Mathematical methods for economic theory

Martin J. Osborne
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6.1.4 Optimization with an equality constraint: conditions under which a stationary point is a global optimum

A stationary point of a concave function is an unconstrained maximizer of the function. For the constrained maximization problem
maxx,y f(x, y) subject to g(x, y) = c,
the concavity of the Lagrangean
L(x, y) = f(x, y) − λ(g(x, y) − c))
guarantees that a stationary point of the Lagrangean is a maximizer.

The argument is simple. If L is concave, given λ, then by a previous result any stationary point of L, given λ, is a maximizer of L. That is, if (x*, y*) is a stationary point of L then

L(x*, y*) ≥ L(x, y) for all (x, y),
or
f(x*, y*) − λ(g(x*, y*) − c) ≥ f(x, y) − λ(g(x, y) − c) for all (x, y).
Now, if (x*, y*) satisfies the constraint then g(x*, y*) = c, this inequality is equivalent to
f(x*, y*) ≥ f(x, y) − λ(g(x, y) − c) for all (x, y),
so that
f(x*, y*) ≥ f(x, y) for all (x, y) with g(x, y) = c.
That is, (x*, y*) solves the constrained maximization problem. Precisely, we have the following result.
Proposition 6.1.4.1
Suppose that f and g are functions of two variables defined on a convex set S that are differentiable on the interior of S, c is a number, and there exists a number λ* and an interior point (x*, y*) of S such that (x*, y*) is a stationary point of the Lagrangean
L(x, y) = f(x, y) − λ*(g(x, y) − c).
That is,
L'1(x*, y*)  =  f'1(x*, y*) − λ*g'1(x*, y*) = 0
L'2(x*, y*)  =  f'2(x*, y*) − λ*g'2(x*, y*) = 0.
Suppose also that g(x*, y*) = c.
  • If L is concave—in particular if f is concave and λ*g is convex—then (x*, y*) solves the problem maxx,y f(x, y) subject to g(x, y) = c.
  • If L is convex—in particular if f is convex and λ*g is concave—then (x*, y*) solves the problem minx,y f(x, y) subject to g(x, y) = c.
Proof  
Given that f and g are differentiable, so is L, so the fact that L is concave means that by a previous result any stationary point of L is a maximizer of L. The argument before the statement of the result establishes that any maximizer of L that satisfies the constraint g(x, y) = c solves the constrained maximization problem.
If g is linear then λ*g is both convex and concave, regardless of the value of λ*. Thus this result has the following corollary.
Corollary 6.1.4.1
Suppose that f is a function of two variables defined on a convex set S that is differentiable on the interior of S, g is a linear function of two variables defined on S, c is a number, and there exists a number λ* and an interior point (x*, y*) of S such that (x*, y*) is a stationary point of the Lagrangean
L(x, y) = f(x, y) − λ*(g(x, y) − c).
Suppose also that g(x*, y*) = c.
  • If f is concave then (x*, y*) solves the problem maxx,y f(x, y) subject to g(x, y) = c.
  • If f is convex then (x*, y*) solves the problem minx,y f(x, y) subject to g(x, y) = c.
Example 6.1.4.1
Consider the problem
maxx,y xayb subject to px + y = m,
considered previously. We found that there is a value of λ* such that
(x*, y*) =  left parenthesis
am
(a + b)p
, 
bm
a + b
right parenthesis
is a stationary point of the Lagrangean and satisfies the constraint. Now, if a ≥ 0, b ≥ 0, and a + b ≤ 1 then the objective function is concave; the constraint is linear, so from the result above, (x*, y*) is a solution of the problem.