Equalising the twin markups in a monopolistically competitive macroeconomy

The first markup is the markup of Price over Marginal Cost, required for individual firms' profit-maximisation. This is related to the elasticity of an individual firm's demand curve. The formula is: P/MC = [1/(1-1/e)].

The second markup is the markup of Average Total Cost over Marginal Cost. Free entry and exit of firms requires a zero profit equilibrium, which requires Price equals Average Total Cost. For the simple case where an individual firm's Marginal Cost is independent of its output y, the formula is ATC/MC = 1 + (TFC/y.MC) where TFC is Total Fixed Cost.

In intro microeconomics we talk about a "Short Run" equilibrium, where the first markup is satisfied (firms in the monopolistically competitive industry are maximising profits). And we talk about "Long Run" equilibrium, where the second markup is also satisfied (firms enter or exit the industry until profits are zero). In Long Run equilibrium both markups are equal to each other: P/MC=ATC/MC. And we tell a story about how positive profits encourage entry of new firms into the industry which causes existing firms' demand curves to shift left until their demand curves are tangent to their ATC curves, and the MR curve cuts the MC curve exactly below that tangency point, equalising the twin markups.

I want to do the same thing for macro. And I don't want to talk about "industries". Because "industry" is a pretty fuzzy concept when each firm produces a good which is different (in some way) from the goods produced by any other firm. And because we don't need to talk about "industries". And because we are talking about new firms entering the whole economy, not just one particular "industry". And I want to recognise that the "Short Run vs Long Run" distinction in macro means something very different from the "Short Run vs Long Run" distinction in micro.

[This post is my version of Miles Kimball's recent post. I am very much on the same page as Miles; except Miles talks about "industries" and I don't. This is mostly a "teaching" post, but macroeconomists might find it useful as a way of organising their thoughts.]

Here is the simplest possible case I can think of:

Labour is the only input, and each firm has a production function y = l - f. You need f units of labour (per period) just to keep the firm operating (the fixed cost) and each additional unit of labour produces one additional unit of output.

Each firm faces a downward-sloping demand curve with constant elasticity e.

If all firms set the same price, then all sell the same quantity of output.

[These three assumptions let me do representative firm analysis. It would be misleading to say that firms are "identical" (because they produce different goods); it would be better to say they are "symmetric".]

I'm also going to assume a downward-sloping Aggregate Demand curve (maybe the central bank targets NGDP), and a competitive labour market with an upward-sloping labour supply curve (as a function of the real wage W/P). Because this post is not about those things.

Solving the model for the symmetric Nash equilibrium is easy (but I will probably mess up the math, as usual). Since (real) Marginal Cost equals the (real) wage W/P (because one extra unit of output needs one extra unit of labour) the first markup determines the real wage as W/P= [1-1/e]. And that real wage in turn determines total employment L, via the labour supply curve L = L^s(W/P). Equality between the twin markups determines the output of an individual firm as y = (e-1)f. [Edited to fix typo/math error duh.] With both L and y determined, the number of firms n is determined via the aggregate production function Y = ny = L - nf, so that n = L/(y+f) = L/ef. With Y determined by n and y, Y = ny = L(e-1)/e, the AD curve then determines the price level P. [Update: One rather neat result of this very simple model is that aggregate output Y depends only on the elasticity of demand e, and not on the fixed costs f. That's because a doubling of f halves the number of firms, and leaves economy-wide fixed costs unchanged.]

But that is just the "Long Run" (in both micro and macro senses) equilibrium. How do we get there? What happens in the "Short Run"? And what about the distinction between the micro "Short Run" (number of firms fixed) and macro "Short Run" (price level fixed)?

I find this triptych useful:

 

The top picture is familiar to anyone who has taken Intro Micro. The only difference is that I have been careful to be explicit about relative price being on the vertical axis, and have set the equilibrium relative price equal to one, since I am doing representative firm analysis.

The bottom picture is familiar to anyone who has taken Intro Macro. The SRAS curve assumes that the nominal price level is fixed in the Short Run ("sticky prices").

But I will need to explain the middle picture.

