What, in your opinion, is the shape of the Long Run Phillips Curve?
Supplementary: if you use the words "full-employment" in your answer, can you do your best to explain what you mean by this. (This is not a "gotcha!" question. I know that's a tall order, because it's not always easy to come up with a simple theoretically and empirically watertight definition of theoretical concepts; many orthodox economists have equal difficulty explaining what they mean by the "natural rate of unemployment".)
I think that Modern Monetary Theorists want "orthodox" macroeconomists to engage them more. I'm not sure that I'm really orthodox, but anyway; I sympathise with their desire, and this is an attempt to meet it.
Here is my "rational reconstruction" of the MMT answer. By that I mean here is what I can best assume they believe if I want to make sense of what they say:
"The Long Run Phillips Curve in {inflation, unemployment} space is L-shaped. Alternatively, the Long Run Aggregate Supply curve in {Price level, real income} space is reverse-L-shaped. The price level, or rate of inflation, is exogenous with respect to Aggregate Demand until you hit "full employment" (or "full employment output"). Then it turns vertical. So increases in Aggregate Demand (for example, due to money-financed increases in government spending, transfers, or tax cuts) will cause output to increase and unemployment to fall, with no effect on the price level or inflation, until you hit "full-employment". Thereafter they only affect inflation."
That's what I used to believe in 1972, when I was studying A-level economics and politics. (I only got a D; but perhaps, I hope, it was because I did very badly on the politics part?)
In 1972 I couldn't define what I meant by "full employment". But now i think I can rationally reconstruct what I would or should have meant by "full employment". Here it is:
"An economy is at "full employment" when an increase in Aggregate Demand won't increase employment even in the short run."
To put it another way, I made no distinction between the Long and Short Run Phillips and Aggregate Supply curves.
OK. Now for the comments.
Worthwhile Canadian Initiative 
Besides the modern veracity of a ‘Phillips Curve,’ the problem is that interest rates and employment have globalized. I greatly admire Prof. Pettis for his work and quote his blog entry here:
http://mpettis.com/2010/04/the-rmb-and-the-magic-of-accounting-identities/
The connection might not be obvious at first, but if you start thinking about the current and capital accounts and their effect on unemployment you might get a new perspective on the matter in question. I do not presume to completely understand it.
I read the Pettis piece, and I am a Pettis fan. He posits the China problem with the lack of reliable information in China about China. So the Phllips curve in China will be highly volatile as the information is not available to hedge the future.
In the USA, we can hedge against inflation for longer due to more accurate information. But eventually, constraints of important inputs appear, and we restructure. Hence the long term Phillips curve defines the non linear restructuring, that is the definition of long term is that Phillips no longer holds.
Here is mine on page 20 of this .pdf.
Click to access MoneyControlPosenSubmission.pdf
Figure 1 The backward bending Phillips curve showing the Minimum Inflation Rate of
Unemployment (MURI).
IMO, the price inflation rate that minimizes the unemployment rate is ABOVE 2%.
Not sure if the link showed up correctly.
http://www.thomaspalley.com/docs/research/
MoneyControlPosenSubmission.pdf
What if there is a shortage of something else that causes price inflation (going vertical) before “full employment”?
Here is an example. What if “full oil” occurs before “full employment”?
By Phillips curve — do you mean the relationship between unemployment and wages, or the relationship between unemployment and inflation? Phillips drew several curves..
It’s certainly possible to have low unemployment without rising unit labor costs — many nations adopted a disinflation policy that consisted of negotiations with labor unions rather than using the CB to bankrupt large swathes of the private sector — c.f. Wassenaar Agreement — and these policies worked!
Volcker’s approach was not the best, or third best, way to control inflation. At the same time, it’s possible for there to be inflation without the cause being rising unit labor costs. A large expansion of credit can lead to demand-led inflation, for example, and low unemployment would not be to blame. In fact, the casual chain might be: unwarranted expansion of credit –> increased prices, with decreased unemployment as a side-effect, rather than a cause, of the inflation.
So the topic is a bit like asking how much liquor do you need to drink before you get pregnant. The whole question is loaded with a lot of assumptions that many reasonable people would object to, even though there is no doubt a statistical relationship between unwanted pregnancy and liquor consumption.
