Chapter twenty Cost Minimization 成本最小化
Chapter Twenty Cost Minimization 成本最小化
Structure The cost minimization problem Average costs Returns to scale and total and average costs Short run and long run costs
Structure The cost minimization problem Average costs Returns to scale and total and average costs Short run and long run costs
Cost minimization A firm is a cost -minimizer if it produces any given output level y >0 at smallest possible total cost cly)denotes the firm's smallest possible total cost for producing y units of output cly)is the firm's total cost function 总成本函数)
Cost Minimization A firm 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 （ 总成本函数）
Cost minimization When the firm faces given input prices w=(Wi,W2,--,Wn) the total cost function will be written as C(W Wny)
Cost Minimization When the 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 Consider a firm using two inputs to make one output The production function is y=f(x1,x2) Take the output level y 20 as given Given the input prices W, and w2, the cost of an input bundle x1, x2 is W11+W2X2
The Cost-Minimization Problem Consider a 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 For given Wi, W2 and y, the firm's cost-minimization problem is to solve min W1X1+w2X2 X1,X2≥0 subject to f (X1, X2)=y
The Cost-Minimization Problem For given w1 , w2 and y, the firm’s cost-minimization problem is to solve min x ,x w x w x 1 2 0 1 1 2 2 + subject to f(x ,x ) y. 1 2 =
The Cost-Minimization problem The levels x,*W1,W2,y) and x, W,, W2,y) in the least-costly input bundle are the firm's conditional demands for inputs 1and2(条件要素需求) The(smallest possible) total cost for producing y output units is therefore c(W1,w2,y)=w1X1(w1,w2,y) +W2X2(W1,w2,y
The Cost-Minimization Problem The levels 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 （条件要素需求）. The (smallest possible) total cost for producing y output units is therefore c w w y w x w w y w x w w y ( , , ) ( , , ) ( , , ). * * 1 2 1 1 1 2 2 2 1 2 = +
Conditional Input demands Given W1, W2 and y, how is the least costly input bundle located? And how is the total cost function 成本函数) computed?
Conditional Input Demands Given w1 , w2 and y, how is the least costly input bundle located? And how is the total cost function （ 成本函数）computed?
Iso- cost lines(等成本线) A curve that contains all of the input bundles that cost the same amount is an iso-cost curve E.g given W, and w2, the $100 iso cost line has the equation w1X1+w2X2=100
Iso-cost Lines （等成本线） A curve 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 equation w1 x1 + w2 x2 = 100
Iso-cost Lines Generally, given W, and w2, the equation of the sc iso-cost line is W1X1FW2X2=C .e W1 X2 w2 X2 Slope is -W,w2
Iso-cost Lines Generally, given w1 and w2 , the equation of the $c iso-cost line is i.e. Slope is - w1 /w2 . x w w x c w 2 1 2 1 2 = − + . w x w x c 1 1 + 2 2 =
