Four MC curves and two PPFs

Just another "Teaching" post. Because it's needed. (Though I have never taught micro beyond first year.)

Marginal Cost means the change in Total Cost per extra unit of output. MC=dTC/dQ. (Not to be confused with Average Cost, which is the level of Total Cost divided by the level of output. AC=TC/Q.)

Do MC curves slope up? (Is MC an increasing function of Q?) If so, why? If not, so what?

These questions matter because: profit-maximising firms set Q where Marginal Revenue=MC; for perfectly competitive firms Price is the same as Marginal Revenue, so they set Q where P=MC and the MC curve is the Supply curve; for imperfectly competitive firms MR=[1-(1/E)]P where E is Elasticity of Demand, so they set Price as a markup over Marginal Cost according to the formula P={1/[1-(1/E)]}MC, and the MC curve plus markup is a "pseudo supply curve".

So in asking whether MC curves slope up we are asking whether supply curves (or pseudo supply curves) slope up.

The answer is: it depends. Because there are four different MC curves that matter. (And if I wanted to do Keynesian/Monetarist macro I would add another two supply curves to the list.)

There's the Short-Run vs the Long-Run; there's the individual firm experiment vs the market experiment; so that's four in all.

But, at the market level, it's much easier to stop asking "why does the MC curve slope up?". We should instead ask: "why is the Production Possibilities Frontier bowed out?" It's the same question.

In the SR, at least one input is held fixed. In the LR, all inputs can be varied. We make this distinction because: it might take longer to vary some inputs than others; even if it doesn't, it's usually easier to teach stuff when you first change one thing at a time, and then teach about envelopes.

1. The individual firm's SR MC curve. I'm pretty sure this one (eventually) slopes up. Malthus knew why. Malthus knew that as you add more and more labour to a given amount of land, with a given technical knowledge, the extra output per extra worker (eventually) diminished. He called this "diminishing returns". We call it "diminishing Marginal Product". If this weren't true, you could grow the same total amount of wheat by using the same total amount of labour and having them all grow wheat in the same window box. SR MC equals Wage/Marginal Product of Labour. For a given W, MC slopes up for the very same reason MPL slopes down.

2. The market SR MC curve. The first-year textbooks say that the market SR MC curve (the market supply curve) is the horizontal summation of the individual firms' SR MC curves, holding the number of firms fixed. If so, it would have the same elasticity (not the same slope) as the average individual firm's SR MC curve.

But that isn't quite right. The market SR MC curve will be steeper (less elastic) than the individual firm's SR MC curve. Because if all firms want to hire more labour, and if the supply curve of labour slopes up, wages will rise. The rise in W, plus the fall in MPL, cause MC to rise as firms increase employment and output.

3. The individual firm's LR MC curve. Under Constant Returns to Scale (where a doubling of all inputs causes a doubling of output), the individual firm's LR MC curve will be perfectly elastic. It will be flat. Under Increasing Returns to Scale the individual firm's LR MC curve will slope down, so we won't have perfect competition [updated: because the second order conditions for profit-maximisation will fail if the MR curve is horizontal]. Instead we will have natural monopoly, or monopolistic competition, or oligopoly. Only under Decreasing Returns to Scale will the individual firm's LR MC curve slope up.

4. The market LR MC curve. This is the one that really matters. Because it tells us whether supply curves (or pseudo supply curves) slope up in the long run. It tells us whether an increase in demand for a good will cause the price to rise.

You might think that under constant returns to scale, the market LRMC curve, like the individual firm's LR MC curve, must be flat. You would be wrong. Here's a simple counterexample. Suppose any individual farm can double labour and double land and double the production of wheat. [updated: Or, if individual farms have U-shaped cost curves, we could double the number of farms and do the same thing.] Any individual farm can do this, but all farms in aggregate cannot do this, if the total supply of land that can grow wheat is fixed. If they all attempt to use more land they will increase the demand for land and push up land rents, and so push up the MC of wheat. The market LR MC curve would be exactly like the market SR MC curve. Both would hold land fixed.