And as an aside, when asking “Is low unemployment a consequence of a booming economy with rising prices, or does low unemployment cause the rising prices?” The first place to look as unit labor costs, which do not lead rising prices, but are a coincident indicator, so there is evidence that the same forces that cause demand to increase also cause unemployment to fall at the same time, bolstering the excess credit as a driver of inflation argument, rather than rising unit labor costs.
I remember once looking at this relationship and have dug up an old chart:)
TMF: Employment is maximized at any stable inflation target. To deny this, really is to deny 30 years of progress.
Nick: If low unemployment leads to hard nominal bargaining, then employment leads to wage inflation, and the data need not admit any possibility of inflation boasting employment even in the short-run.
RSJ: “By Phillips curve — do you mean the relationship between unemployment and wages, or the relationship between unemployment and inflation? Phillips drew several curves..”
Whatever you (they) like. Put some real variable (unemployment, employment, output..), on the horizontal axis, and a nominal variable (prices, wages, their rates of change…) on the vertical axis, and describe the curve(s) traced out in the long run when aggregate demand varies. Call it the Phillips Curve, Aggregate Supply curve, whatever.
We must ralize that any long run relationship between inflation and full employment of resources not just labor but also productive capacity and other materials depends on the level of inefficiencies/inadequacies of resources. These are variable with policies that expand human capital embedded in labor,skill development,technology/innovation of existing productive capacity, infrastructure that reduces frictions, etc. For example, capital adequacy instead of being measured only in monetary value it should also be measured in terms of its real capacity to produce. For a given financial capital measurement, there is a variable adequacy to produce output and reduce inflationary pressures. This can be viewed as an inflation at risk estimation of capital adequacy!In any event, at the long run the curve is shifting with private spending in R&D, education and training programs and discretionary fiscal spending as development policy.
“mployment is maximized at any stable inflation target. To deny this, really is to deny 30 years of progress.”
At nearly 10% unemployment, it is long past time to revisit this.
Nick,
Ok, I put y/y change in unit labor costs on the y-axis, and unemployment on the x-axis, and find no correlation in the data. Theoretically, you could assume it would be “L” shaped for very low levels unemployment that are in practice unreachable, and in times of depression unit labor costs would be falling while unemployment is rising, but generally there is no relationship.
The real question is, why single out labor, if not for political reasons? Do we require 5% of homes to be vacant? 5% of the population to be homeless? 5% of capital to be idle? 5% more oil to be produced than is sold? We are able to distinguish between a market clearing and the price. But for some reason we are afraid of the labor market clearing and are not afraid of the other factors of production clearing. So I object the casual relationship implied by NAIRU, rather than the act of plotting nominal variables against real ones to infer that an economy suffers from nominal effects.
RSJ: “The real question is, why single out labor, if not for political reasons?”
None. “Unemployment” could mean unemployment of any resource, if you like. Labour is the most important, quantitatively, but all resources can be unemployed. And if you put output, or real income, on the x axis, then all productive resources are implicitly included.
Put it another way: if you believe in a natural rate of unemployment, you almost certainly have to believe in a natural rate of unemployment of any resource, not just labour.
“Put it another way: if you believe in a natural rate of unemployment, you almost certainly have to believe in a natural rate of unemployment of any resource, not just labour.”
Agreed! But in that case, you are believing that there is a “natural” failure rate of the market to clear. Now — there is nothing wrong with believing in that per se, but why would this natural failure to clear rate be dependent only on the price of the good in question, and not on the quantities/prices of the other factors of production?
So this is again an aggregation problem. If you believe that there is an aggregate production function with the following characteristics:
* Adding an additional unit of labor without increasing the capital stock will result in a positive marginal increase in output.
* Capital and labor are substitutable for each other
* All combinations of labor and capital will result in non-zero output
Then indeed you can argue for a NAIRU.
But, at the micro-level, all of the above are false. At the micro level, businesses lose money, cannot substitute labor for capital, and cannot increase output by increasing only a single factor of production. So the quantity and price demanded for each factor depends on the quantity and price demanded for all other factors. A business short of capital cannot hire more labor. And a business short of labor cannot deploy more capital. Because of this constraint, if the cost of capital is too high, then a business will not hire more labor even if labor is fairly priced. As a result, you have an explanation of why the aggregate labor market would fail to clear.
