Nov 09 2009

## A Micro problem for the advanced Econ student

Greg Mankiws Blog: Take Out Your Pencils

I love that Harvard Economics professor Gregory Mankiw blogs, but I hate that has de-activated the comments on his blog. Yesterday he posted a question from his own Harvard introductory economics class.  Since he doesn’t allow comments though, I cannot tell if I’m solving it correctly. So I will re-publish it here and ask my readers to solve the problem in the comment section.

IB and AP students who have studied microeconomic should be able to put some of their basic algebra skills to work to solve this one.

Only one firm produces and sells soccer balls in the country of Wiknam, and as the story begins, international trade in soccer balls is prohibited. The following equations describe the monopolist’s demand, marginal revenue, total cost, and marginal cost:

Demand: P = 10 – Q
Marginal Revenue: MR = 10 – 2Q
Total Cost: TC = 3 + Q + 0.5 Q^2
Marginal Cost: MC = 1 + Q

where Q is quantity and P is the price measured in Wiknamian dollars.

a. How many soccer balls does the monopolist produce? At what price are they sold? What is the monopolist’s profit?

b. One day, the King of Wiknam decrees that henceforth there will be free trade—either imports or exports— of soccer balls at the world price of \$6. The firm is now a price taker. What happens to domestic production of soccer balls? To domestic consumption? Does Wiknam export or import soccer balls?

c. In our analysis of international trade in Chapter 9, a country becomes an exporter when the price without trade is below the world price and an importer when the price without trade is above the world price. Does that conclusion hold in your answers to parts (a) and (b)? Explain.

d. Suppose that the world price was not \$6 but, instead, happened to be exactly the same as the domestic price without trade as determined in part (a). Would anything have changed when trade was permitted? Explain.

Post your solutions below, I really want to know if I have solved it correctly! About the author:  Jason Welker teaches International Baccalaureate and Advanced Placement Economics at Zurich International School in Switzerland. In addition to publishing various online resources for economics students and teachers, Jason developed the online version of the Economics course for the IB and is has authored two Economics textbooks: Pearson Baccalaureate’s Economics for the IB Diploma and REA’s AP Macroeconomics Crash Course. Jason is a native of the Pacific Northwest of the United States, and is a passionate adventurer, who considers himself a skier / mountain biker who teaches Economics in his free time. He and his wife keep a ski chalet in the mountains of Northern Idaho, which now that they live in the Swiss Alps gets far too little use. Read more posts by this author

### 12 Responses to “A Micro problem for the advanced Econ student”

1. Khairul Rusydion 10 Nov 2009 at 11:37 pm

a. Assuming production at Profit Maximizing condition of MR=MC, 10-2Q=1+Q; Q=3.

At this Q, P=10-3=7.

Total Profit = Total Revenue – Total Costs = (PQ)-(3+Q+0.5Q^2) = (21)-(10.5) = 10.5

b. At P=6, Qty dd w/ trade, 6=10-Q; Q=4; Domestic Consumption will increase by 4-3=1

At P=6, Qty ss by domestic firms w/ trade = Indeterminate since market SS Schedule not given. [I gave up at this part after a few hours. Reasoning as follows.]

On one hand, MC of Monopoly = SS Market since the one monopoly firm constitutes the entire market. Hence MC = SS = 6 = 1+Q since Price Taker; Q=5. This implies that if it the firm is purely a Price Taker and goes ahead and produces for \$6 regardless of conditions, it would have an incentive to produce 5 units. This is impossible because it assumes that the firm would produce even if it incurs subnormal profits, and because 5 units lies to the right of the initial equilibrium quantity of 3. Which defies the Law of Supply that states that a lower price means that producers have lesser incentive to produce.

