Download A Revealed Profitability Proof of the Laws of Supply and Demand for

A Revealed Profitability Proof of the Laws of Supply and Demand for a Single Firm
This note provides a proof that input demand curves slope downward and supply curves slope
upward for any profit maximizing firm. The proof is elementary in the sense that it does not rely
on calculus, linear algebra, etc.; all that is needed is a basic understanding of vector multiplication
(i.e., inner product) and an assumption of existence of a solution to the firm’s profit maximization
problem.
Let Y ⊂ <n be the feasible production set for a firm. An element of Y is called an input-output
vector, and is simply an n-tuple such as (y1 , y2 , . . . , yn ). If the ith component of a feasible inputoutput vector is positive, then the ith component is an output for the firm. Productive inputs are
measured as negative numbers. This convention for representing inputs and outputs is convenient
because it allows us to represent profit as the inner (or “dot”) product of a feasible input-output
vector and a vector of prices.
Let y ∈ Y be a feasible input-output vector and let p ∈ <n+ be a nonnegative vector of prices.
The inner product of p and y is
p ◦ y ≡ (p1 , p2, . . . , pn ) ◦ (y1 , y2 , . . . , yn )
n
X
=
pi yi
i=1
= p1 y1 + p2 y2 + · · · + pn yn .
Notice how the convention for representing inputs and outputs works here. If the ith component
of y is positive then pi yi is the revenue from the ith output. If the ith component of y is negative
then pi yi is a cost, which is subtracted in the calculation of profit.
Assume a solution to the profit maximization problem exists and suppose y ∈ Y is the profitmaximizing input-output vector when the price vector is p. Similarly, suppose y 0 ∈ Y is the
profit-maximizing input-output vector when the price vector is p0 . Since y maximizes profit at the
prices given by p, p ◦ y ≥ p ◦ y 0 for all y 0 ∈ Y . Similarly, since y 0 maximizes profit at the prices
given by p0 , p0 ◦ y 0 ≥ p0 ◦ y for all y ∈ Y . Rearranging terms gives the following inequalities:
p ◦ y − p ◦ y0 ≥ 0
p0 ◦ y 0 − p0 ◦ y ≥ 0.
These can also be written as
p ◦ (y 0 − y) ≤ 0
p0 ◦ (y 0 − y) ≥ 0.
Subtracting the first of these inequalities from the second yields
p0 ◦ (y 0 − y) − p ◦ (y 0 − y) ≥ 0,
or
(p0 − p) ◦ (y 0 − y) ≥ 0.
Assume all the prices except the ith are unchanged. In this case all of the components of the
vector p0 − p are zero except the ith. Carrying out the vector multiplication yields
(p0 − p) ◦ (y 0 − y) = (0, 0, . . ., 0, p0i − pi , 0, . . ., 0) ◦ (y10 − y1 , y20 − y2 , . . . , yn0 − yn )
= ∆pi ∆yi ≥ 0.
This means ∆pi and ∆yi are the same sign (i.e., pi and yi are moving in the same direction). If yi
is an output this means a higher price results in more production – the law of supply. If yi is an
input this means a higher input price results in a decrease in the amount of the input used – the
law of demand.