[STOP freaking out about the Demand Curve in the middle picture being horizontal at a relative price of one! It means all firm's have the same income-elasticity of demand (one), and does not mean they have infinite price-elasticity.]

Suppose we start in full equilibrium in all three pictures. Now hold all firms' nominal prices constant, hold the number of firms constant at n*, and then shift the AD curve to the right, so the economy moves along the SRAS curve. Every firm sees its (downward-sloping) demand curve shift right by the exact same percentage (by the symmetry assumption needed for representative firm analysis). If every firm satisfies that increase in demand, by increasing output by the same percentage, what happens to its MR, MC, and ATC (the points on the MR, MC, and ATC curves on the top picture, which shift when all firms expand output together)? That's what the curves in the middle picture tell us.

The MR curve in the middle picture could slope either up or down. It will slope up if the elasticity of an individual firm's demand curve increases in a boom, slope down if elasticity decreases in a boom, and is horizontal (as drawn) if elasticity stays constant.

The MC curve in the middle picture probably slopes up. It would only be horizontal if the individual firm's MC curve in the top picture is horizontal, and if all firms could hire more labour (and other inputs) without needing to increase real wages to attract more labour (the economy-wide labour supply curve is horizontal). Otherwise it will slope up (and will be vertical if the economy-wide labour supply curve is vertical).

The ATC curve in the middle picture could slope either way. Because the individual firm's ATC curve in the top picture slopes down, but if real wages increase when all firms expand output and employment that causes the ATC curve in the top picture to shift up.

If the MC curve is steeper than the MR curve in the middle picture, then what microeconomists call "short-run equilibrium" (because we're holding number of firms fixed at n*) and what macroeconomists call "long-run equilibrium" (because prices adjust) will be stable. If the AD curve shifts right, with prices initially fixed, each individual firm will want to raise its relative price because MR < MC, so the general price level rises and sales fall as we move back along the AD curve to the original LRAS curve. (Yes I'm assuming the AD curve slopes down, which is a big assumption, but Off-Topic for this post).

OK, what about the number of firms? What process (if any) ensures that the MR and MC curves in the middle picture (eventually) cross at exactly the same level of output as the Demand and ATC curves?

Suppose initially that all firms are maximising profits, but profits are positive. (This must mean that the MR and MC curves in the middle picture intersect at a lower level of output than where the ATC and D curves intersect.) So new firms enter. The entry of new firms may drive down nominal prices (move the economy down along the AD curve) but that is irrelevant for real profits, since symmetry ensures their relative prices stay the same (at one). But the entry of new firms will drive up real wages, as the economy moves along the labour supply curve. So individual firms find MR < MC, so they cut output. And it is this cut in output by individual firms that increases the second markup (of ATC over MC) until it equals the first markup (of P over MC).

The entry of new firms (holding output per firm constant) raises W/P. And the rise in W/P raises MC and ATC by exactly the same percentage (given my simplifying assumptions about the production function). But the ATC curve is always more elastic than the MC curve in the middle picture, simply because ATC includes the Fixed Costs and MC doesn't. So an increase in the number of firms always shifts the middle picture's ATC-D intersection left more than it shifts the MC-MR intersection left. [I think I've said that right.]

[It's all very different from micro, where entry of new firms into an "industry" causes a fall in the relative price in that "industry".]

But Miles Kimball raises an interesting complication. What happens if the entry of new firms causes demand elasticity to increase? (This is plausible if new firms increase the density of the existing product-space, rather than expanding the product space along new dimensions.) If this effect is strong enough, so that an increase in n causes e to increase by a large enough amount, the entry of new firms would cause the MR curve in the middle picture to shift up by more than the MC curve, so the entry of new firms causes existing firms to expand rather than contract output. Hmmm.

Macro is hard. I think I will stop there.

Comments

[Not Trampis]

Good to have you back Nick. We plebs need you

[Nick Rowe]

Not Trampis: Thanks! Yes, I was a long time away from blogging. Various reasons.

