y for y input 1 * * x* ( y )  x ( y )  x 1 1 1 *x1 ( y )A Cobb-Douglas Example of Cost MinimizationFor the production function/ 3 2/ 3 y  f ( x1 , x 2 )  x1 1 x2the cheapest input bundle yielding y output units is* x* ( w , w , y ), x 1 1 2 2 ( w1 , w 2 , y )  w  2/ 3  2w  1/ 3  1   2  y,  y .    2w 1     w 2  A Cobb-Douglas Example of Cost MinimizationSo the firm’s total cost function is* c( w 1 , w 2 , y )  w 1x* ( w , w , y )  w x 1 1 2 2 2 ( w1 , w 2 , y )A Cobb-Douglas Example of Cost MinimizationSo the firm’s total cost function is* * c( w 1 , w 2 , y )  w 1x1 ( w 1 , w 2 , y )  w 2x 2 ( w 1 , w 2 , y ) 2/ 3 1/ 3  w2   2w 1   w1  y  w2   y  2w 1   w2 A Cobb-Douglas Example of Cost MinimizationSo the firm’s total cost function is* c( w 1 , w 2 , y )  w 1x* ( w , w , y )  w x 1 1 2 2 2 ( w1 , w 2 , y ) w2   w1    2w 1   1    22/ 32/ 3 2w 1  y  w2   w2 1/ 3y/ 3 2/ 3 1/ 3 1/ 3 2/ 3 w1 w y  2 w1 w 2 y 1 2A Cobb-Douglas Example of Cost MinimizationSo the firm’s total cost function is* c( w 1 , w 2 , y )  w 1x* ( w , w , y )  w x 1 1 2 2 2 ( w1 , w 2 , y ) w2   w1    2w 1   1    22/ 32/ 3 2w 1  y  w2   w2 1/ 3y/ 3 2/ 3 1/ 3 1/ 3 2/ 3 w1 w y  2 w1 w 2 y 1 21/ 3 2  w 1w 2    3   y.4A Perfect Complements Example of Cost Minimization Thefirm’s production function isy  min{4x1 , x 2 }. Inputprices w1 and w2 are given.  What are the firm’s conditional demands for inputs 1 and 2?  What is the firm’s total cost function?A Perfect Complements Example of Cost Minimizationx2 4x1 = x2min{4x1,x2}  y’x1A Perfect Complements Example of Cost Minimizationx2 4x1 = x2min{4x1,x2}  y’x1A Perfect Complements Example of Cost Minimizationx2 4x1 = x2 Where is the least costlyinput bundle yielding y’ output units?min{4x1,x2}  y’x1A Perfect Complements Example of Cost Minimizationx2 4x1 = x2 Where is the least costlyinput bundle yielding y’ output units?x2* = ymin{4x1,x2}  y’x1* = y/4x1A Perfect Complements Example of Cost MinimizationThe firm’s production function is y  min{4x1 , x 2 } and the conditional input demands arey * x1 ( w 1 , w 2 , y )  4* and x 2 ( w 1 , w 2 , y )  y.A Perfect Complements Example of Cost MinimizationThe firm’s production function is y  min{4x1 , x 2 } and the conditional input demands arey * x1 ( w 1 , w 2 , y )  4* and x 2 ( w 1 , w 2 , y )  y.So the firm’s total cost function is * c( w 1 , w 2 , y )  w 1x1 ( w 1 , w 2 , y )*  w 2x 2 ( w 1 , w 2 , y )A Perfect Complements Example of Cost MinimizationThe firm’s production function is y  min{4x1 , x 2 } and the conditional input demands arey * x1 ( w 1 , w 2 , y )  4* and x 2 ( w 1 , w 2 , y )  y.So the firm’s total cost function is * c( w 1 , w 2 , y )  w 1x1 ( w 1 , w 2 , y ) w 2x* 2 ( w1 , w 2 , y ) y  w1   w1  w 2y    w 2  y.  4  4Average Total Production Costs Forpositive output levels y, a firm’s average total cost of producing y units is c( w 1 , w 2 , y ) AC( w1 , w 2 , y )  .yReturns-to-Scale and Av. Total Costs Thereturns-to-scale properties of a firm’s technology determine how average production costs change with output level.  Our firm is presently producing y’ output units.  How does the firm’s average production cost change if it instead produces 2y’ units of output?Constant Returns-to-Scale and Average Total Costs Ifa firm’s technology exhibits constant returns-to-scale then doubling its output level from y’ to 2y’ requires doubling all input levels.。