DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS                                           ECONOMICS 21
                                   Dartmouth College, Department of Economics: Economics 21, Summer‘02
                                   Dartmouth College, Department of Economics: Economics 21, Summer‘02
                                   Dartmouth College, Department of Economics: Economics 21, Summer‘02
                                          Topic 2: Theory of the Firm
                                          Topic 2: Theory of the Firm
                                                Economics 21, Summer 2002
                                                          Andreas Bentz
                                           Based Primarily on Varian, Ch. 18-25
                                  The Setup
                                  The Setup
                                   „ A firm 
                                       ‹ produces output y, which it can sell for price p(y)
                                           » p(y) is the inverse market demand function
                                       ‹ from quantities of inputs (factors): x , x , …
                                                                            1   2
                                       ‹ input cost (per unit): w , w , …
                                                                1  2
                                   „ How can this firm produce?
                                       ‹ technology
                                   „ How should this firm produce?
                                       ‹ cost minimization
                                   „ How muchshould this firm produce?
                                       ‹ profit maximization
                                                                                                     2
                © Andreas Bentz                                                                                  page 1
             DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS                        ECONOMICS 21
                           Dartmouth College, Department of Economics: Economics 21, Summer‘02
                           Dartmouth College, Department of Economics: Economics 21, Summer‘02
                           Dartmouth College, Department of Economics: Economics 21, Summer‘02
                                             Technology
                                             Technology
                                                Production
                           Intro: Production
                           Intro: Production
                            „ In our problem, the firm’s production 
                              technology is given;
                            „ and: the technology is independent of the 
                              market form (market structure):
                               ‹in particular it has nothing to do with competition or 
                                 firm behavior.
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             © Andreas Bentz                                                            page 2
               DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS                                    ECONOMICS 21
                               Production Function
                               Production Function
                                „ A production function tells you how much 
                                   output (at most) you can get from given 
                                   quantities of inputs (factors).
                                    ‹Example (Cobb-Douglas): f(x , x ) = x a x b. 
                                                                      1   2    1   2
                                                                                0.5  0.5
                                                                     ‹Here: x      x    .
                                                                               1     2
                                                                                             5
                               Short-Run Production Function
                               Short-Run Production Function
                                „ In the short run, not all inputs can be varied: at least 
                                   one input is fixed.
                                    ‹ Suppose input 2 is fixed at x = x : y = f(x , x )
                                                                2   2      1  2
                                „ We can still vary output by varying input 1. This is the 
                                   short-run production function.
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               © Andreas Bentz                                                                          page 3
              DARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS                              ECONOMICS 21
                             Marginal Product
                             Marginal Product
                              „ Suppose input 2 is held constant: how does output 
                                 change as we change input 1?
                                  ‹ The marginal product of input 1 is the partial derivative of the 
                                    production function with respect to input 1.
                                  ‹ Example: Holding x constant at x = 2, how does u change as 
                                                    2           2
                                    we change x by a little, i.e. what is the slope of the blue line?
                                              1
                                                                                      7
                             Marginal Product, cont’d
                             Marginal Product, cont’d
                              „ Formally, the marginal product of input 1 of the 
                                 production function f(x1, x2) is:
                                                   +∆         −             ∂
                                              f (x     x ,x )  f (x ,x )     f (x ,x )
                                 MP = lim        1      1  2       1  2  =       1  2
                                     1  ∆ →
                                         x1 0            ∆                     ∂
                                                           x1                   x1
                              „ That is, at which rate does output increase as 
                                 this firm uses more of input 1?
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              © Andreas Bentz                                                                    page 4