5. The LR PPF. This is much simpler. Plus it forces us to think in general equilibrium as opposed to partial equilibrium. Draw a PPF, with quantity of wheat produced on the vertical horizontal axis and quantity of apples produced on the horizontal vertical axis. The slope of the PPF is the Marginal Cost of wheat in terms of apples. The inverse of the slope is the MC of apples in terms of wheat.

If the PPF is a straight line, then both MC curves would be flat. If there were only one input (call it "labour"), and if all labour were identical, and if there were constant returns to scale in both wheat and apples, the PPF would be a straight line, and both MC curves would be flat.

But those are very extreme assumptions.

The PPF is much more likely to be bowed out.

If there is both labour and land, and if wheat is more land-intensive and apples are more labour-intensive, (or vice versa) the PPF will be bowed out. (Because if you gave the wheat farmers relatively more land and relatively less labour you could produce more wheat and more apples than if you split the land and labour equally between the wheat farmers and the apple orchards.)

If labour (or land) is different, with some labour (or land) having a comparative advantage in apples and other labour (or land) having a comparative advantage in wheat, the PPF would be bowed out. (Because if you give the wheat farmers the good wheat land and the apple orchards the good apple land you could get more wheat and more apples than if you split all the land equally between the wheat farmers and the apple orchards.)

If the PPF is bowed out, the slope will change as we move along it. The MC of wheat will increase as we produce more wheat (and fewer apples). The MC of apples will increase as we produce more apples (and less wheat).

If wheat is more land-intensive than apples, and so apples are more labour-intensive than wheat, land rents will rise, and wages will fall, if we produce more wheat and fewer apples. Plus, wheat farmers won't be increasing land and labour in proportion as they produce more wheat. Because they can't, in aggregate. Because they are taking land and labour away from apples, and apple orchards use proportionately more labour and less land than wheat farmers.

So even if the individual wheat farmer has constant returns to scale and a flat LR MC curve, the market LR MC curve for wheat will slope up.

6. The SR PPF. This is an afterthought. If we wanted to, we could draw a SR PPF assuming land cannot switch between apples and wheat in the short run. We can only switch labour. The SR PPF would be even more bowed out than the LR PPF. The LR PPF would be the envelope of all the SR PPFs. The slopes of those SR PPFs would tell us the market SR MC curves, and explain why they slope up. Because the SR PPF would be more bowed out than the LR PPF, the market SR MC curve would be less elastic than the market LR MC curve.

OK. That's me done, I hope. Back to macro/money.

Comments

[Gene Callahan]

Nick, I have tremendous appreciation for these posts of yours on teaching economics: I linked to the previous one with praise. And I'm sure as I carefully work through this one, I will gain similar insights. However, a quibble: "Marginal Cost means the change in Total Cost per extra unit of output."

But then you take the derivative of Q. But surely if Q comes in units we cannot take its derivative, right? The derivative of a discontinuous function is undefined, isn't it? Now, I'm not sure what practical impact that has, and perhaps it is unproblematic to treat a discrete function as if it were continuous. Nevertheless, I think we ought to speak precisely to the extent we can.

[Patrick]

Gene: FWIW, it's been a while but I think the discrete equivalent of a derivative is the forward difference operator.

[Frances Woolley]

"Though I have never taught micro beyond first year."

Except my 2nd year micro class this afternoon! How did it go? Excellent talk by Larry Kotlikoff here in Calgary - his version of the fiscal future of the US is so gloomy I'm imagining Greece with guns.

[JCE]

since you're going back to macro/money, you should reply to this post by david glasner http://uneasymoney.com/2012/11/25/its-the-endogeneity-redacted/

in particular, what do you think of this: "So whether Nick Rowe and Bill Woolsey are right that inflation and recession are caused by a monetary disequilibrium involving an excess demand for, or an excess supply of, the medium of exchange, or whether Scott Sumner is right that monetary disequilibrium is caused by an excess demand for, or an excess supply of, the medium of account depends on whether the supply of inside money is endogenous or exogenous."

does the model you work with implicitly assume exogenous money?

