Question

Exercise 5 Cobb-Douglas and Decreasing Returns to Scale (Farming)

Exercise 5. Cobb-Douglas and Decreasing Returns to Scale (Farming) There are over 2 million farms in the United States, covering almost a billion acres of agricultural land. Consider farming output, y (measuring thousand bushels of corn), as a function of short-term factors, such as water, fertilizer, seeds, considered as a composite input, X, and land as a long-term factor, L. Both are necessary, and they each present diminishing marginal returns. Assume that the production function is y=XXL. a) What is the marginal product of short-term inputs, dy/ox? Holding land fixed, why would increasing water, seeds and fertilizer have diminishing marginal returns? b) What is the marginal product of land, dy/OL? Holding other inputs, such as water, seeds and fertilizer, why would increasing land have diminishing marginal returns? c) A farm has L=1000 acres, and uses X=256 in inputs. How many thousands bushels of corn, y, does it produce? If it doubles both Land X, does y double, more than double, less than double? d) The price of each thousand bushel of corn is p=$3000, while each composite unit of X costs w $750. There is also an opportunity cost for land use, as a farmer could be using acres for other profitable purposes (other crops, or ranching). Assume the opportunity cost for each acre of land, L, is r=$250. Since ne py-WX-L, how much profit does the farm in partc make? e) in the short run, the farm cannot change land usage, so acres are fixed at L=1000. How much X should the farm use to profit maximize? What would outputy be? How much profit can this farm make? f) in the long run, can also adjust the acreage used (assume it has acres at its disposal that can be dedicated to alternative use). What would the level of and be for long run profit maximization? What is the corresponding y? How much profit is the farm making now?

Answer 1

a) Page No Y = xul2 MP x = 24 = t J 4 , keep L fixed, aMPx co as his fixed & we continue to. additional add inputs, the olp produced wener will continue to fall MPL = QY/L = V W X TL as x is fixed, then when more land is made available, then additional olp unite produced will decrease C) L= 1000, X=256, Y = (1000)'a (256)2 y = 505.964 And Now y (al, ax) = a12+12 Y Cho L, X) = XYL, X) So as inputs & land double, y will also double , as Prodo funch exhibit CRS. d) p = 3000 P=8000, Px = w = 450, r = 250. Tapy-wx-8L = 3000 (50 5.964) - 750 x 256-250x1000 = $ 1075,892 ans gun e) L=1000, NOW T = P x 12 (1000) 12. W x-7 (1000) O P (1000)" - w sob ota ox 2JX P (1000)) = (x) 12 x 1(1000)"? | Y= x12 (1009) - Yu = p (1000) - ly* = 500p/60 | I Ama awn TE PY - COXX - 1000g T = 500 p² - P. (1000) 12 - 1000r. Any W 2 Qw

Its Mandatory To answer only first 4 parts