The cost-Minimization problem At an interior cost-min input bundle 21(a)f(x1,x2)=yand (b slope of isocost slope of isoquant; i.e. W1_TRS MP MP at(X1, X2). w2 2 f(x1x2)≡y X1 X1
The Cost-Minimization Problem x1 x2 f(x1 ,x2 ) º 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 f(x1,x2)=x13x23 + Input prices are W, and w2 o What are the firm's conditional input demand functions?
A Cobb-Douglas Example of Cost Minimization uA firm’s Cobb-Douglas production function is uInput prices are w1 and w2 . uWhat 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,*, x2*)which minimizes the cost of producing y output units: a y=(x1)1/3 、2/3 )°(x2) and (b) W1=0y/8x1(1/3)(x1)-213(x2)23 W20y/0x2(2/3)x1)13(x2)-13 *2 2x1
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 (a)y=(x1 1/3/2/3 2 W2 2x1 From(b),2 2w1 W2 Now substitute into(a to get 2/3 (x)l/3/ 2W 2w W2 W2 2/3 w2 So X1=\2w 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 1 2 * * (a) (b) . From (b), x w w x 2 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