Economics 204–Final Exam–August 21, 2007, 9am-12pm
Each of the four questions is worth 25% of the total
Please use three separate bluebooks, one for each of the three Parts
Part I
1. State and prove the Supremum Property.
2. Prove that if a finite set Xhas exactly nelements, then 2X, the set of all subsets of
X, has exactly 2nelements. Hint: use induction.
Part II
3. Consider the function
f(x, y)=x3−y3+2x2+2xy +8y2−9x−15y−2
(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 the eigenvectors of D2f(x0,y
0) and provide an orthonormal basis for R2
consisting of eigenvectors. 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 (e) to describe the level sets of
fnear the point (x0,y
0); include the shape, the directions of the principle axes,
and (where appropriate) the lengths of the principal axes.
Part III
4. Let F:C×Rp→Rnbe a continuous function, where C⊆Rn.
(a) Given ε>0, define
Ψε(ω)={x∈Rn:|F(x, ω)|<ε}
Show directly from the definition that for each ε>0, Ψεis a lower hemicontinuous
correspondence. (We are using correspondence in the sense of the Lectures, so that
we do not require that Ψε(ω)=∅).
(b) Let
Ψ(ω)={x∈Rn:F(x, ω)=0}
Show directly from the definition that if Cis compact, then Ψ is an upper hemicon-
tinuous correspondence. (We are using correspondence in the sense of the Lectures,
so that we do not require that Ψ(ω)=∅). Hint: The proof is by contradiction.
Suppose that Ψ is not upper hemicontinuous at some ω0; this tells you that there
is a sequence ωn→ω0with certain properties.
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