Question

please help with part D-H!

Problem 2 Mr. Lee runs an orange grove, SweetOrange. It's harvest season and each week using labor services (L) and equipment (K) SweetOrange can bring q(L,K) = 250L1/3 K1/3 pounds of oranges to market. a) Does technology at SweetOrange display diminishing returns to labor? Does it display diminishing returns to equipment? Explain. b) Does technology at SweetOrange display increasing, constant, or decreasing returns to scale? c) If SweetOrange employs 2 workers and uses 3 pieces of equipment, what is the firm's Marginal Rate of Technical Substitution? What is the economic interpretation of this number? This week, labor hands cost $100 each while renting equipment costs $100 per piece and SweetOrange wishes to produce 1,000 pounds of oranges at the lowest cost possible. d) Write SweetOrange's cost minimization problem, and using the Lagrangian method, find the cheapest technique that allows SweetOrange to produce 1,000 pounds of oranges. d.1) Write the Lagrangian function d.2) Write the first order conditions for the Lagrangian problem. d.3) Solve the first order conditions and find L and K^. e) What is the lowest cost at which the company can produce 1,000 oranges? f) What share of the amount you found in parte) is represented by labor cost? What share is represented by the cost of using equipment? What relationship do you notice between these shares and the parameters in the production function? g) What is the Average Cost of a pound of oranges? h) Suppose one week SweetOrange wished to harvest four times as many oranges (q' = 4,000). What would happen to the Average Cost of a pound of oranges?

Answer 1

ƏMP q = 250443 KY3 Page 1 9) MP = 2q = 250 k'o * ¢"3! = 83.33 ila kis SOT = 83.33 (-3) 1*-1k"co So, there are diminishing returns to Labor, MP = 250 x L’s x 4k.219 = 83.33 X '3 k-243 other 83:33 (13) _'* *?! 60 So, there are diminishing, returns to equipment b) 'Augment production function by factor 2., q - A F(Ak,al) = 250 (22)'3 (AK) "3 - 250 223 4 V3 KMS = 20.64 F(K,L) if FAK, AL) = F(,L) & atB<, then F(x, 2) exhibits diminishing retrons. So, Technology exhibits diminishing returns as at B = 0.67€ 1. K = 3 = 1,5 c) MRTSWK - MP - 43.33 ("sk" K 87.33 _'3 x -213ī Ž = 1.5 So, at the rate of 165 units, a single unit of Labor can be substituted with a unit of equipment. - d) w = 100, r= 100 q=1000

Page 2 C = WLtok = 100L+100k dil) A = 250 '3 k"5 + 2 (0-100L-100k) d.2) FOC 24 0,450, R = 0 d.3) 40 = 83.33 (213K!3 - 1001 = 0 7 0.83 (+?\5''3 = 2 - 0 then, atzo → 83.33 ('3 k-213_1009=0 = 0.83_3K-213 = 2 - © using © & © [-213x3 = 1'3x-213 [k= Put in production function, q= 250 ('3 !3 → 1000 = 143 = (498/2 = * 5) (24)!= L* » l* = 8 = K* Age So, I & K" = (8,8) Ano

e)

Lowest TC = w*L + r*K

TC = 100 * 8 + 100 * 8

TC = $1600

f)

share represented by labor cost = w*L

= 800

share represented by equipment = r*K

= 800

We observe that both shares are equal as the parameters given in the production function are also equal (1/3)

g)

Average Cost = TC / q

= 1600 / 1000

= 1.6

h)

Since, AC = TC / q

Increase in the denominator from 1000 to 4000 will reduce the average cost

new AC = 1600 / 4000

= 0.4

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