Problem set 4 due Tuesday June 19 by 2:45 pm in class or in the mailbox.
Problem 1. Linear regression - Least squares approximation
Correlated quantities x and y are measured together many (n) times, with results xi and yi . You guess that
the relation between x and y is roughly an affine linear one :
y ≈ ax + b
To find the best values for a and b, we thus consider the function
f (a, b) =
n
X
(yi − axi − b)2
i=1
which we will try to minimize.
~ = 0).
(a) Find the critical points of f (i.e. the points where ∇f
~
For specific values of the data, the equation ∇f = 0 has infinitely many solutions. Can you understand why ?
Otherwise, show that the unique critical point (a∗ , b∗ ) of f is a global minimum point.
(b) Your values for (xi , yi ) are (0, 1), (1, 2), (2, −1), (3, 2) and (4, 1). Find (a∗ , b∗ ). What do you conclude on
the quantities x and y ?
Problem 2.
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Find the maximum of f (x, y) = ex
+2xy+2y 2
on the square −1 ≤ x, y ≤ 1.
Hint : Don’t do any computation.
Problem 3. Cobb-Douglas
We want to maximize the Cobb-Douglas utility function
u(x) = u(x1 , x2 , · · ·, xn ) = xa1 1 xa2 2 · · · xann
subject to the constraints
p · x ≤ w,
x1 ≥ 0, x2 ≥ 0, · · · , xn ≥ 0
where ai , pi , w are positive constants (and p = (p1 , · · ·, pn )).
If you find n variables confusing, try first with two variables : u(x, y) = xα y β .
(a) Show a priori that a solution to the problem exists and that it satisfies p · x = w.
(b*) Use Lagrange multipliers to solve the problem. (You can do this either before or after parts (c)-(f)).
(c) We now solve the problem using another method.
Consider a positive valued function f , and suppose that x∗ is a global maximum point of f . Let a and c be
two positive reals.
What can you say about the maxima of the two functions f (x)a and f (cx) ?
(d) When a1 + · · · + an = 1, use the Jensen inequality for the function ln to prove that
u(x) = xa1 1 xa2 2 · · · xann ≤ a1 x1 + · · · + an xn
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When do we have equality ? Write down in a nicer way what you get in the special case a1 = · · · = an = 1/n.
This is called the inequality of arithmetic and geometric means.
(e) Use the above equation to solve our problem when the vectors (a1 , · · ·, an ) and p = (p1 , · · ·, pn ) are proportional.
Hint : First do this in the special case a1 + · · · + an = p1 + · · · + pn = 1 so that pi = ai . For the general case,
rescale both p and w, and raise f to an appropriate power to reduce to the special case above.
(f) Solve the general case where (a1 , · · ·, an ) and (p1 , · · ·, pn ) may not be proportional by using an appropriate
change of variables x̃i = ci xi to reduce to the previous case.
(g) Denoting by x∗ the solution to the problem, and letting u∗ = u(x∗ ), compute
quantities positive or negative ? How do you interpret this ?
∂x∗
i
∂pi
and
∂u∗
∂w .
Are these
Hint : The xi denote quantities of n different types of goods, pi their prices, w your wealth and u is your
happiness function.
Problem 4. Resource splitting
Let f (x) be a strictly convex function of one variable.
We want to minimize the function g(x, y, z) = f (x) + f (y) + f (z) subject to the constraint x + y + z = 1.
(a*) Use Lagrange multipliers to solve the problem.
Hint : note that f 0 is an increasing function.
(b) Give another way to solve the problem, using the convexity of f .
(c) At what point does the n variable function ln(x1 )+···+ln(xn ) attains its global maximum, when subjected
to the constraint x1 + · · · + xn = 2 ?
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