Therefore in aggregate, you would expect U(Labor) to also be a function of U(capital) and of the price of capital. And in that case, you lose the ability to say that a low U(labor) will “cause” the price level to increase. It could just be that at the micro level, capital and not labor tends to be the constraining factor on production (because the ratios are constrained), and so you can max out labor utilization without maxing out production and therefore prices.
Or, it could be the other way around, and labor is more constrained. So if you aggregate from the bottom up, U(L) just doesn’t supply enough information to draw a relationship between it and the price level. But I agree that when you put “Output” on the x-axis, you can get a relationship — but perhaps not a very interesting one.
RSJ: “But I agree that when you put “Output” on the x-axis, you can get a relationship — but perhaps not a very interesting one.”
I could see an Austrian taking that position, because they don’t think aggregates are meaningful and interesting (in some sense). But I don’t interpret MMTers as thinking that. As far as I know, since they do commonly talk about aggregate concepts, like aggregate demand, output, employment, etc., they do think aggregates are meaningful. (Of course, when pressed, no doubt like any sensible people they would agree that aggregation can leave out some other interesting information.)
You have sketched out a theory of the natural rates of unemployment for each of the various inputs (and even “labour” can be disaggregated into the millions of different types of labour), based on Leontieff (y=min{k,l} ) production functions at the micro (individual firm) level. I disagree with your theory. But that’s a bit beside the point of this post. Unless you are saying that your theory is also the MMT theory of the natural rate?
“if you believe in a natural rate of unemployment, you almost certainly have to believe in a natural rate of unemployment of any resource, not just labour.”
Really?
Adam P: Yes, (allowing of course that some of those natural rates might be zero).
You get a natural rate of unemployment under neutrality and super-neutrality of money. If money were non-neutral, then it would be a big coincidence if some real variables (like some unemployment rates) changed but other real variables stayed the same. Mostly, in General Equilibrium, everything depends on everything else.
I suppose if there were a zero natural rate of unemployment in some resource, (due to zero search and other frictions etc.), then that zero natural rate might stay invariant to other real changes.
Is that what you had in mind?
Nick:
I’m not the MMT expert, but I think everyone believes aggregate quantities are meaningful and important — I’m only disputing the relevance of this particular pair of variables. I know that MMT likes to focus on sectoral balance sheet identities and other aggregate accounting identities. But accounting identities are not “real” identities.
About the min{} functions, yes, this is an example of how a natural failure of the market to clear would arise. But you don’t need to believe in Leontieff production functions per se.
If you are looking at micro-firms that grow in size, then each firm is a price taker for all factors of production other than capital, which is unique to the firm.
Suppose you have a whole distribution of micro-firms. That means you get a distribution of marginal products of labor, for example, and a distribution of marginal products of other factors. At any wage rate, there will be some firms for whom labor is too expensive and other firms for whom labor is too cheap.
You can hope that the firms for whom labor is too cheap will grow in size, and as they do, their MPL will fall to the “equilibrium” wage. And the small firms will shrink so that the MPL will rise to the equilibrium rate. But what if there is more than 1 factor of production? Now, you need either all the factors of production to be perfectly substitutable or you need the production curve to be exactly the same function for each firm. If there is variance in the shape of the production function or some non-infinite elasticity of substitution of factors of production, then a model with more than one firm will in general not be able to reach a state where every firm has the exact same marginal products for all factors of production. And as long as that happens, the market will fail to clear. So this is a much more general result than a requirement that each PF be Leontieff.
And it has nothing to with frictions — it holds with zero frictions or search costs. Variance in production functions and substitutability of factors is sufficient to get the market to fail to clear.
And it’s Saturday, so here is a mini example:
Assume not one, but two production functions in a very general form: f_1(K_1, L_1) and f_2(K_2, L_2). And let the total employed labor be L = L_1 + L_2, and the total deployed capital be K = K_1 + K_2. Then the requirement that the marginal product of f_1 with respect to L_1 = marginal product of f_2 with respect to L_2 = wage rate
forces L_1 to be a function of X(L, K, K_1). And the similar requirement on capital forces K_1 to also be a function of Y(L, K, L_1). These curves will generally intersect in disjoint loci, and each loci will describe a surface Z(L,K). What this means is that a change in the total K forces a change in the total L. But changes in K — dK/dt = Investment — are volatile and depend on expectations, therefore the total size of the labor force is similarly volatile. And this assumes perfect flexibility in wages and no frictions as well as perfectly competitive markets.
