详细题目可参考附件。有会做的同志可以指点一二。不胜感激。
1. Consider the equation
ln (y) + cos y + x3y - yex + y2 = 6
(a) Can you solve this equation to obtain an explicit expression for y as a function of x?
(b) Can you obtain an explicit expression of dy
dx
at a point (xo; yo) that satises the
equation?
2. Let u : R ) R be a C2 function satisfying Du(x) > 0 and D2u(x) < 0 for all x. Let ; I,
and r be positive parameters, and consider the problem:
max
x;y;s
u(x) + u(y)
s.t. x + s = I
y = (1 + r)s
(a) Give a brief economic interpretation of this problem.
(b) By substituting the constraints into the objective function, derive a rst-order nec-
essary condition for a mximum.
(c) Is this condition sucient for a maximum? Explain your answer(Need not a proof).
(d) What can you say about the signs of @s
@r ? Justify your answer.
3. Let u : R2
++ ! R be a C2 function. Let p1; p2; I > 0 and consider the following
equality-constrained maximization problem
max
x1;x2
u(x1; x2) subject to p1x1 + p2x2 = I:
(a) Write the rst-order necessary conditions.
(b) Write the second-order necessary conditions;
(c) derive an expression for @x1
@p1
;
(d) and state the condition you need to satisfy the hypothesis of the IFT in order to
answer 3c.
4. Let m  0, and consider the problem
max
(x1;x2)2R2
+
3px1 + x2 subject to 9x1 + 8x2  m:
For each of the following three values of m, either nd and verify a saddle-point, or prove
that a saddle-point does not exist. Be sure to make your reasoning clear.
(a) m = 24;
(b) m = 9; and
(c) m = 0.
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