[Nick Rowe]

Gene: thanks! Yep. Wheat is smooth down to the grain, but tractors, fields of land, and to some extent workers too (though part-time and overtime smooths a bit) come in lumps. So a lot of "curves" will have irregular steps. I don't sweat this one too much. When I'm talking with my farming siblings we talk about whether they want to get another field. When I'm teaching intro economics I draw a smooth curve. Economists' comparative advantage comes when we look at markets, and by the time we add up lots of lumpy steps they are going to look fairly smooth. We can't (usually) tell individual farmers anything they don't already know a lot better than us. I figure we are probably getting bigger things wrong than smoothing out those lumps.

Frances: yep. Just After I posted I thought "that isn't precisely correct after today"! It went well. 3 hour classes do drain me a bit though, towards the end. I had a momentary panic when I thought "Does Frances do the Hicks or Slutsky decomposition for the income and substitution effects?" The TA said "Hicks", so I did Hicks, and the students seemed to recognise it fine. I was doing demand and supply analysis for Toronto banning plastic bags, but one of the students said go to the corner solution on the indifference curve, which I agreed was the better answer. I had a memory lapse on moral hazard, adverse selection, and dogs. I got it right for buying a dog from the pound, but I forgot what you said about the Kennel Club.

[Nick Rowe]

JCE: I did write a comment, but it wasn't very clear. Yep. I need to think about what David said.

[nemi]

Of course you have increasing MC if you put all a countries resources into the construction of airplanes. A lot of the factors are not particularly suited for that and we would need a lot of specific materials pushing prices upwards. But this is hardly relevant for the reality we live in. In reality, the question is whether the construction of the two hundred and first airplane is cheaper than the two hundred – and it almost certainly is if you allow for time to adjust the semi fixed factors (without me knowing anything about airplane construction). And that is usually the case for most individual products (but not necessarily for groups of products such as agricultural products).

In micro, we do not use PPF curves(at least not in with the same meaning as in macro) but isocost curves when considering the tradeoff between different products (or the output set consistent with some inputs in a multi output production function).

[Nick Rowe]

nemi: I strongly disagree. Scarcity and choice, trade-offs in the allocation of resources between competing ends, opportunity cost, which are all illustrated by a PPF, are the most fundamental concepts in economics that I teach right at the very beginning if intro micro. (Greg Mankiw agrees.) When we decide whether or not to build the 201st airplane we are moving along the PPF and switching resources into airplanes and out of something else, the slope of the PPF is the MC of the 201st airplane, and if the PPF is curved out that MC slopes up.

(If increasing returns to scale were strong enough to offset the comparative advantage effects of heterogenous factors, then the PPF could be bowed in.)

[Kristjan]

Nick, I am sure you are familiar with Piero Sraffa arguments. What do think of them?

[Kristjan]

that is(my last comment): what do you think of them.

[Jacques René Giguère]

Nick : I don't want to talk for nemi but if he is teaching in a business class, as I do during winter, after explaining PPF, we use cost curves and never think back about PPF.

[nemi]

Nick: Really? Wow! And by PPF curves you mean what society actually is able to produce (by e.g. allocation all resources to the construction of airplanes)? Surely you understand how irrelevant that question is (and how impossible it is to estimate it -or to even estimate the resourse costs for the next few years)? Not in the sence that a lot of things in economics is irrelevant, but really really irrelevant. I mean, you are talking about the cost of producing something over a range where the marginal benefit of additional planes is negative (they are just taking up space)

The isocost curve and production possibility set look the same as the PPF, but you do not make assumptions about what would be possible when allocating all resources to e.g. airplane construction.

[Nick Rowe]

Jacques Rene: "...after explaining PPF, we use cost curves and never think back about PPF."

I do the same. But after writing this post, (which came after reading some confused stuff out there, and thinking about it), I'm beginning to think that is a mistake.

Kristjan: empty boxes debate, returns to scale, and perfect competition? A long time ago, and my memory is very hazy about who said what. Yep, with increasing returns to scale at the level of the firm, so the minimum efficient scale is large relative to the size of the total market, we don't get perfect competition. No worries. We just model it as monopoly, monopolistic competition, oligopoly, whatever. I think a much bigger issue is that individual firm's outputs are not perfect substitutes in the eyes of buyers. (Even if we are talking barley, there are many characteristics on which barley can differ, and they can matter if it's malting barley. My brother had a very detailed contract for producing specialty barley for a brewery, at double the price per tonne compared to feed barley. And can the grain merchant easily get his truck to the particular farmer's silo?

