Question

9. Sally has been hired as an interior decorator for a company. She has $1,000 to spend on lamps and flower vases. Lamps cost $10 and flower vases cost $20. Her utility function over lamps (x) and vases (y) is as follows: U(x,y)= 2x + 10y+xy a. What is Sally's objective function? b. What is Sally's budget constraint? c. Set up the Langrangian maximization problem. d. Give the first order conditions. e. Solve the system of equations, and tell how many lamps and vases Sally will buy. Rounding down is okay if you get decimals (that way you can still afford everything.

Answer 1

2 1 sally's utility faction is given by U(x,y)=2x+10ytry ONU ADALAR NOW, as sally is abjective function is to manimise ability .. or mau (2x+10ytry) u, y 20 by Sally has & 1000 to spind on Lamps and blown vasus. ie Ma $ 1.00 Lam r costs $10 ie Py=$10 and flown vases cost $20 ly = Hace sally's budget constraint, м2 xPttXP - 1000 = 10x + 204 e) Now Lagrangian fonction for utility manimising problem ist La 24+1 + + +> [la - 1oz- 248 7 the x= lagrage multiplin CSds can Now in first or de condition for thilify manimisation is

22 2+ y - 10120 an a Yol chto (74) -- = ed =) 10 and 24 = lot a- 20x=0 =) (10+) -- 20 Now, equating both valus ofx from equation () and (1) wc gets 2ty = 27y - loto 1042 : To ☺ ☺ 4+2y = 10+x T = (4+2y-10) = x -- II 5) Now WC get, 1000-103-204 20 =) 1020y = 1000 substituting from equation (ii) x = 4+24-10 10x+20y=1000 10 (4+24-10) +20y=1000 40+ zoy - 100 + 20y = 1000 H04-60-1000 =) a CS hned with CamScanner 404 =1060 =) Y = 26.5 ñ 27

and as, x= 14724-10) = 4 +12 x 26.5) - 10 = 47 ☺ Hnce, sally will buy 47 lamps lamps and 27 flown vass . Scanned with CamScanner