On the other hand, assume that firm is still rationale and aims to maximize profits, and that even if it was a price taker, it is still able to control its output and need not necessarily produce whatever quantity it is enticed to produce according to the SS function above. Hence, it will continue to produce so long as it gains normal profits whereby TR is just able to cover TC i.e. TR=TC; PQ = 3+Q+0.5Q^2. Solving where P=6 since it is a price taker, Q=0.6411 from the quadratic equation. Assuming that the firm produces to the nearest whole units where it does not incur a loss (since 0.6411 soccer balls is impossible to produce), the nearest whole Q value = 0. Hence, the firm would produce 0 soccer balls at a market price of \$6 since it would not risk producing at a loss. In this case, an import of 4 soccer balls from the rest of the world would occur, and the only Wiknam soccer ball producer would cease to exist.

Hence, from either of the above, there is no way to arrive at a Y=mX+C line graph to form the Supply function. This is because there are no co-ordinates that can coincide with (3,7) from the initial equilibrium without trade to form a positive sloping supply curve for the new entire market.

c. The assumption does not hold. Too tired to answer this one in full. Question b) above zapped up most of my energy.

d. No, no trade would have occurred assuming no specialization took place after that etc. At the new World Price where the quantity demanded by domestic consumers still remained unchanged, all quantity demanded was able to be supplied by domestic producers. Hence, no imports were necessary.

2. Jason Welkeron 11 Nov 2009 at 4:12 am

Khairul,

Wow, thanks for your exhaustive answer! It was enlightening to read! I am not SURE it is all correct though! I am waiting to see what some other people have to say before I post my answer! Great job though! Are you an econ student, teacher, or economist? By the way, isn't this question pretty humbling?! Can you imagine freshman at Harvard sitting down to solve this? I thought I knew my economics, but this EC10 question has had struggling so much!!

Jason

3. Matt Bohnenkampon 11 Nov 2009 at 9:06 am

A) Q=3 and P=\$7…I agree with the answer above.

B) If \$6 were the price taken by the firm they would produce where MR (now \$6) equals MC. So…\$6=1+Q and therefore Q=5 and P=\$6.

The domestic demand is P=10-Q so at \$6=10-Q and the domestic demand is therefore 4. Which means they will have produced 5, but only 4 will be demanded domestically so they are exporting 1.

By the way, at the price of \$6 this firm would be making a profit, but since it didn't say anything about this monopoly becoming competitive domestically, I will assume that firms cannot enter and lower the world price in the long run.

C) No the answer doesn't hold because the soccer ball firm in Wiknam is a monopoly and not competitive. Therefore, they would charge a price that is higher than the competitive price because they are protected by barriers to entry and can charge a higher price to maximize their profits. If trade is allowed, they become a price-taking monopolist (weird thought). As a price-taker, this monopoly will actually produce more because now they do not have price setting power and MR=P instead of MR being below the price like it was in part A.

D) Yes…if the world price were \$7 like it was in part A this firm would produce where MR(P)=MC or \$7=1+Q or 6 soccer balls. The domestic demand would be where the price of \$7=10-Q or 3 soccer balls and therefore the firm would export 3 of the 6 they produced.

Once again, the reason why they produce more when they become a price-taker is because now they have no price setting power. In effect, they don't have to account for the "opportunity cost" of lowering the price of soccer balls in order to sell more. I'm not sure if this is the correct terminology to use, but basically I'm saying that now D=MR=P=AR rather than D>MR.

4. Anthony Knoxon 11 Nov 2009 at 9:57 am

Fun stuff….let's give it a shot here.

I'm one of those old-school theory guys who likes tables, so when I did this question, I went to work building the table and the graph too. This is when it is not cool to be a visual guy.

A. So in going through the equations that are provided, it is fairly quick to see that your MC/MR is equal (with 4) when your quantity is at 3 and your price is 7. So that's my answer to A as well, and when applied against the TC equation, I get 10.5 (PQ-10.5). So it looks like I'm in agreement with the comment above.

