Question

Question 2: Consider a ski resort which uses capital K (mountains, ski lifts, lodges, etc.) and labor L (ski instructors, rental shop employees, kitchen staff, etc.) to accomodate q skiers per day according to the production function q= f(L,K) = (Kº + LP) where 0 <p<1. (1) Given the ski resort wants to accomodate q skiers at the minimum possible cost, solve for L*(q) and K*(q) in terms of wages w, price of capital r, and the number p. (2) Suppose w =r=1 and p= 1/3. Find L*(q) and K*(q) for q = 500, q = 1000, and q = 2000. Plot these point on a graph and then sketch the expansion curve. (3) For the same w = r = 1 and p= 1/3, what is the ski resort's cost function C(q)? Graph it. (4) Suppose w increases to w' = 4, but r = 1 and p = 1/3 do not change. Calculate the new L*(q), K*(q) and C(q). Does the ski resort spend more on capital and/or labor when wages are w' instead of w?

Answer 1

Lik 9=f(L,K) = (K +L] The expert cest = WL trek. Min altrek sit. a = (x + 2 ] Ve. Lagrangian: 2-wltrek & & [4-(K + L) ] (Foc de - w - KPU)*!... - of a =R-2 (k² + 2) + f k = 0 – at a = 9-(K + L) 0 - 9 = (x + 2) * Dividing O by @. w. (k+7.3" => (kitt fx 20 >k+C) =>La Coyota © Using & in a ra i n 9 = (x + (+)#x)*]* = k* {1+12y=]] - [***] [1+ ( ) - Using & in @ Lu q ☺ ② If w=r=1, and P = Y and q=500, K* 500 K ri++0713 - 0 - 62.5 and L* = (+) 3-.k = 1x62.5 = 62.5. of q = 1000, kt_ 1000 = 125 and Lt = (+) 5 1 125 - 125 ③ with W=re=1 and 1=1 k* = ? and Lt = 7 => Cost function: c(a) = wkare WLtrck = wara (c(q) = 2 (wtre)

of w'-4, re- and f = t3 K***a **** 0170-nseq and Lot * 0.87369 - 4 x 0.8736 q = 0.125x0.8736q=0.10929 and cost function C(q) = 4 LtK = 4 +0. 10929 + 0.87369 = 0.4368 9 +0.87369 - 1.31049 When wages are w'instead of w, the ski resort spend more on tabor capital, i.e. capital cost = 0.87369 and Labor cost is. 0.4368 9 and 0.8736 9 7 0.43689 since a 70.