The Cost-Minimization problem At an interior cost-min input bundle: < 1(a)f(x1, X2)=yand (b slope of isocost= slope of isoquant; i. e MP TRS at(x1,X2) W2 MP 2 f(x12x2)≡y 大 X1
The Cost-Minimization Problem x1 x2 f(x1 ,x2 ) y’ x1 * x2 * At an interior cost-min input bundle: (a) and (b) slope of isocost = slope of isoquant; i.e. f(x ,x ) y * * 1 2 = − = = − w w TRS MP MP at x x 1 2 1 2 1 2 ( , ). * *
A Cobb-Douglas Example of cost Minimization A firm's Cobb-Douglas production function is y=f(x1,x2)=x13x23 Input prices are w, and w2 What are the firm's conditional input demand functions?
A Cobb-Douglas Example of Cost Minimization A firm’s Cobb-Douglas production function is Input prices are w1 and w2 . What are the firm’s conditional input demand functions? y = f(x ,x ) = x x . / / 1 2 1 1 3 2 2 3
A Cobb-Douglas Example of cost Minimization At the input bundle x* X 2 )which minimizes the cost of producing y output units: a 2、2/3 y=(x1)(x2) and (b) 2/3 ay/ox1(1/3)(X1)2°(x2) W2y/0x2(2/3(x1)13(x2)-13 2
A Cobb-Douglas Example of Cost Minimization At the input bundle (x1 *,x2 *) which minimizes the cost of producing y output units: (a) (b) y = (x ) (x ) * / * / 1 1 3 2 2 3 and − = − = − = − − − w w y x y x x x x x x x 1 2 1 2 1 2 3 2 2 3 1 1 3 2 1 3 2 1 1 3 2 3 2 / / ( / )( ) ( ) ( / )( ) ( ) . * / * / * / * / * *
A Cobb-Douglas Example of cost Minimization 1/32/3 W1 X2 (a)y=(x1) x2) w2 2X 2w1☆ From(b), w2 Now substitute into(a to get 2|3(2w 2/3 兴、1/3 y=(X1 wxy W2 2/3 W2 So X1=\2w1 y is the firm's conditional demand for input 1
A Cobb-Douglas Example of Cost Minimization y = (x ) (x ) * / * / 1 1 3 2 2 3 w w x x 1 2 2 2 1 = * * . (a) (b) From (b), x w w 2 x 1 2 1 * 2 * = . Now substitute into (a) to get y x w w x w w = x = ( ) . * / * / / * 1 1 3 1 2 1 2 3 1 2 2 3 1 2 2 x w w 1 y 2 1 2 3 2 * / = So is the firm’s conditional demand for input 1