B. Using the assumption that the monopolist is profit maximizing in keeping with the Law of Supply, we would see the marginal revenue go to 6 units each, which makes your MC/MR point become 6, setting up a new equilibrium at 5 units, which on the basic level COULD be your domestic production (Unit 4 MR 6/MC 5, Unit 5 even @ 6). So there is a justification for 5 units of soccer balls produced. The problem here is that the demand price for the 5th unit is 5 Wiknamian dollars, so that unit won't sell (firm incurs deadweight losses). Consumption will be 4 due to the world price, and since the firm is the industry, it is very tough to justify producing beyond 4 units. This is where both options make your head spin.

C. In this case, the price without trade was 7, and the world price was 6, which according to the question made Wiknam an importer. I don't think this holds true when the marginal analysis supports one more unit.

D. If the world price went to the 7 MDollar price I answered in A, the incentive for the local producer is still there to keep production there. As a result, I don't believe anything changes. The producer still produces, but now it is at higher profits.

5. Khairul Rusydion 11 Nov 2009 at 10:52 am

Hi Jason,

I'm an econs student in Singapore awaiting entrance to university next year. I'm re-taking my A Level econs examination NEXT WEEK D: as a private candidate, and this blog has been a brilliant learning aid! Wish I had stumbled upon it much, much earlier! IB and AP Economics has a lot of common ground with A Level Economics, and I'd personally think it'll be fantastic if we could get A Level Authors and Readers to join in as well. Will tell all my juniors about this site.

Yup, this question is pretty challenging. Prima facie looking at the first question you'd think it would be a piece of cake hahaha. But its from Harvard so I'd like to elect that the Prof made the ensuing confusion deliberate, rather than it being a co-incidence.

Hi Matt,

b) Thanks! I think I missed out the part of MR (now \$6) in my previous answer. Mathematically, it is interesting to note that if you solve the quadratic equation of TR-TC=0 where P=6, maximum value occurs where P=6, Q=5 and Max Profit=9.5. So even as a price taker, it is still maximising profits at this expanded output, which I couldn't reconcile earlier.

I think I forgot to differentiate the quadratic equation to find the maximum point (i.e. max profits) rather than the x-intercept.

Hi Anthony,

d) Thanks! Yup, I don't think that there would be any change to the status quo. Its basically the same situation as before right? Where the market clears domestically with national qty demanded = national quantity supplied at the same conditions of P=7, Q=3.

b) Seems that we're all in agreement that there is indeed justification to produce at 5. But profits at P=6,Q=5 is 9.5, and profits at P=6,Q=4 is 9. So rationally, they would produce at Q=5, but there is no justification for production for beyond Q=4, since it would not be demanded domestically. So the question is whether the firm produces at 5 to maximize profits and produce an extra unit for nothing (which happens in the real world, i suppose). Or if it produces at 4 since there is a demand for only up to 4. Or produces 5, sells 4 and exports the last unit? There is no demand curve for the rest of the world, so whether they export (and whether there is a demand for this last unit from the rest of the world) is indeterminate, right?

6. Spencer Coeon 12 Nov 2009 at 2:31 am

Jason,

I gave the first two parts of this question to my students on Monday. It sounds like you and I had similar results…that is that the students (most) did well with the algebra of part A but many struggled with the transition from monopoly to perfect competitor.

As for my own thoughts to the answers. I am in agreement with Matt Bohnenkamp above although I prefer to think of the firm as one (of many) perfectly competitive firms in a global market instead of as a "price-taking monopolist" in one nation/kingdom.

7. Chrison 12 Nov 2009 at 5:51 am

A. I agree with everybody above. Set MR=MC (profit max) to find Q=3, plug that into demand equation to get a price of 7. Plug that into TC to get a profit of 10.5

B. If the firm becomes a price taker at the world price of 6 then MR=6 always (just like a perfectly competitive firm P=MR). Set that equal to MC to find Q of 5, Plug the price of 6 into the demand equation to get the domestic consumption of 4. Export 1.