Personally, I usually think in terms of monopolistic competition, for that reason. Different products.

More generally on Sraffa though. The standard Sraffian model has labour as the only ultimate factor, and all labour is identical!!! That plus CRS gives you a linear PPF. And if he wants us to assume the vector of outputs stays the same when r changes, he is implicitly assuming identical homothetic Leontieff (L-shaped) preferences!! No way.

[Michael]

Nick, You have written an nice summary, but if the economic state is such that a firm or a market or a sector is not on the respective curve, what is the usefulness of these ideas? Your results assume both full employment of resources and full utilization of capacity. We know that most firms plan production below full capacity so the relevant marginal costs do not increase with output. Industries are not even on their supply curves unless all the firms are in profit maximizing equilibrium, and firms are not in this equilibrium when they choose to produce below full capacity. I enjoy reading your posts because you are a good neoclassical economist and you are open mined. So why do you believe that firms and markets are best represented by positions that are on the marginal cost and supply and PPF curves rather than states off these curves? To me your post is a nice summary, but it's based on a big fudge?

[Jacques René Giguère]

Nick: in fact, we don't totally leave aside PPF. Marginal rate of Technical Substitution curves are a kind of PPF, though written and viewed differently. Do you have something about that in mind?
Writing fast between gradings and last advices and couselling for the term paper...

[nemi]

Isn’t the technical rate of substitution the slope of the input requirement set? The slope of the production possibility set is, I think, usually called the marginal rate of substitution.

The frontier of the production possibility set is, of course, the same thing as the macro PPF (and its interior) but "THE" PPF is a function of all of societies resources while the PPF:s I use only contain (and is defined over) a small subset of the resources. I would define it over a range between maybe zero and 200 % of current output - which is far wider than I generally need to explain any relevant micro principle (or applied use of that principle) - and narrow enough so that you, atleast sometimes, can keep stuff exogeneous.

What I mean by e.g. increasing returns to scale (or decreasing MC) would also only be defined over this range.

[nemi]

I´m going to stop spamming this thread now, but I think I was unclear.

In a micro production possibility set, the decreasing return of scale would be seen through the distance between PPF:s with different amounts of inputs, not through the slope of the PPF between the products under consideration (depending on assumptions off course).

[Nick Rowe]

Jacques Rene: You mean isoquants? I don't think about them much. I'm not sure if they help much here, since we are varying Q, and looking at how costs change as we do that.

nemi: I'm trying to remember terminology. I think:
"Marginal Rate of Transformation" is the slope of the PPF (number of extra apples produced if you produce one less banana)
"Marginal Rate of Substitution" is the slope of an indifference curve (number of extra apples consumed if you consume one less banana holding utility constant).
"Marginal Rate of Technical Substitution" is the slope of an Isoquant (number of extra workers if you have one less acre of land holding output of apples constant).

Michael: It's the job of monetary policy to ensure that the "full employment" assumption is true and that standard micro is true! Yep, I would have needed two more supply curves if I were to open this up to macro/money considerations.

Remember, the main point of this post: the elasticity of the market MC curve is not the same as the elasticity of the individual firm's MC curve. The first could slope up even if the second slopes down.

What you say about capacity depends on what you mean by "capacity". An individual firm's MC curve can slope up even if it is not producing at capacity, if by that we mean the point where the ATC curve stops falling.

[nemi]

Apparently I was not going to stop spamming yet.

Nick:
1: that terminology sounds about right (why is it so hard to remember?).

2: Monetary policy is hardly going to put firms on their respective PPF, and thus not the aggregate of those firms output on the PPF either.

Michael:
Every micro economic efficiency study (and there are a lot of them) finds a vast majority of the firms way inside the PPF (technically inefficient) as well as in the “wrong” part of the PPF (economically inefficient) so I think people, at least in applied micro, are well aware that we never are anywhere near the PPF (and if you look at actual management reasoning, tools, calculations and plans, as is done in special fields such as e.g. forest or fishery economics, it is often outright bizarre).

