minwhere_{z}_{1}_{,z}_{2}w_{1}z_{1}+w_{2}z_{2}subject toy=F(z_{1},z_{2}),

Denote the amounts of the two inputs that solve this problem by *z*_{1}*(*y*, *w*_{1}, *w*_{2}) and *z*_{2}*(*y*, *w*_{1}, *w*_{2}). The functions
*z*_{1}* and *z*_{2}* are the firm's **conditional input demand functions**. (They are conditional on the output *y*, which is taken as given.)

The firm's minimal cost of producing the output *y* is *w*_{1}*z*_{1}*(*y*,*w*_{1}, *w*_{2}) + *w*_{2}*z*_{2}*(*y*,*w*_{1},
*w*_{2}) (the value of its total cost for the values of *z*_{1} and *z*_{2} that minimize that cost). The function TC defined by

TC(is called the firm'sy,w_{1},w_{2}) =w_{1}z_{1}*(y,w_{1},w_{2}) +w_{2}z_{2}*(y,w_{1},w_{2})

for some value ofw_{1}z_{1}+w_{2}z_{2}=c

The cost-minimization problem of the firm is to choose an input bundle (*z*_{1}, *z*_{2}) feasible for the output *y* that costs as little as possible. In terms of the figure, a cost-minimizing input bundle is a point on the *y*-isoquant that is on the lowest possible isocost line. Put differently, a
cost-minimizing input bundle must satisfy two conditions:

- it is on the
*y*-isoquant - no other point on the
*y*-isoquant is on a lower isocost line.

Another example of a firm's cost-minimization problem is given in the following figure. In this case the isoquant does not have the "typical" convex-to-the-origin shape; instead, it is bowed out from the origin. The cost-minimizing bundle is, as before, the bundle on the isoquant that is on the lowest possible isocost curve. This bundle is indicated by the large black dot. (Note that the point
at which an isocost line is tangent to the isoquant **maximizes** the cost of producing the output *y* along the isoquant.)

Now, the equation of an isocost line is

which we can rewrite asw_{1}z_{1}+w_{2}z_{2}=c

so that we see that is slope isz_{2}=c/w_{2}(w_{1}/w_{2})z_{1}

If the isoquants are smooth and convex to the origin and the cost-minimizing input bundle(z_{1},z_{2})involves a positive amount of each input, then this bundle satisfies the following two conditions:

- (
z_{1},z_{2}) is on they-isoquant (i.e.F(z_{1},z_{2}) =y) and- the MTRS at (
z_{1},z_{2}) isw_{1}/w_{2}(i.e. MRTS(z_{1},z_{2}) =w_{1}/w_{2}).

The condition that the MRTS be equal to *w*_{1}/*w*_{2} can be given the following intuitive interpretation. We know that the MRTS is equal to MP_{1}/MP_{2}. So the condition that the MRTS be equal to
*w*_{1}/*w*_{2} is equivalent to the condition

orw_{1}/w_{2}= MP_{1}/MP_{2},

MP_{1}/w_{1}= MP_{2}/w_{2}:

Examples and exercises for cost-minimization problems

Copyright © 1997 by Martin J. Osborne