6= 1+Q (MR=MC)

Q= 5 (domestic production)

P=10-Q (market demand)

6=10-Q

Q=4 (domestic consumption)

5-4=1 (export 1)

C. The conclusion does not hold because the industry was not competitive to begin with. If the industry had been competitive to begin with (or if the monopoly would have charged where P=MC, which would be the competitive result) Then, the MC curve becomes the industry supply curve so set supply = demand (MC= demand equation) to get:

1+Q=10-Q

Q=4.5 plug this into the market demand to find P

P=10-Q

P=5.5 which is below the world price so the conclusion in part c would have held.

D. IF the world price is 7 and the firm is a price taker at 7, then:

MR=MC

7=1+Q

Q=6 (domestic production)

plug 7 price into market demand

7=10-Q

Q=3 (domestic consumption)

export 3

8. Danon 20 Nov 2009 at 2:13 pm

Thanks for the great blog post (+ helpful comments). As a former Ec 10 student, I can confirm that Matt B got the answers Mankiw was looking for.

Looks like Mankiw has another post up ( http://gregmankiw.blogspot.com/2009/11/take-out-y… ), also without answers. I'd love to get in on the discussion if you make a new post!

9. Jason Welkeron 21 Nov 2009 at 12:52 am

Dan,

Thanks for the link to the latest Mankiw problem! And kudos to Matt B for solving the last one! I have just published the latest question and solved it myself on some scrap paper. I am dying to see if I solved it correctly! Please check it out and let me know your solution! http://welkerswikinomics.com/blog/2009/11/20/mank
Jason

10. pat tocheon 29 Nov 2009 at 8:07 pm

Matt, I agree with you, by and large, but one of your comments could be pursued a little further. You write "it didn’t say anything about this monopoly becoming competitive domestically, I will assume that firms cannot enter and lower the world price in the long run."

You raise two points, 1) will the monopoly remain a domestic monopoly under trade, 2) what is the effect of free entry and exit on the domestic market structure?

1) One interpretation, like yours, is that the domestic firm becomes a price-taker but remains a monopoly domestically. Note, however, that this is a result rather than an assumption because if the world price had been low enough (much lower than 6 dollars) domestic demand would have exceeded domestic supply and the domestic country would have to resort to imports from the rest of the world to satisfy domestic demand — the monopoly would no longer be the sole supplier of the good domestically and could not legitimately be called a monopoly anymore.

2) What if firms were free to enter and exit the domestic and world markets? If the world market (i.e. the pre-trade non-domestic market) is competitive, the world price must be equal to the minimum of world average total cost. Since the world price is assumed to be 6 in part (b), we can infer that the minimum average total cost must be 6 in the rest of the world. The minimum average total cost of the domestic firm, on the other hand, is approximately 3.5 by my calculations (no time to double-check), so that the domestic monopoly is more efficient than the foreign competitors. If the domestic technology is patented the domestic firm should be able to win a price war, forcing other firms to exit the industry, and in turn become a world monopoly. We don't know anything about world demand so we cannot infer the monopoly price under free trade and a world monopoly. If the domestic technology is non-patented free trade will result in the rest of the world adopting the superior technology and the new competitive world price would fall to about 3.5 (1+sqrt(6), if I'm not mistaken, must sign off will double-check this later).

Do you guys agree with my analysis in 1) and 2) above?

Khairul, I think you miss the point that there is now international trade, so that NO, it is not true that "the market clears domestically with national quantity demanded = national quantity supplied." Rather, the domestic market does not clear and the excess supply is exported (the world market, whatever that is, presumably does clear, so that world quantity demanded = world quantity supplied).

11. pat tocheon 30 Nov 2009 at 6:23 am

Here's my take on the problem, please leave feedback and comments below (I will be checking back here), especially if I've made errors, committed omissions or typoed. Cheers,

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A Micro problem for the advanced Econ student | Economics in Plain English