Question

2. CONSTRAINED OPTIMIZATION WITH MULTIPLE PRODUCTS Cruise Shoes has the capacity to produce up to x pairs of shoes weekly (see ch.6 case). It currently produces women's shoes with a cost structure given by Cw = 120,000+ 0.0025Qwand faces a perfectly competitive market which prices women's shoes at Pw = $40 per pair. Cruise retains you to advise it about adding a line of girls' shoes, production costs for which are given by CG = 20,000+ 0.0025QG? Demand for these girls' shoes is estimated to be: QG = 9,600 - 200PG. a) Calculate Cruise's maximum profit when making both types of shoes, with its current capacity of 8000 shoes. b) What production capacity would maximize profits for Cruise Shoes? c) What if Cruise Shoes increases its capacity to 9,000 shoes?

Answer 1

Profit from Producing women's shoes = TR-TC =40 QW - 190,000 - Ooo as aul . { Pw = $40} Q Gid - 9600 - 2000 Gr Pa = 9600- agd , 48 - Goud 200 :: Profit from producing Girls shoy = TR-TC ={usered) and -20000 -0.coas a gde : Total Profit = Profit from women's shost profit from GIRLS SHOES = 400W +48 QGd - QGd" -120000 - 20000 -0.0025 Qw4-00025 200 - QGd² (We know 8000 - QW +Qad) = HO Qwt 48 (8000-QW) - (8000-Qw) 140000 -0.coasawt -0.0025 (8006- Qw? 200

Differentiate Total profit with respect to aw. = 40 - 48 - Doo - 80 -0.01aw -4020 100 @w = 6u00 , QG = 1600 © Present these value in total profit function. we get total profit = $82400 profit maximising quantity of " 8000" for women and 3a00' for gionls. We get these values when we marimise the profit function of Women and giols separatly. 80, total capacity = 8000 + 3200 7 11 200 If the capacity is un creased to "2000" The total profit function now change as 19000 = Qut GG) © Tp' = 40 QW +48 (9000- Qw) - (2000-(w.) -140000 – 0.gasaw -0.0025 (2000- aus? 2TP 40-48 - 06-90-001 aw -45-0 100 dQu540- aw = 71501 @G = 1850