Economics 204–Final Exam–August 29, 2005, 6-9pm
Each question is worth 20% of the total
Please use separate blueb ooks for each of the three Parts
Part I
1. (a) Define the term “contraction.”
(b) State the Contraction Mapping Theorem.
(c) Give the following portion of the proof of the Contraction Mapping Theorem:
Given a contraction, start with an arbitrary point, and show how to construct a
sequence of points and prove that the sequence converges to a limit which is a
candidate fixed point. You need not prove that the limit is in fact a fixed point.
2. Suppose T∈L(Rn,Rn) is a linear transformation, and let Vbe any basis of Rn. Show
that λis an eigenvalue of Tif and only if λis an eigenvalue of MtxV(T).
Part II
3. Consider the function
f(x, y)=x3+y3+6x2+8y2−2√3xy +(2
√3−15)x+(2
√3−19)y
(a) Compute the first order conditions for a local maximum or minimum of f. Show
that the first order conditions are satisfied at the point (x0,y
0)=(1,1).
(b) Compute D2f(x0,y
0) and give the quadratic Taylor polynomial for fat the point
(x0,y
0).
(c) Find the eigenvalues of D2f(x0,y
0) and determine whether fhas a local max, a
local min, or a saddle at (x0,y
0).
(d) Does fhave a global max, a global min, or neither, at (x0,y
0)?
(e) Find an orthonormal basis for R2consisting of eigenvectors D2f(x0,y
0). Rewrite
the quadratic Taylor polynomial for fat the point (x0,y
0) in terms of this basis.
(f) Use the quadratic Taylor polynomial found in part (d) to describe the approximate
shape of the level sets of fnear the point (x0,y
0).
Part III
4. Show that a compact subset Sof a metric space (X, d) is closed. To get full credit, you
must directly use the open cover definition of compactness.
Please turn over
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