So even if you start out with generic micro production functions, in which capital and labor are substitutable, you still end up with the conclusion that only certain types of (capital, labor) combinations are feasible — regardless of the wage rate or interest rate.
And that is just with 2 production functions, and no requirements that they be Leontieff.
There are many such emergent effects when you disaggregate production that explain well the aggregate economic phenomena that we see, but are completely hidden and inexplicable if you assume the entire economy is a single corn farm, or that all production functions are identical.
Even at full employment of resources a variable adequacy of these resources to produce can reduce inflationary pressures and expectations.In my research the adequacy factor is calibrated to incorporate random/stochastic steps and jumps with an asymptotic hyperbolic function in order to avoid decay with size. Why assume that there are no inadequacies of resources? These can be assisted and remedied by discretionary fiscal policy of allocative nature such as reorganization, innovation, skills formation and other qualitative measures of development policy that reduce frictions.These measures are given the stock of resources and productive capacity. On the other hand, growth policy is thew discretionary fiscal measures that expand the stock of resources. Finally,the automatic endogenous fiscal policy performs the role of stabilizing the effective utilization of these resources at or near full employment.In this hypothesis with a variable adequacy to produce, the relationship between full employment of resources and inflation is unstable and shifting. Actually there is a case of inflation at risk given the degree of inadequacy.
RSJ: I just don’t see it.
Assume there are 2 farms, that both produce the same good, corn, and wages and capital rentals are measured in corn. (You assumed that the output prices are the same for both production functions, and equal to one, so I’m just echoing your implicit assumptions here.) Assume the 2 farms have different production functions (because, say, one has sandy and one clay soil). OK.
So the competitive equilibrium (and the solution to the social planner’s problem of maximising total output of corn) is defined by:
Farm 1’s MPL1(L1,K1) = Farm 2’s MPL2(L2,K2)
Farm 1’s MPK1(L1,K1) = Farm 2’s MPL2(L2,K2)
L1+L2 less than or = L
K1+K2 less than or = K
4 equations and 4 unknowns, so look good so far.
I can imagine production functions where we don’t get an interior solution, i.e. where one or both of the resource constraints is non-binding (is a strict inequality). That would be where the MPL and/or MPK is zero in equilibrium. So it would be efficient for the social planner to leave labour and/or capital unemployed in equilibrium. But in competitive equilibrium that would mean that wages and/or capital rentals are zero. In other words, labour and/or capital isn’t scarce, so some is unemployed.
Is that what you are saying?
If so:
1. You can get that result with a single farm; you don’t need 2 farms. Just assume strongly diminishing returns and abundant labour.
2. I don’t think it’s realistic. Unless you are talking about really Malthusian economies, marginal labour seems to be capable of producing some extra output, and would be employed if the firm could sell the extra output and hire the worker at a low enough real wage.
OK, a couple of points:
What do you mean by “boundary”? For example, in the special case of two CRS CDPF with exponents a and b, then the result is
Labor 1 = lambda_1 Capital 2
Labor 2 = lambda_2 Capital 1
where lambda_i is some ugly function of a and b. The point is that the total labor demand is a function of the size of the capital stock, regardless of the wage rate.
But the rate of change of the capital stock is investment, which is volatile. Even if the the total labor demand were maximal, any variation in investment will push the utilized labor away from maximum. So you get into the situation that you previously dismissed, which is fixed capital/labor ratios leading to unemployment independent of the wage rate due to a shortage of capital. In this case, these ratios are fixed because there is more than one production function. By the way, I am not saying that in the “real” economy the ratios are fixed, but I am saying that a shortage of capital can force unemployment regardless of the wage rate, and you don’t need leontieff PF for this. standard constant return to scale production functions will do the trick, and rapidly diminishing production functions will work even more strongly, as in that case you get bounded solution sets — e.g. ellipses, etc.
re: point 2
“marginal labour seems to be capable of producing some extra output, and would be employed if the firm could sell the extra output and hire the worker at a low enough real wage.”
That is the point! From whence would this “extra” spending power come?
You can start out with perfectly flexible wages, and still have equilibria that have unemployment if your model consists of a flow of (overlapping) production functions, each of which have rapidly diminishing returns. In fact, you can have a growing economy and still experience these multiple equilibria in case of shocks to the expected return of each micro-production function.
