Chapter Twenty Cost Minimization
Chapter Twenty Cost Minimization
Cost Minimization A firm is a cost-minimizer if it produces any given output level y 0 at smallest possible total cost. c(y)will denote the firm's smallest possible total cost for producing y units of output. c(y)is the firm's total cost function
Cost Minimization A firm is a cost-minimizer if it produces any given output level y 0 at smallest possible total cost. c(y) will denote the firm’s smallest possible total cost for producing y units of output. c(y) is the firm’s total cost function
Cost Minimization When the firm faces given input prices w =(w1,w2,...,wn)the total cost function will be written as C(W1,-.-,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 firm's production function is y=fx1,x2). Take the output level y 0 as given. Take the input prices w and w2 as given.Then the cost of an input bundle (X1,X2)is W1X1+W2×2
The Cost-Minimization Problem Consider a firm using two inputs to make one output. The firm’s production function is y = f(x1 ,x2 ). Take the output level y 0 as given. Take the input prices w1 and w2 as given. Then the cost of an input bundle (x1 ,x2 ) is w1x1 + w2x2
The Cost-Minimization Problem For given w,w,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 *(w1,w2,y) in the least-costly input bundle are the firm's conditional demands for inputs 1 and 2. The firm's total (smallest possible) 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. The firm’s total (smallest possible) 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 For given w1,w2 and y,how are the levels of inputs 1 and 2 in the least costly input bundle located? And how is the firm's total cost function computed?
Conditional Input Demands For given w1 , w2 and y, how are the levels of inputs 1 and 2 in the least costly input bundle located? And how is the firm’s total cost function computed?
Iso-cost Lines A curve which contains all of the input bundles which cost the same amount is an iso-cost curve. E.g.,given wi and w2,the $100 iso- cost line has the equation W1X1+W2x2=100
Iso-cost Lines A curve which contains all of the input bundles which 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 In general,given w and w2,the equation of the $c iso-cost line is W1X1+W2X2=C which rearranges to 1x1+ c X2=- w2 w2 The slope is -w /w2
Iso-cost Lines In general, given w1 and w2 , the equation of the $c iso-cost line is which rearranges to The slope is - w1 /w2 . x w w x c w 2 1 2 1 2 = − + . w x w x c 1 1 + 2 2 =
Iso-cost Lines X2 C”≡W1X1+W2X2 C'≡WX1+W2X2 X1
Iso-cost Lines c’ w1x1+w2x2 c” w1x1+w2x2 c’ < c” x1 x2