Question

Agricultural Household Model

a. Assume there is a farming household which has to make a decision about how much food to produce. The marginal cost of producing an extra unit of food, where Q is the total quantity of food produced, is MC = 2*Q. If the price of food, PFood, = $12, how much will the farm produce? What is the farm’s profit?

b. The price of other stuff is POther = $6. Assume the household has consumption preferences such that the household utility = XFood2 * XOther, where XFood is the amount of food consumed and XOther is the amount of other stuff consumed. That is if XFood = 5 and XOther = 10, the household’s utility is 52*10 = 250.

First, draw the budget curve for the household, assuming that all of its profits from part (a) are exactly equal to its income. Put the amount consumed of other stuff on the Y-axis and food on the X-axis. Second, determine the amount of food (XFood) and other stuff (XOther) that the household consumes given the household’s utility function. (The household can buy fractions of the goods, e.g. if it has $2, it can buy 1/5 of a unit of food.) Big Hint: These are Cobb-Douglas utility functions. Optimal consumption will be where the indifference curve is just tangent to the budget constraint. That is, where the price ratio = the marginal rate of substitution: PFood / POther = 2* XOther/ XFood. Other Hint: remember that the HH must also satisfy the budget constraint: PFood * XFood + POther * XOther = $30.

c. Assume the price of food increase to PFood = $16. What is the new amount of food that the farm household produces and what is the farm’s profit?

d. Assume the price of other goods is unchanged. Does the amount of food consumed go up or down? Comparing the situation where PFood = $12 vs PFood = $16, how much net food is produced for the market, that is, what is total amount of food produced minus amount of food consumed in each case?

Answer 1

Solution:

a) Food produced = 6 , Profits = 36

b) Budget curve =

12Xf + 6Xo <= 36

Optimal consumption = (xf, xo) = (2,2)

c) New food produced at Pf = 16, is 8 with profits = 64

d)New optimal consumption (Xf', Xo' ) =(2.667, 3.556)

Amount of consumption of food has increased by 0.667 units

Net food produced in 1st case with price of f= 12 is = 6-2 = 4

Net food produced in 2nd case with price of f= 16 is 8-2.667 = 5.333