Homework Assignment \#6 ECO 4401, Fall Semester 2024
Due: Friday, December 6th at the beginning of class.
1. For each of the following functions, find the coordinates of all critical points in the indicated space. Then, for each critical point, check to see if the second order, sufficient condition for a local maximum or minimum is met.
a. \( y=f(x)=-2 x^{2}+8 x+25 ; x \in \mathbb{R} \)
b. \( y=f(x)=2 x^{3}-45 x^{2}+300 x+500 ; x \in[0,+\infty) \)
c. \( y=f(x)=x^{4}-4 x^{3}+4 x^{2}+4 ; x \in \mathbb{R} \)
d. \( z=f(x, y)=8 x^{3}+2 x y-3 x^{2}+y^{2}+1 ;(x, y) \in \mathbb{R}^{2} \)
e. \( z=f(x, y)=x^{2}+y^{2}(2 x+2) ;(x, y) \in \mathbb{R}^{2} \)
f. \( z=f\left(x_{1}, x_{2}, x_{3}\right)=-4 x_{1}-28 x_{2}+4 x_{3}-4 x_{1}^{2}-2 x_{2}^{2}-3 x_{3}^{2}-2 x_{1} x_{2}- \) \( 2 x_{2} x_{3} ;\left(x_{1}, x_{2}, x_{3}\right) \in \mathbb{R}^{3} \)
2. Find all second-order derivatives of each of the following functions. Then, for each function, use its second derivatives to determine whether the function is convex at all points in its domain, concave at all points in its domain, or neither.
a. \( y=f(x)=\sqrt{(6+x)} ; \quad x \geq-6 \)
b. \( z=f(x, y)=3 x^{4}+5 x^{2} y-y^{3} ; \quad(x, y) \in \mathbb{R}^{2} \)
c. \( z=f(x, y)=-5 x^{2}-y^{2}+2 x y+6 x+2 y+7 \quad(x, y) \in \mathbb{R}^{2} \)
3. Consider a perfectly competitive market in which a tax of \( t_{0} \geq 0 \) is collected from sellers for each unit sold. Price and quantity are determined by the following supply and demand functions:
- \( Q^{d}=1000-20 P \)
- \( Q^{S}=-200+100\left(P-t_{0}\right) \)
a. Write down the matrix equation \( A x=\boldsymbol{b} \) that describes the equilibrium combination of price and quantity when \( x \) is defined as \( x \equiv\left[\begin{array}{l}P^{*} \\ Q^{*}\end{array}\right] \). In particular, what are \( A \) and \( b \) ?
b. Use Cramer's rule to find the function that relates the equilibrium quantity ( \( Q \) ) to the per-unit tax \( \left(t_{0}\right) \).
c. Write down the equation that describes total tax revenue as a univariate function of \( t_{0} \). Hint: by definition, \( T T R \equiv Q^{*} \times t_{0} \).
d. Find the first and second derivatives of the total tax revenue function.
e. How large would the tax need to be in order to maximize total tax revenue?