So I am not saying that full employment is impossible, but that whether or not there is full employment depends on the capital stock size, which itself depends on expectations, even with perfectly flexible wages.
oops, should have previewed:
it should read: labor 1 = lambda_1capital_1, etc.
The point is that with a single PF, K^aL^1-a, if you take MPL and set it to w, you get:
K^a/L^a = w
so even with a very small K, you can make w small enough so that L is at full employment. And people took this toy and climbed mount everest with it.
But as soon as you have *two production functions, then w drops out and you get a relationship of the form
K_i= lambda_i * L_i –> and no “w” appears to bail you out and provide you with full employment by being small enough.
— regardless of the form of your production function. Only in the degenerate case where all production functions have the same exponent can you fall back on the “let’s lower wages” to boost employment argument.
In all other cases, you need to increase capital to boost employment. And we all know how volatile capital is.
Hope that makes sense…
Guys,
I am sure you are familiar with the problems of heterogeneity and indivisibilities of resources, the Cambridge aggregation controversy and the Samuelson reswitching issue of non monotonicity. The relation between the employment of resources and their remumeration is not so clear and stable, and their markets cannot clear.
Panayotis: I used to know something about the Cambridge^2 Capital Controversy, 30 years ago!
RSJ: OK. I think I follow you now.
First off, forget my “farm” example above. Since you are assuming Constant Returns to Scale in Labour and Kapital, there obviously cannot be a third fixed factor, land. So your firms are not “farms” (because farms have land).
OK. So we have 2 firms, both producing the exact some good (so the two firms have the same price of output), each with a different technology. Both firms’ technologies are Constant returns to Scale Cobb-Douglas:
Y=L^a.K^b where a+b=1
But one firm has a more labour intensive technology than the other. (One firm has a higher a/b ratio than the other).
Perfect competition.
Those are your assumptions, right?
Since I am crap at math, I’m going to “solve” for the aggregate labour demand curve intuitively.
Hold aggregate K constant for the moment, while we vary L, and so vary aggregate L/K.
Imagine we start of with zero L, and slowly increase L, and watch what happens to MPL as we do this.
With very small L, and so very small L/K, only the capital intensive firm produces output. The labour intensive firm shuts down. With only one firm operating, we get the standard downward-sloping MPL curve as L/K rises.
When L (and L/K) increases past a certain point, call it L’, both firms start operating. As L increases further, each firm holds its K/L ratio constant, but the capital-intensive firm produces less and less output (using less and less K and L), and the labour-intensive firm produces more and more output (using more and more K and L). The MPL at each firm is constant; it does not diminish as aggregate L increases.
(This is where, as you say, it looks as if each firm has a fixed coefficients technology, even though they don’t).
When L (and L/K) increases still further, past a certain point, call it L”, the capital-intensive firm shuts down completely, and only the labour-intensive firm operates. As L increases past this point, we again get the standard diminishing MPL as more and more L is added to a fixed K.
Under perfect competition, the MPL curve is the same curve as the labour demand curve. So what does this curve look like, under your assumptions?
Holding aggregate K constant, the labour demand curve slopes down between 0 and L’. Then it becomes horizontal (perfectly elastic) between L’ and L”. Then it slopes down again as L increases past L”. But the MPL is always positive.
What happens if aggregate K increases? The whole labour demand curve shifts horizontally to the right. If K doubles, for example, L’ and L” will both double. We get double the quantity of labour demanded at any given wage (or MPL). (That’s because, under CRS, only the ratio K/L matters for MPL.)
Sure, it’s a weird labour demand curve. It has a kink at L’ and at L”. And it’s horizontal between L’ and L”. But that labour demand curve cannot create unemployment. You would only get unemployment if: wage was stuck above MPL; or there’s a shortage of Aggregate Demand. But if wages were above MPL, or there were a shortage of AD, we could get unemployment even with a regular labour demand curve.
(If we drop the assumption that both firms produce the exact same good, so the relative price of the two goods varied as their relative outputs varied, we would get rid of the 2 kinks and the horizontal section. the labour demand curve would be a regular downward-sloping one.)
You don’t get unemployment under your assumptions.
Plus, I don’t think this is what the MMTers are talking about anyway.
Yep: Micro theory is slowly coming back to me.
Draw the unit isoquant for each technology in {K,L} space. Two convex curves, that intersect, so the envelope of those two curves is not convex. But it can be convexified easily.
