Question

Complete parts a-e.

1. Consider the following (Cobb-Douglas) utility function: U = xayB And budget constraint: MZ PeX+PY *Treat Px, P, M, a, and B as positive constants. Note, a + B < 1. Using these equations, please answer the following questions: a. Formally state this consumer's utility maximization problem and write down the relevant Lagrangian. (6 pts) b. Using your work from part "a.", derive demand curves for X and Y. Show all work. (6 pts) C. Show that the demand curves derived in part b. satisfy the rule of equal marginal utility per dollar spent (i.e. MU: = y). (6 pts) d. Show that the law of demand holds for both X and Y. (6 pts) e. Show that both X and Y are normal goods. (6 pts) Py

Answer 1

The answers have been explained in the answer sheet.

(a) u=X y op , Maximize u-xay B. sit. M-Px x + PyY. The Lagrangean can be Dorétten as, L = x + x[m- P₂ X – Pyy] aç = axml 180=10 pay for 9.27 0 = MR X-QyY=0 — Guven that 30C is satisfied, 0 ☺ Po > RATB x xxx- .: M = Bax + PyY B =BX+XR: | - 6x + = M = Pxx. att. OM ! - atp. Po

xBx = y Py. M=13 X+ Oy y M = y. 7. & + Pyry 'M= Pry. 7 p + 17 y are the respective demand fore. o prove that Mux ! U=xzß px alu Besong Differentiating u wrax, we get, Mlly = au = ax = XX- Differentiating lu co.sty, we get ! Muy = = x By х ху - [from = 480-ly B. X (O+B) ay at ß ' [ from 6) BM . y 2+3 Brdy B y (x+B) = BM

. Mus Muly [prooved] Px 7 "Pay The love of demand shows that quantity demanded increases as price Halls. from s v am Ixtß P Differentiating x w .n.it ex, me get Hence, the law if dein and holds Again, from 6 x ßm ut X + B Py Differentiating y w.rit Poy, we get, . BM to y = - 2 +8 Sh of demand holds for 7. .. Hence, the Law