Consider the following consumer maximization problem. maximize u(x, y) subject t

Question

Consider the following consumer maximization problem. maximize u(x, y) subject to p_xx + p_yy lessthanorequalto I; x, y greaterthanorequalto 0, where p_x and p_y are prices of commodities and y and I is income (budget). We assume u(x, y) = x^2y and let p_x = 2, p_y = 1 and I = 10. (a) Solve the problem of maximizing u(x, y) subject to 2x + y lessthanorequalto 10. What is the optimal values of x and y? (There are non-negativity constraints x, y greaterthanorequalto 0 but we can ignore these so long as all points satisfying the necessary condition for maximum are positive). (b) Let (x, y) be the solution to problem b. Letting u = u (x, y), solve the following minimization problem (observe that minimizing a function f is equivalent to maximizing -f.): min 2x + y subject to u(x, y) greaterthanorequalto u.

Explanation / Answer

u(x,y) = x2y

2x+y=10 gives y = 10-2x

u(x) = x2(10-2x) = 10x2-2x3

u'(x) = 20x-6x2

u"(x) = 20-12x

Set first derivative to 0

x=0 or x = 10/3

u"(0) = 20>0 and u"(10/3) <0

Maximum when x= 10/3 and y = 10-2(10/3)

i.e. x= 10/3 and y = 10/3

------------------------------

x=0 gives minimum

i,.e x=0 and y =10 gives minimum

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.