Draw the isocost line that is tangent to both those unit isoquants. The slopes of the rays from the origin to those two tangency points points define the K/L ratios of each technology, under the assumption that both technologies are being used. They will both be used if the aggregate K/L ratio lies somewhere between the slopes of the two rays. By varying the ratios of the outputs of the two technologies, you can vary the aggregate K/L ratio even if both technologies hold their K/L ratios fixed. So that constructs an aggregate unit isoquant that is weakly convex. (The aggregate unit isoquant is a straight line between those two tangency points, and outside those two tangency points it simply follows the convex locally superior technology, because only one technology is being used.)
So the aggregate production function is well-defined, and weakly concave (“weakly”, because it has a triangular-shaped flat section in the middle). And the aggregate labour demand curve is also weakly downward-sloping (it has a horizontal section in the middle). And the aggregate capital demand function is also weakly downward-sloping.
OK, good. Now I point out that
1. Kink region — the non-convex region — is the general case
* The economy does have more than 1 firm. It has many industries, even.
* The labor pool, or occupations if you prefer, service many different industries, so there can never be 1 firm.
* Capital of all firms competes
* All firms in all industries compete for Land, energy, etc.
So we are always operating in the “kink” region in which the economy has more than 1 firm. Moreover, it is a pretty reasonable assumption that the production characteristics of firms are not the same.
2. You need equilibrium in both markets simultaneously.
Suppose you are at a point (K_1,K_2, L_1, L_2) such that:
MPL1 > MPL2 and MPK2> MPK1.
I claim that in this situation, there does not exist a wage that allows both markets to clear. As firm 2 sheds labor to make MPL2 rise to equal MPL1, and firm 1 takes on labor to make MPL1 fall to MPL2, MPK1 will also rise and will not equal MPK2.
So there is a feasibility region of certain tuples of (K_i, L_i) that are needed for the market to clear. And this feasibility region is actually a surface that can be described as
Total Labor Employed = f(Total Capital).
This is a simple implicit function result — as long as the gradients {(MPL1, MPK1), (MPL2, MPK2)} are not identically the same, they will be equal on a set of dimension 2, which can be viewed as a curve in (Total K, Total L) space. Independent of the interest rate or wage rate, simultaneous equilibrium in both markets requires the firms to operate at some point on this curve. And this curve will intersect “full employment” only for a single magical size of the total capital stock. Any other size of the capital stock, and employment will not be full.
If the number of firms > then the number of factors of production, then this result still holds — the total quantity employed of each factor of production can be viewed as a function of the total capital stock, so that
Total Labor = f_1(Total K),
Total Land = f_2(Total K),
etc.
What this means is that if you start at full employment, say, but the level of total capital decreases, then the level of total labor employed must decrease as well. And no change in the wage rate will help you, unless somehow a shift in the wage rate causes the capital stock size to increase. This is important because, in practice, the level of the capital stock is volatile and dependent on forward looking expectations. A change in these expectations, lowering K, will force a certain level of unemployment even if wages are perfectly flexible.
3. It seems to me that you are trying to find equilibrium in the labor market, and then say that you can find equilibrium in the capital market, and just assume that this means you can simultaneously find equilibrium in both. You can’t do this except in the degenerate case of a single production function. In the general case, you will only be able to choose pairs (equilbrium wage, equilibrium return on capital) in which lowering the wage rate also requires lowering the MPK, and this forces you to hold certain capital/labor combinations.
None of this requires Constant Return to Scale, but it’s easy to explicitly solve for the constraints in that case. All it requires is that production functions be independent (e.g. that their gradients not be multiples of each other). This is all just calc 101 or wherever you learn the implicit function theorem, which I admit is a deep result, as it requires a contraction mapping argument, but nevertheless should not be up for dispute.
Ugh, it should read
“As firm 2 sheds labor to make MPL2 rise to equal MPL1, and firm 1 takes on labor to make MPL1 fall to MPL2, MPK1 will fall and MPK2 will rise, so the two will diverge. In that case, both firms need to add large amounts of capital to bring MPK into equilibrium. Therefore decreasing the wage rate forces the equilibrium MPK to decrease, and it forces K to increase.”
