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# C Dcostminimization

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### C Dcostminimization

1. 1. Cost Minimization Chapter Twenty
2. 2. <ul><li>Given w 1 , w 2 and y, how is the least costly input bundle located? </li></ul><ul><li>How can the conditional factor demands be derived? </li></ul><ul><li>And how is the total cost function computed? </li></ul>The Cost-Minimization Problem
3. 3. The Cost-Minimization Problem The first-order condition for cost minimization is thus Before we have had pMP 1 = w 1 pMP 2 = w 2 Eliminating p yields MP 1 / MP 2 = w 1 / w 2
4. 4. Deriving Cost Functions <ul><li>Solving the problem of cost minimization yields the cost functions </li></ul><ul><li>Cost functions will depend on technology </li></ul>
5. 5. A Cobb-Douglas Example of Cost Minimization <ul><li>A firm’s Cobb-Douglas production function is </li></ul><ul><li>Input prices are w 1 and w 2 . </li></ul><ul><li>What are the firm’s conditional input demand functions? </li></ul>
6. 6. A Cobb-Douglas Example of Cost Minimization At the input bundle (x 1 *,x 2 *) which minimizes the cost of producing y output units: (a) (b) - w 1 /w 2 = - MP 1 /MP 2 and
7. 7. A Cobb-Douglas Example of Cost Minimization At the input bundle (x 1 *,x 2 *) which minimizes the cost of producing y output units: (a) (b) - w 1 /w 2 = - MP 1 /MP 2 and
8. 8. A Cobb-Douglas Example of Cost Minimization At the input bundle (x 1 *,x 2 *) which minimizes the cost of producing y output units: (a) (b) - w 1 /w 2 = - MP 1 /MP 2 and
9. 9. A Cobb-Douglas Example of Cost Minimization (a) (b) From (b), Now substitute into (a) to get So is the firm’s conditional demand for input 1.
10. 10. A Cobb-Douglas Example of Cost Minimization is the firm’s conditional demand for input 2. Since and
11. 11. A Cobb-Douglas Example of Cost Minimization So the cheapest input bundle yielding y output units is
12. 12. A Cobb-Douglas Example of Cost Minimization So the firm’s total cost function is