Question

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 part e) 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

Problem - 2 concept of marginal Produc Stats That as a factor of Production increases how does The Production of good change. matnmatically it is calculated as da where do q = CC, K) dL ( Jactors diminishing Returns it is decrease in marginal Product as the amount of a single of foodcection is increased dh > O > 0 (a) q (L,K) = 250 L 13 k 113 marginal product of daboy - mpe de de 2500 1 2 3 4 13 so & L, k 30 da = How does marginal Product change as d2² a change in unitoy Labor -513 113 1 = 2500 1 / (-22) L K co +, L,K >O WI- dz 02

d22 'Since da shows how does marginal Product of Labor respond as we increase The unit af Labor so There Is diminishing Return to Labor yor Now nequilment 1/3 -5/3 Коу J2q = Lo VK, LO dk² So equipment also has diminishing return. (b) for economics of Set Scale, if double. The factor oy Production double me Produc Hon men There will be constant returns to scale, or if Production increase by Less - Than double then decreasing Returns To Scale. if Production increase more-than double then there will be increasing returns to Scale.

numerically and for our cause q (L,K) = 250 113 113 Let we increase the Labor and cquilment by 8 Times, SO 9(82, 8K) = 250 0843" SKR) " = 250 (8) 43 1/3 (8) 13 43 = 250 (2) (2) (1/3 kl13 = 4 (250 L 13 K3 ) 9 (81, 8K) = 49(L,K) So q (86, 8K) L 82 (L,K) there is decreasing returns So to scale. те, (C) MRTS = MP = 250 250 113 250.413 k213 - 200 (213 k13 260 (113 k213 = k113 k2/3 [ 13.(2/3 MW

MRTS (marginal Jute oy Technical Substitution ) = ? Stats That How much amount of capital input has To be given up in order to use an adaitional Unit of Labor so that output Remain constaint: In our case MRTS = 3 = 105 Labor of confinal we have to So To use one more accution Jorgo 105 units of capital. 113 113 (d) PL = $ 100 pk = $100 q=1000 units 1000 = 250 L KB (del) Lagrangian becht kko K = C iso-Cost Line 100Lt 100 k= 2= 100 Lt look td { 100-250 L 13 k137 (2.2) -2/3 113 ad - 100 - ( 13 ) 250 (k o

- 2/3 = K 0 Too- d 20 de 100-125 18 13 =0 (to 3) th = - 100-1 250 27/3 + 113 - 0 ► 202*43 KMG - 300 2501-13 K 113 , 300 250(113 k 43 = dk 4 0 250 (1( 3 equating (i) and (ii) 213 300 (2/3 = 300k 250 K 113 250 L'13 (2/3 (3 - 2/3 kl 13 L=k

1000 = 250 L 13 K 13 113 K=L 113 1000 = 250KK 1000 = k 3 250 coule (413 - 7 (4) + 3 = x 11 X (2)3 =k 8 =K=1