Basically, if you don’t want to look at the implicit function argument, the “pictorial” way to see this dynamic is that with more than 1 firm, the equilibrium wage rate moves in the same direction as the equilibrium MPK. So if you need a low wage rate to make the labor markets clear, you will get a low MPK as well. But a low MPK requires a high total capital stock. So full employment will only be possible with a high enough level of the total capital stock, and changes in the capital stock will cause unemployment.
With only a single firm, MPK becomes decoupled from MPL, and you can make the labor markets independently of the capital markets by adjusting the wage rate for any given equilibrium MPK.
And I agree that I have hijacked this thread — sorry! If you want, you can move these posts to some other thread? The minimum wage thread?
RSJ: “And this curve will intersect “full employment” only for a single magical size of the total capital stock. Any other size of the capital stock, and employment will not be full.”
Nope.
Start at that magical size of the total capital stock. Now remove one unit of capital, holding the labour force constant, so the aggregate capital/labour ratio falls. Here’s how we stay at full employment:
The capital-intensive firm (with the higher K/L ratio) contracts both K and L, holding K/L constant, and so MPL and MPK constant.
The labour-intensive firm (with the lower K/L ratio) expands both K and L, holding K/L constant, and so MPL and MPK constant.
Intuitively, the aggregate K/L ratio is a weighted average of the high K/L ratio in the capital-intensive firm and the low K/L ratio in the labour-intensive firm. By varying the relative size of the two firms, you vary the weights in the weighted average, and get to full employment.
“Start at that magical size of the total capital stock. Now remove one unit of capital, holding the labour force constant, so the aggregate capital/labour ratio falls. Here’s how we stay at full employment:”
No, here is an example of the math:
L_1 = lambda_1 K_1 (lambda_1 is the ratio for firm 1)
L_2 = lambda_2 K_2 (lambda_2 is the ratio for firm 2)
You do agree that the “equilibrium” in both markets solution requires this, right?
Now, subtract 1 unit of capital — say from firm 1.
Then L_1 must fall by lambda_1 in order to stay at equilibrium in both markets. You have reduced the capital stock and created unemployment as a result. let’s see if we can get back to full employment without increasing the capital stock:
You add that labor to firm 2. But in order to stay at equilibrium, then K_2 must increase by 1/lambda_2. So you have only achieved full employment as a result of increasing the total capital stock from the level it was when you had unemployment.
“Intuitively, the aggregate K/L ratio is a weighted average of the high K/L ratio in the capital-intensive firm and the low K/L ratio in the labour-intensive firm. By varying the relative size of the two firms, you vary the weights in the weighted average, and get to full employment.”
Yes, but you are still constrained by the largest ratio. You cannot vary the size of the firms arbitrarily and obtain full employment independent of the size of the total capital stock.
In this case, say lambda_1 = 2 and lambda_2 = 3. Then if the capital stock is 1/4, nothing you can do will give you full employment. There does not exist a wage rate that can achieve full employment in this case.
This is unlike the situation in which you have just a single firm. In that case, even with a capital stock of .000001, you can lower the wage rate to a level that achieves full employment.
In the more realistic case of a large number of firms that are not constant return to scale (e.g. monopolistic competition), you have more complicated curves instead of straight lines — e.g. hyperbolas
L_1^2 – lambda_1K^2 = 1
…
L_N^2 – lambda_NK^2 = 1, etc.
and get into a situation in which no shift in relative firm sizes will give you a lower capital/labor ratio, and yet still have a large number of firms.
I forgot to finish the example
Drop 1 unit of capital from firm 1, and you lose lambda_1 units of labor
To add lambda_1 units of labor back to firm 2, you need to add lambda_2/lambda_1 units of capital.
In other words, only if lambda_1 = lambda_2 — the production functions are the same function — can you keep the capital stock constant and add or subtract labor to the economy.
Now, in general, you can argue that this process will get rid of one of the firms. But it wont — as competing firms die out in a given industry (or sub-industry), the remaining firms begin to operate under monopolistic competition, not constant returns to scale. So you will no longer be in a situation in which there are fixed ratios and one firm will be more capital intensive than another only for some levels of K, and it will be more labor intensive for other values of K. In other words, the ratios are no longer fixed, and if you remove “too much” capital from one industry, it becomes the most labor intensive industry. So you are stuck with multiple firms as long as you have multiple industries (or subindustries), or geographic regions, etc.
