北京大学：《微观经济学》Chapter Twenty Cost Minimization 成本最小化

Chapter twenty Cost Minimization 成本最小化

Chapter Twenty Cost Minimization 成本最小化

Structure o The cost minimization problem ◆ Average costs e Returns to scale and total and average costs o Short run and long run costs

Structure uThe cost minimization problem uAverage costs uReturns to scale and total and average costs uShort run and long run costs

Cost minimization cA firm is a cost -minimizer if it produces any given output level y 20 at smallest possible total cost. cly)denotes the firm's smallest possible total cost for producing y units of output ocl) is the firm's total cost function (总成本函数)

Cost Minimization uA firm is a cost-minimizer if it produces any given output level y ³ 0 at smallest possible total cost. uc(y) denotes the firm’s smallest possible total cost for producing y units of output. uc(y) is the firm’s total cost function （总成本函数）

Cost minimization When the firm faces given input prices w= W1, W2,, wn) the total cost function will be written as y

Cost Minimization uWhen 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 e Consider a firm using two inputs to make one output The production function is y f(xi x2) ◆ Take the output level y≥0 as glven 4 Given the input prices W, and w2, the cost of an input bundle(x1, x2)is W1X1+ W2X2

The Cost-Minimization Problem uConsider a firm using two inputs to make one output. uThe production function is y = f(x1 ,x2 ). uTake the output level y ³ 0 as given. uGiven the input prices w1 and w2 , the cost of an input bundle (x1 ,x2 ) is w1x1 + w2x2

The cost-Minimization problem For given W,, W2 and y, the firms cost-minimization problem is to solve min W1X1+W2X 2 X1,X2≥0 subject to f(x1, X2)=y

The Cost-Minimization Problem uFor 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 ³  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) o 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 uThe 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）. uThe (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 4 Given W1, W2 and y, how is the least costly input bundle located? c And how is the total cost function 成本函数) computed?

Conditional Input Demands uGiven w1 , w2 and y, how is the least costly input bundle located? uAnd how is the total cost function （ 成本函数）computed?

Iso- cost lines(等成本线) PA 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） uA curve that contains all of the input bundles that cost the same amount is an iso-cost curve. uE.g., given w1 and w2 , the $100 iso￾cost line has the equation w x w x 1 1 2 2   100

Iso-cost lines Generally, given W, and w2, the equation of the Sc iso-cost line is WiX1+W2X2=C Le W1 X2 X1+ w2 2 ◆ Slope Is-WW2

Iso-cost Lines uGenerally, given w1 and w2 , the equation of the $c iso-cost line is i.e. uSlope is - w1 /w2 . x w w x c w 2 1 2 1 2    . w x w x c 1 1  2 2 