[ltr]

I look forward to your posts, but I have no idea what this post is about. A brief non-jargon explanation of what this post is about would be ever so welcome. (I have a doctorate from a fine, fine school and am a professor, but I am lost here and would like to be found.)

[Nick Rowe]

ltr: sorry. Maybe I've been away from blogging too long, and my writing/presentation skills have got rusty. Maybe I will re-write it someday.

Let me try this:

In Intro micro when we do monopolistic competition we tell students what the long run equilibrium looks like and how we get there, by firms producing where MR=MC to maximise profits, and firms entering or exiting the "industry" to bring profits to zero. I want to do the same for macro. It's not just one "industry" (whatever precisely that means) that is monopolistically competitive; it's the whole economy.

Does that help?

[ltr]

In Intro micro when we do monopolistic competition we tell students what the long run equilibrium looks like and how we get there, by firms producing where MR=MC to maximise profits, and firms entering or exiting the "industry" to bring profits to zero. I want to do the same for macro. It's not just one "industry" (whatever precisely that means) that is monopolistically competitive; it's the whole economy.

Does that help?

[ This is a fine help and just what I at least needed.

Thank you, and do blog happily on and I will look forward to each post. ]

[Egmont Kakarot-Handtke]

Marginal madness
Comment on Nick Rowe on ‘Equalising the twin markups in a monopolistically competitive macroeconomy’

Your treatment of profit is partial and marginalistic. Marginalism is defined by the Walrasian axiom set=microfoundations. Because the Walrasian axioms are provable false your treatment of profit is false. Marginalism has already been dead in the cradle 140+ years ago.#1

For the determination of monetary profit of the economy as a whole one has to start with the most elementary case of a pure consumption economy without investment, government, and foreign trade.#2 In this elementary economy three configurations are logically possible: (i) consumption expenditures are equal to wage income C=Yw, (ii) C is less than Yw, (iii) C is greater than Yw.

In case (i) the monetary saving of the household sector Sm≡Yw-C is zero and the monetary profit of the business sector Qm≡C-Yw, too, is zero.
In case (ii) monetary saving Sm is positive and the business sector makes a loss, i.e. Qm is negative.
In case (iii) monetary saving Sm is negative, i.e. the household sector dissaves, and the business sector makes a profit, i.e. Qm is positive.

It always holds Qm+Sm=0 or Qm=-Sm, in other words, loss is the counterpart of saving and profit is the counterpart of dissaving. This is the most elementary form of the Profit Law. Total profit is distributed among the firms that comprise the business sector.

Profit for the economy as a WHOLE has NOTHING to do with productivity, the wage rate, the working hours, exploitation, competition, capital, power, monopoly, waiting, risk, greed, the smartness of capitalists, or any other subjective factors. Total profit/loss is objectively determined in the most elementary case by the change of the household sector’s debt.#3

Egmont Kakarot-Handtke

#1 For details see ‘First Lecture in New Economic Thinking’
http://axecorg.blogspot.de/2017/05/first-lecture-in-new-economic-thinking.html

#2 The macrofoundations approach starts with three objective-systemic (= behavior-free) axioms: (A0) The objectively given and most elementary configuration of the economy consists of the household and the business sector which in turn consists initially of one giant fully integrated firm. (A1) Yw=WL wage income Yw is equal to wage rate W times working hours. L, (A2) O=RL output O is equal to productivity R times working hours L, (A3) C=PX consumption expenditure C is equal to price P times quantity bought/sold X. For a start it holds X=O.

#3 For more details see cross-references Profit
http://axecorg.blogspot.de/2015/03/profit-cross-references.html

[Nick Rowe]

Egmont:

Capitalists may consume too. Workers may save too. And it's saving in the form of money that matters for the AD curve. And this post is not about the AD curve anyway.

[Egmont Kakarot-Handtke]

Nick Rowe

You say: “Capitalists may consume too. Workers may save too.”

True. In this case the Profit Law reads Qm≡Yd+Sm. This, though, does not alter the fact that your treatment of profit is false.*

Egmont Kakarot-Handtke

* For details see cross-references Profit
http://axecorg.blogspot.de/2015/03/profit-cross-references.html