SOLUTIONS ECONOMICS 172B
Midterm exam, Wednesday, January 30, 2008
1. (18 minutes, 20 points) Recall but do not write the definitions of convex set, concave function, and
upper contour set. Prove that every upper contour set of a concave function is a convex set, noting
where the various definitions are used in the proof.
Let f be any concave function. Let t be any real value and consider the upper contour
set defined by t. Let x and y be any two points in the upper contour set, and let λ be any
value in [0, 1]. In order to show that the upper contour set is convex, it must be shown
that λx + (1 − λ)y is also a point of the upper contour set. By definition of the upper
contour set,
f (x) ≥ t and
f (y) ≥ t
((*))
Because the function is concave
f (λx + (1 − λ)y) ≥ λf (x) + (1 − λ)f (y)
From relation (*) the right hand side of this relation is ≥ λt + (1 − λ)t = t. Therefore
f (λx + (1 − λ)y) ≥ t
and therefore λx + (1 − λ)y is in the upper contour set. Since these facts are true for
all choices of concave f, real t, x, y, and λ in [0, 1], every upper contour set of a concave
function is a convex set.
The proof is much shorter if more mathematical notation is used and you repeat the
obvious less frequently than I did.
2. (18 minutes, 20 points) Consider the bisection method of maximizing a concave function of one variable.
In 100 words or fewer, explain how the method works. You may use as many diagrams or equations
as you wish, or you may describe the spreadsheet that applies the method.
Initialize by choosing xlow and xhigh so that f 0 (xlow ) > 0 and f 0 (xhigh ) < 0. By concavity of
f , the maximum lies in the interval (xlow , xhigh ). Choose a small > 0 as the convergence
criterion.
Step: Let x0 be the midpoint of the interval (xlow , xhigh ). Evaluate f 0 (x0 ). If f 0 (x0 ) > 0
replace xlow by x0 .If f 0 (x0 ) < 0 replace xhigh by x0 . Now if xhigh − xlow < the iterations
are complete. and you take the midpoint of the interval as the approximate solution. If
xhigh −xlow ≥ repeat the step using, of course, the new interval (xlow , xhigh ). The maximum
of f is always in the interval (xlow , xhigh ), which is progressively narrower. Eventually it
must converge. (82 words)
A diagram would be nice.
f(x)
xlow
x’
x*
xhigh
3. (9 minutes, 10 points) Consider the function f (x1 , x2 ) = −x21 − 3x22 .
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(a) Write the gradient.
µ
¶
¶ µ
f1 (x1 , x2 )
−2x1
∇f (x1 , x2 ) =
=
f2 (x1 , x2 )
−6x2
(b) In 50 words or fewer, explain the significance of the gradient of a concave function.
The gradient vector lies in the direction of steepest ascent. In seeking to maximize
a concave function f by choice of x1 and x2 , it is useful to explore in that direction,
that is to look at
g(t) = f (x̄1 − 2x̄1 t, x̄2 − 6x̄2 t)
One follows g (t) to its maximum and then starts over from that point. (46 words)
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