Microeconomics, Module 5, “The Behavior of Firms”

Homework assignment

(The attached PDF file has better formatting.)

Profit Maximization

A firm faces a demand curve of P = 130 – 5Q. The marginal cost for this firm is 20 + Q. The fixed costs are 500. Assume that the firm produces a quantity and charges a price to maximize profits.

A.    What is the marginal revenue curve facing the firm?
B.    What is the quantity produced by the firm?
C.    What is the equilibrium price charged by the firm?
D.    What are the total variable costs of the firm?
E.    What are the total costs of the firm?
F.    What is the total revenue of the firm?
G.    What is the net profit of the firm?

Are we supposed to get a formula for the marginal revenue curve, or just make a table with revenue, cost, and quantity and sketch it?

[NEAS: Either method is fine.  The homework shows you have read the textbook and understand the material.  We are not looking for specific solution formats.]

Just to make sure I'm understanding this stuff, I thought I'd post my answers to the homework. Do others agree with me, or did you get something else? I have:

A) Marginal Revenue=130-10Q.   B) 10.   C) 80.   D) 250.   E) 750.   F) 800.   G) 50.

[NEAS: Correct for the continuous analysis, which is the preferred method for real problems.  Landsburg's text uses discrete tables, since many of his college readers can't handle calculus.]

I'm thrown off by the Cost portion of this question. Is the 20, in the 20+Q, a fixed cost along with the $500? Can someone confirm these answers, they are slightly different than Rick's?

A) 130-10Q
B) 10
C) $80/UNIT
D) $775-$520= $255
E) $520+$255= $775
F) 10x$80 = $800
G) $800-$775= $25

[NEAS: See the postings below regrading the $20, which is not a fixed cost.]

Bjabb,

I followed the steps shown for a very similar problem in Exercise 5.10 from the "Practice Problems" posting for this same module. For part D of this homework, I took the definite integral of the marginal costs from 0 to the quantity produced. So, I have Total Variable Costs = ∫₀¹⁰ (20 + Q) dQ  =  [20Q + Q²/2] ₀¹⁰  =  200 + 100/2 - 0 - 0 = 250. That's how I got my answer of 250 for Part D.

For Part E, I added Fixed Costs of 500 to this 250 to get Total Costs of 750.

For Part G, Net Profit = Total Revenue - Total Costs = 800 - 750 = 50.

To answer your question, I don't think that the 20 in the 20+Q represents an additional fixed cost. It represents part of the marginal cost for each unit produced. Does this make sense?

[NEAS: Correct; the 20 is a marginal cost, as stated in the problem.  It is 20 for each unit, not a fixed cost of 20 for all  units combined.]

I mostly agree with Bjabb:

a) MR = 135-10Q
b) 10
c) 80
d) 21+22+...+30 = 255
e) 500+255=755
f) 800
g) 800-755=45

My one difference with Bjabb is the 500 vs 520 in part e. In the book, example 5.3 (I think) shows that the marginal cost isn't counted at Q=0.

[NEAS: The 255 total variable cost is correct for the discrete analysis.  See also the Jacob - Rachel dialogue below.]

Jacob: Two candidates posted different solutions on the discussion forum. Which is right?

Rachel: We can solve the homework assignment two ways: a discrete perspective and a continuous perspective.

With a discrete perspective, we form a table. The marginal cost is 21 for the first unit, 22 for the second unit, and so forth. The candidate has an equilibrium quantity of Q = 10, so the total variable costs are 21 + 22 + 23 + … + 30 = 255.

For a continuous perspective, we use calculus. We integrate 20 + Q from 0 to 10. This gives 20Q + ½ Q² from 0 to 10 which equals 20 × 10 + ½ × 10² = 250.

Jacob: Which method should we use?

Rachel: For the homework assignment, either method is fine.

~ Firms producing jumbo jets or super-computers or satellites might make only ten units a year. For these firms, we use the discrete analysis.

~ Other firms produce thousands of goods a year. If the firm makes shirts, Q may be in thousands of units.

Landsburg uses a discrete analysis with tables, since his readers are first year college students, many of whom can not handle calculus. Earlier editions of Landsburg’s text used calculus in the text; in the current edition, the calculus is in an appendix.

Actuaries know calculus. Most real problems use continuous functions. You gain more from the course if you use calculus whenever possible.

Jacob: What parts of the solution change between the discrete and continuous versions?

Rachel: The variable costs change in Part D. This changes the total costs in Part E and the net profit in Part G.

Jacob: Why do the two perspectives give different solutions?

Rachel: Suppose we want the marginal cost for Q = 3.

~ The discrete perspective says the marginal cost is 20 + 3 = 23.

~ The continuous perspective says Q is in thousands, and we want the marginal cost for the third thousand. This is the average marginal cost between Q = 2 and Q = 3:

½ × (22 + 23) = 22.5

Each marginal cost is reduced by 0.5, so the 10 × 0.5 = 5, which is the difference between 250 and 255.

Should a question like part D of this homework arise on the exam, will there be two correct answers, or should we do both the discrete and continuous methods to determine the correct answer?

[NEAS: If the exam problem gives a function, use the continuous method. If the exam problem gives a table with discrete numbers of goods (1, 2, 3, etc), use the discrete method. The choices in the exam problem will always be sufficiently different that both methods give the same answer.]

Shouldn't the answer to part A. be 110 - 11Q? Why would the Marginal Revenue not be impacted by the marginal cost of the last unit produced? Additionally, this function means that the equilibrium quantity (Q = 10) minimizes the Marginal Revenue function and thus maximizes total revenue, which would seem to me to be correct. What am I doing wrong by adding the marginal cost to the marginal revenue equations posted by others?