Question

For the purposes of this problem, the quantity produced (?) will be measured in units of 100,000 pounds. Prices and costs will be measured in $thousands. Note that the units do not affect your method of solving the profit-maximization problem as long as the units are consistent. However, you should convert the quantities and prices back to single pounds and dollars at the end. The wholesale price of milk (?) is 19 ($thousand). As a milk producer, you face various variable, fixed, and sunk costs as follows. The cows cost three thousand dollars per 100,000 pounds of milk produced (? = 3?). You own the cows but the cows can easily be resold, so the (? = 3?) is like a wage paid on your investment in the cows. Your workers cost an amount per cow that increases as the size of your herd grows because it becomes more difficult to manage (? = ? + 1 2 ? 2 ). Care for the cows costs one thousand dollars per 100,000 pounds of milk produced (? = ?). You own your land and cannot readily resell it. The mortgage for the land costs 80 thousand dollars (? = 80). You also own outdated machinery used to collect and store the milk that cost 100 thousand dollars when you purchased it, but the machinery cannot be resold or used in any way (so it is a sunk cost).

a) Categorize each of the costs as variable, fixed, or sunk. What is your profit-maximizing output? What are your profits (or losses) in the short-run? Would you shutdown in the short-run? If other firms face the same costs as your farm, do we expect the industry to experience entry or exit?

b) What happens to the profit maximizing output you produce if the rent on the land falls 25% to $60,000? What happens to the profit maximizing output you produce if you had paid $200,000 for your milking machinery? Describe why these results intuitively make sense in this setting.

c) Graph the short-run supply curve for your farm. On a separate graph make a short-run supply curve for the market if it contains 1,000 farms that have the same costs as your farm.

Answer 1

a) The cost of the cows(which can be resold) which is proportional to the Quantity of the milk produced from it(Like a wage paid) C=3Q: Variable cost

Wages paid to the workers which increases along with the increase in cows (C=Q+12Q²) : Variable cost

Costs for taking care of the cows(C=Q) is also proportional to the quantity of milk produced: Variable cost

Mortgage for the land(C=80) is a fixed cost that needs to be paid for the production to occur but can be recovered later(but not readily): Fixed cost

Cost of the outdated machinery at purchase(C=100) which cannot be resold or used in any other way : Sunk cost

Profits= Revenue-Costs= 19Q-(3Q+Q+12Q²+Q+80+100)= -Q²+14Q-180

For maximizing profit, we differentiate and equate to 0

-2Q+14=0 (Also note that second derivative -2 is negative so it is indeed maximum)

Q=7 (or Quantity of milk produced is 700,000 pounds)

Inserting this value back in the profits
Profits= -7*7+14*7-180= 49-180=-131

Since profits is negative, there are losses of 131 (131,000$) in the short run. I would not shut-down in the short run since the losses are of fixed costs and sunk costs. The revenue is still higher than the Variable costs. The profits after excluding the fixed and sunk costs (=49) is still positive. Leaving now would cause a loss of 180 unlike the 131 happening now.

If other face firms face the same costs as my farm, we expect the industry to experience to exit because of sustained losses and firms will eventually downsize and leave

b)The profits in the first case of rent changing will become 19Q-(3Q+Q+12Q²+Q+60+100)= -Q²+14Q-160 and in the second case will become 19Q-(3Q+Q+12Q²+Q+80+200)= -Q²+14Q-280 in the second case had we paid 200,000$ for the machinery. However in each cases,the profit maximizing condition will still be

-2Q+14=0 ( FIrst derivative where the constant has no role in determining the outcome) As fixed and sunk costs have already been paid and do not affect the decision about how much to produce. Profit maximizing condition only considers Price and those costs which are affected by the level of production to decide the profit-maximizing output

c) For the Short-run supply curve,we will get the profits as

PQ-(3Q+Q+12Q²+Q+80+100)= -Q²+(P-5)Q-180

And the profit maximizing output as a function of price P becomes

-2Q+P-5=0

Q=(P-5)/2 (For supply curve of my farm) (first graph with Q in 100,000 on x axis and Price in thousands of $ on y axis)

Now for supply curve of 1000 farms

Aggregate supply will be 1000 times that of my farm a the same price P

so Q_aggregate=1000Q=1000(P-5)/2=500P-2500 (Same graph graph with Q in 100,000,000 on x axis and Price in thousands of $ on Y axis)