## Examples and exercises on the output expansion path

### Example: a production function with fixed proportions

Consider the production function F (z1z2) = min{z1z2}. 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 z2 = z1.

### Example: a production function with fixed proportions

Consider the production function F (z1z2) = min{z1/2, z2}. 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 2y 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 z2 = z1/2.

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

A general production function with fixed proportions takes the form F (z1z2) = min{az1bz2} 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 z2 = (a/b)z1.

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

Consider the production function F (z1z2) = z1 + z2, in which the inputs are perfect substitutes. Given the shape of its isoquants, its output expansion path is
• the z2 axis if w1 > w2 (if the price of input 1 exceeds that of input 2 then the firm uses none of input 1)
• the z1 axis if w1 < w2 (if the price of input 2 exceeds that of input 1 then the firm uses none of input 2)
• the set of all pairs (z1z2) if w1 = w2 (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 (z1z2) = az1 + bz2, for some constants a and b. Given the shape of its isoquants, its output expansion path is
• the z2 axis (the firm uses none of input 1) if w1 > (a/b)w2
• the z1 axis (the firm uses none of input 2) if w1 < (a/b)w2
• the set of all pairs (z1z2) (every combination of inputs is optimal) if w1 = (a/b)w2. [That is, in this case there is not a single output expansion path.]

### Example: a Cobb-Douglas production function

Consider the Cobb-Douglas production function F (z1z2) = z11/2z21/2. The conditional input demand functions for this production function are
z1 = y(w2/w1)1/2 and z2 = y(w1/w2)1/2.
Thus the output expansion path satisfies z2/z1 = w1/w2, or
z2 = (w1/w2)z1.
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.