## Combinations of inputs that yield the same output: isoquants

The set of all pairs (*z*_{1}, *z*_{2}) of inputs that yield the output *y* is the *y*-**isoquant**. Given the production function *F* ,
the *y*-isoquant is thus the set of all pairs (*z*_{1}, *z*_{2}) for which
*y* = *F* (*z*_{1}, *z*_{2}).

## "Typical" isoquants

Isoquants may take a wide variety of forms. When we draw a "typical" one we usually assume that it is smooth and convex to the origin, as in the following figure. It is easy to give examples of technologies for which the isoquants do *not* take this form, but in the theory we treat such isoquants as typical.

The absolute value of the slope of the isoquant through the input pair (*z*_{1}, *z*_{2}) is called the **marginal rate of technical substitution** (MRTS) between input 1 and input 2 at (*z*_{1}, *z*_{2}). Its
value tells us how many extra units of input 2 we need to use in order to produce the same output as before when we reduce the number of units of input 1 by 1.
For a typical production function, with isoquants convex to the origin, the MRTS diminishes as more of input 1 is used. We say that such a production function has a **diminishing marginal rate of technical substitution**.

Hold the amount *z*_{2} of input 2 fixed and decrease the amount *z*_{1} of input 1 by *z*_{1}. Then the amount of output decreases by
MP_{1}*z*_{1} units,

where MP_{1} is the marginal product of input 1.
Now increase the amount of *z*_{2} of input 2 by *z*_{2}. Then the amount of output increases by

MP_{2}*z*_{2} units,

where MP_{2} is the marginal product of input 2.
Along an isoquant output is constant, so the decrease in output caused by the decrease in *z*_{1} must be exactly equal to the increase caused by the increase in *z*_{2}. That is,

MP_{1}*z*_{1} = MP_{2}*z*_{2},

or
*z*_{2}/*z*_{1} = MP_{1}/MP_{2}.

The ratio *z*_{2}/*z*_{1} is the absolute value of the slope of the isoquant, or the MRTS. Thus we conclude that
MRTS = MP_{1}/MP_{2}:

the marginal rate of technical substitution is equal to the ratio of the marginal products.
Examples and exercises on isoquants and the marginal rate of technical substitution

Copyright © 1997 by Martin J. Osborne