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||'''Unit'''|| ECC5650 ||
||'''Topics'''|| Concavity, Convexity, Optimization ||

''Try first to attempt the following problems without looking up a textbook.''

'''Concavity, Convexity, Quasiconcavity, Quasiconvexity'''
 1. (Chiang) Given the definition of a ''concave (convex) function'', three theorems can be deduced:
  1. '''Theorem 1 (linear function)''' ''If $f(x)$ is a linear function, then it is a concave funcation as well as a convex function, but not strictly so.''
  2. '''Theorem 2 (negative of a function)''' ''If $f(x)$ is a concave function, then $-f(x)$ is a convex function, and vice versa. Similarly, if $f(x)$ is a strictly concave function, then $-f(x)$ is a strictly convex function, and vice versa.''
  3. '''Theorem III (sum of functions)''' ''If $f(x)$ and $g(x)$ are both concave (convex) functions, then $f(x)+g(x)$ is also a concave (convex) function. If $f(x)$ and $g(x)$ are both concave (convex) and, in addition, either one or both of them are strictly concave (strictly convex), then $f(x)+g(x)$ is strictly concave (strictly convex).''
 '''=>''' Prove these theorems.
 2. Check $z = x_1^2 + x_2^2$ for concavity or convexity, and hence determine whether the function has a unique maxima or minima.
 3. (Chiang) Check $z = x^2 (x \geq 0)$ for quasiconcavity and quasiconvexity.
 4. (JR A1.48) Let $f(x_1,x_2) = -(x_1 - 5)^2 - (x_2 - 5)^2$. Prover that ''f'' is quasiconcave.

'''Brouwer's Theorem'''
 0. Prove that the empty set $\emptyset$ and entire set $\mathbf{R}^n$ are both open and closed.
 1. (JR, A1.38) Let $f(x) = x^2$ and suppose that $S = (0,1)$. Show that ''f'' has no fixed point even though it is a continous mapping from ''S'' to ''S''. Does this contradict Brouwer's Theorem? Why, or why not?

'''Lagrange Multiplier Method'''
 1. Find the extremum of $z = xy$ subject to $x+y = 6$.
 2. Find and characterise the critical points of $f(x,y) = 3x-y+6$ subect to $x^2+y^2 = 4$.

'''Kuhn-Tucker Conditions'''
 1. Check the first and second-order conditions for the following problem:
  * Minimize $C = (x_1 - 4)^2 + (x_2 - 4)^2$
  * subject to:
   * $x_1 + x_2 \geq 5$
   * $ -x_1 \geq -6$
   * $ -2x_2 \geq -11$
   * $ x_1,x_2 \geq 0$

'''Envelope Theorem'''
 1. (Riley) To produce ''q'' units of output, a profit-maximising firm requires $\frac{1}{2}q^2$ units of the single input. The price of the output is ''p'' and the price of the input is ''r''.
  1. Write down an expression for profit $\pi(q,p,r)$.
  2. Solve for the profit maximizing output $q^*$ and hence show that $$\frac{\delta \pi}{\delta p}(q^*,p,r)=q^*=\frac{p}{r}$$
  and $$\frac{\delta \pi}{\delta r}(q^*,p,r) = -\frac{1}{2}\left(\frac{p}{r}\right)^2 \quad .$$
  3. Show also that maximized profit for different input and output prices is $$\Pi(p,r)=\pi(q^*(p,r),p,r)=\frac{1}{2}\frac{p^2}{r}$$
  4. Confirm that $\frac{\delta \Pi}{\delta p} = \frac{\delta \pi}{\delta p}(q^*,p,r)$ and $\frac{\delta \Pi}{\delta r} = \frac{\delta\pi}{\delta r}(q^*,p,r)$.