## Examples and exercises on the output expansion path

### Example: a production function with fixed proportions

Consider the production function *F* (*z*_{1}, *z*_{2}) = min{*z*_{1}, *z*_{2}}. Given the shape of its isoquants, the output expansion path of this production function is a ray from the origin, as in the
following figure. For any input prices, the firm uses *y* units of each input to produce *y* units of output (see its conditional input demands), so that its output expansion path is the line *z*_{2} = *z*_{1}.

### Example: a production function with fixed proportions

Consider the production function *F* (*z*_{1}, *z*_{2}) = min{*z*_{1}/2, *z*_{2}}. Given the shape of its isoquants, the output expansion path of this production function is a ray from the origin, as in the
following figure. For any input prices, the firm uses 2*y* units of input 1 and *y* units of input 2 to produce *y* units of output (see its conditional input demands), so that its output expansion path is the line *z*_{2} = *z*_{1}/2.

### Example: a general production function with fixed proportions

A general production function with fixed proportions takes the form *F* (*z*_{1}, *z*_{2}) = min{*a**z*_{1}, *b**z*_{2}} for some constants *a* and *b*. Given the shape of its isoquants,
the output expansion path of this production function is, as in the previous case, a ray from the origin. For any input prices, the firm uses *y*/*a* units of input 1 and *y*/*b* units of input 2 to produce *y* units of output (see its conditional input demands), so that its output expansion path is the line
*z*_{2} = (*a*/*b*)*z*_{1}.

### 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. Given the shape of its isoquants, its output expansion
path is
- the
*z*_{2} axis if *w*_{1} > *w*_{2} (if the price of input 1 exceeds that of input 2 then the firm uses none of input 1)
- the
*z*_{1} axis if *w*_{1} < *w*_{2} (if the price of input 2 exceeds that of input 1 then the firm uses none of input 2)
- the set of all pairs (
*z*_{1}, *z*_{2}) if *w*_{1} = *w*_{2} (if the prices of the inputs are the same then every combination of inputs is optimal). [That is, in this case there is not a single output expansion **path**.]

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

The general form of a production function in which the inputs are perfect substitutes is *F* (*z*_{1}, *z*_{2}) = *a**z*_{1} + *b**z*_{2}, for some constants *a* and *b*. Given the shape of
its isoquants, its output expansion path is
- the
*z*_{2} axis (the firm uses none of input 1) if *w*_{1} > (*a*/*b*)*w*_{2}
- the
*z*_{1} axis (the firm uses none of input 2) if *w*_{1} < (*a*/*b*)*w*_{2}
- the set of all pairs (
*z*_{1}, *z*_{2}) (every combination of inputs is optimal) if *w*_{1} = (*a*/*b*)*w*_{2}. [That is, in this case there is not a single output expansion **path**.]

Consider the Cobb-Douglas production function *F* (*z*_{1}, *z*_{2}) = *z*_{1}^{1/2}*z*_{2}^{1/2}. The conditional input demand functions for this production
function are
*z*_{1} = *y*(*w*_{2}/*w*_{1})^{1/2} and *z*_{2} = *y*(*w*_{1}/*w*_{2})^{1/2}.

Thus the output expansion path satisfies *z*_{2}/*z*_{1} = *w*_{1}/*w*_{2}, or
*z*_{2} = (*w*_{1}/*w*_{2})*z*_{1}.

As in all the previous examples, the output expansion path is thus a ray from the origin.
**NOTE**: The output expansion path is not *necessarily* a ray from the origin! It happens to take this form for all these examples, but in general it can take a wide variety of forms.

Copyright © 1997 by Martin J. Osborne