Of course you will get a lot of errors from the methodology (generally data envelope analysis (DEA) with bad data and questionable assumptions) but when I say far from the PPF I mean FAAAAR from the PPF (and way wrong mix as well).

The efficiency of the market economy (or any other economic system) is obviously a silly fairytale. My guesstimate of the technical efficiency of the system is at about 50 % of the theoretical maximum. Add economic efficiency and we are down to 30 %. If the social welfare function is the sum of the log of individual consumption baskets, the social efficiency is at 0 percent (but I would probably put the social efficiency at 5 %)

It would be really interesting to hear other poples guesstimates.

[Frances Woolley]

Nick "But that isn't quite right. The market SR MC curve will be steeper (less elastic) than the individual firm's SR MC curve. Because if all firms want to hire more labour, and if the supply curve of labour slopes up, wages will rise. The rise in W, plus the fall in MPL, cause MC to rise as firms increase employment and output."

I'm not sure that I quite understand. By steepness, I think you mean, say, the increase in marginal costs with a 100 unit increase in quantity produced?

The question is: why would it make a difference whether that 100 unit increase came from one firm producing 100 units more or 100 firms producing one unit more?

The difference doesn't come from the impact on wage rates - it won't take any more labour to produce the extra output if it's spread over 100 firms, rather than concentrated in one.

In fact, if anything, I would expect the firm's SR MC curve to be steeper than the industry one. Suppose the firm is initially producing at the minimum point on its short run average cost curve. Recall that in the short run capital (or some input) is fixed. For a single firm to produce 100 units more output, it will have to do it entirely by hiring more labour, which will cause it to move relatively far from its cost-minimizing capital/labour ratio, causing marginal costs to rise. However if that increase can be spread over 100 firms, each firm won't have to move that far from its optimal capital/labour ratio.

[Frances Woolley]

Ah, I understand. You don't mean steeper. You mean less elastic. Then the point on wages goes through.

See your previous post on demand curves.

[HM]

The argument that heterogenous input requirements mean that the production possibility frontier bows out seems reasonable.

Endogenous technical change seems like a major threat to the reasoning though: the demand for one good determines the demand for technology in that sector, and thus the growth rate of productivity.

I suppose it is an economies of scale argument, but only in the very long run.

[Nick Rowe]

Frances: yep, it only works with elasticity, not slope. It's funny how deeply "elasticity" is built into our (economists'?) subconscious. I don't know whether it's because we think spatially??

HM: good point. Yep, technology is non-rival, so it's natural monopoly.

[Nathanael]

"1. The individual firm's SR MC curve. I'm pretty sure this one (eventually) slopes up. "

Unless you're manufacturing ideas.

So, OK, let's try to get this right, for goods with rival cofactors of production (such as land for the ag example) it (eventually) slopes up.

Not all goods and services have rival cofactors of production, though.

[andrew lainton]

But isnt it sloppy terminology and a 'micro' textbook fallacy to refer to an individual firms supply curve as

'even if the individual wheat farmer has constant returns to scale and a flat LR MC curve, the market LR MC curve for wheat will slope up'

The MC curve as experienced by a firm in the future and by the market are the same if you assume no arbitrage.

Rather it is better to say that a firm perceives falsely the supply curve it will experience because that curve will shift to the right over time because of macroeconomic factors and a fallacy of composition.

Both of which would invalidate rational expectations and so called 'micro (built on sand) foundations'

Either way we must throw away the micro textbooks and use much less confusing terminology that is founded on actual costs experienced by firms outside mythical perfect competition and the interaction of firms across multiple markets.

But as firms always plan to maximise productivity over their investment horizon I dont think that either would be good explanation.

[Nick Rowe]

Andrew: the individual firm's MC curve tells you what happens to MC if that one firm increases output holding other firms' outputs constant. The market MC tells you what happens to MC if *all* firms increase output. Let MC(y1,y2,y3, etc). The elasticity of the individual firm's MC is the elasticity of MC wrt y1, setting dy2=dy3 etc=0. The elasticity of the market demand curve is the elasticity wrt Y=y1+y2+y3+etc.

(And elasticity =/= slope, BTW).