Question

5. A company owns an inventory of 100 units of a good. It must sell the entire inventory over the next three years. The profit function for sales within any one period is: T(x) = 50x4 -0.5x} t = 1,2,3 It wishes to maximize the present value of profit: V = a (x1)+ (x2) (x3) 1+r. (1+r) Suppose r = 0.25. Find the optimal values of X1, X2, and X3. (Hint: Use the condition that X1 + X2 + x3 = 100 to substitute in for X3 or solve via the Lagrange multiplier approach.]

Answer 1

Given : T (X) = 5o W -0.5x42, =1,2,3 max : V = Tal+ T (12) + (23) litro (ltro? to at 12+2₂=100 setting up Lagring ean L= V ++ [ 100–56-342-23] 2= T(x1+ T (22) + TT (43) +[loo-M-22-n] (1+) (1+r)? 2 = (50%,-0.5') + (5092-0-50?) + (soxg-0.504) Hd (100-2,-*= 3] (14 ) (trg First order Londitions De = 50-34, -dzo .. 0 Pantone [5-*=+] 2 - 4 = 50 - 22 ad zo 7 50-12 e 77 (177) L . ad=0-> - d .. 5o - Rz (1+r) 2 Sorkz i tr) oh - 100-91-92-1270 a) loos nita2 taz... ① Taking ratio od os 50-81= - => 50-24,Seiten 50-12/147? (1+r) (50-*,) = 50-dz 12 = 50 - (1+r (50-21) 1 2 = ? - or tru (12= (+) - sor]

Taking ratio of ② 40 50-022 Sonne Titro (to² (50-21)= 56-23 Px3 = 50- (1487°/60-2007] substituting where of de a az in equation (1) dit 92+ 2z=100 it (tria, 5ort son CHR(50-21)=100 ait(1.25)", - sol 0,25) +50-(1.25)² (56-02)=100 2.25 %, + 37.5 -11.5625 ) (90-92-100 2.25n, + 37.5 - 78.125 +1.56254 =100 - 3.8125n, no. 625 5100 % = 144.625 140.625 436.89 3.8125 na = (17712, -50% = (1.25)/360897-5610-28) [^2 = 33.61] do 233 low-,-32= loom (36.89 433.61) [3.= 29.5] optimal vekses are [> 26.04; n2= 33.61333 =298]