## Example of long run and short run cost functions

### Example: a production function with fixed proportions

Consider the fixed proportions production function *F* (*z*_{1}, *z*_{2}) = min{*z*_{1}, *z*_{2}} (one worker and one machine produce one unit of output). The long run
total cost function for this production function is given by
TC(*y*,*w*_{1},*w*_{2}) = *w*_{1}*y* + *w*_{2}*y* = (*w*_{1} + *w*_{2})*y*.

Its short run total cost of production when the amount of input 2 is fixed at *k* is
STC_{k}(*y*) = | *w*_{1}*y* + *w*_{2}*k* | if *y* *k* |

| | if *y* > *k*. |

These functions are shown in the following figure.

### Example: a production function in which the inputs are perfect substitutes

Consider the production function *F* (*z*_{1}, *z*_{2}) = *z*_{1} + *z*_{2}, in which the inputs are perfect substitutes. The long run total cost function for this
production function is given by
| *w*_{1}*y* | if *w*_{1} < *w*_{2} |

TC(*y*,*w*_{1},*w*_{2}) = | *w**y* | if *w*_{1} = *w*_{2} = *w* |

| *w*_{2}*y* | if *w*_{1} > *w*_{2} |

Its short run total cost of production when the amount of input 2 is fixed at *k* is
STC_{k}(*y*) = | *w*_{2}*k* | if *y* *k* |

| *w*_{1}(*y* *k*) + *w*_{2}*k* | if *y* > *k*. |

Note that in this case if *w*_{1} < *w*_{2} then there is **no** level of output for which short run total cost is equal to the long run total cost. In this case the firm would like to use only input 1 to produce any output, so if *k* > 0 there is no output for which the short run total cost is equal to the
long run total cost. The short and long run cost functions in this case are shown in the following figure.

### Example: a Cobb-Douglas production function

Consider the production function *F* (*z*_{1}, *z*_{2}) = *z*_{1}^{1/2}*z*_{2}^{1/2}. The long run total cost function for this production
function is given by
TC(*y*,*w*_{1},*w*_{2}) = 2*y*(*w*_{1}*w*_{2})^{1/2}.

Its short run total cost of production when the amount of input 2 is fixed at *k* is

STC_{k}(*y*) =*w*_{1}*y*^{2}/*k* + *w*_{2}*k*.

Note that TC is a linear function of *y* while STC is a quadratic function. (Remember that *w*_{1} and *w*_{2} are fixed.) The functions are shown in the following figure.

Copyright © 1997 by Martin J. Osborne