RSJ: “Yes, but you are still constrained by the largest ratio. You cannot vary the size of the firms arbitrarily and obtain full employment independent of the size of the total capital stock.
In this case, say lambda_1 = 2 and lambda_2 = 3. Then if the capital stock is 1/4, nothing you can do will give you full employment. There does not exist a wage rate that can achieve full employment in this case.”
Yes you can.
Just to repeat your question, to show I’ve understood it, and to clear up some notational difficulties:
Suppose that, with both firms operating, the K/L ratio in the labour intensive firm is 1/2, and the K/L ratio in the capital-intensive firm is 2.
So if the ratio of the total supplies of capital to labour is between 1/2 and 2, we can always find a mix of output of the two firms that makes the weighted average K/L ratio of the two firms equal to the ratio of the total supplies, and so get to full employment.
But what happens, you ask, if the ratio of the total supplies of K/L is (say) 1/4?
Answer: the capital-intensive firm stops production. We are only left with one firm operating. And it operates at a K/L ratio of 1/4. And as the K/L ratio varies between 0 and 1/2, with only that one technology being used, the MPL/MPK ratio varies accordingly, and so does the wage/rental ratio.
The reason the capital-intensive firm stops production is that it will have higher costs per unit of output than the labour intensive firm. So it will make losses and shut down, You can see this from the isoquant/isocost diagram.
(In {K,L} space, a “unit isoquant curve” defines the combinations of K and L that will produce 1 unit of output. With Cobb-Douglas production functions Y=L^a.K^(1-a) the unit isoquant will be convex to the origin. If ‘a’ is different between the two firms, each will have a different unit isoquant, and the two curves will cross once. The isocost lines are downward-sloping straight lines that show combinations of K and L that cost the same to rent. The slope of an isocost line will equal the wage/rental ratio. Competitive equilibrium is at a point where the isocost is at a tangent to the isoquant.)
Just to add some intuition:
RSJ’s “lambda’ is defined (I think) as the L/K ratio. The convention in economics is the inverse, the K/L ratio, which we denote by lower case k.
Under Constant Returns to Scale, MPL and MPK are a function of k only.
There are 3 types of equilibrium in this case:
1. Both firms operating. In this case, MPL1=MPL2=W, and MPK1=MPK2=R (where W is wage and R is rental on labour). In this case, the aggregate k lies somewhere between k1 and k2. As you vary aggregate k (by varying the total supplies of K and L), neither k1, k2, W, or R change. The only thing that changes is the relative output of firm 1 to firm 2. Define k1* and k2* as the K/L ratios for the two firms when both are operating. Assume k1* is less than k2* (so firm 1 is labour intensive and firm 2 is capital intensive).
2. If aggregate k is less than k1*, then only firm 1 operates. W=MPL1 and R=MPK1. Firm 2 would make losses at this W and R, so does not operate. W is greater than MPL2 and R is greater than MPK2, for any k2. As k increases in this range, MPL=W rises, and MPK=R falls.
3. If aggregate k is greater than k2*, then only firm 2 operates. W=MPL2 and R=MPK2. Firm 1 would make losses at this W and R, so does not operate. W is greater than MPL1, and R is greater than MPK1, for any k1.
“But what happens, you ask, if the ratio of the total supplies of K/L is (say) 1/4?
Answer: the capital-intensive firm stops production.”
No, because the firms are in different industries — or sub-industries, and there is a large number of them. As competitors go out of business the production characteristics of the remaining firms in each industry change, shifting more towards monopolistic competition in which it is no longer the case that the ratio of capital to labor is constant, but it is a curve. In that case — the general case of non-constant returns to scale and changing production functions — does not allow you to get rid of all of the firms but one — you can get rid of some, of course, but you still do not reach the single firm-single production function case, and so you are still stuck with the fact that a downward shift in the equilibrium wage rate forces a downward shift in the MPK. You cannot independtly tweak wages in order to get the labor market to clear without also tweaking the MPK downwards as well. And the latter is a volatile function of expectations.
We do live in an economy with a large number of sectors and a large number of firms. Although this line of reasoning might be a good argument as to why monopolistic competition is inevitable even if you start with perfect competition — i.e. this is an argument that perfect competition is not “stable” if there are multiple markets all drawing from the same pool of resources.
I would add that this is a bit similar to your earlier “black hole” question. You are taking a model applicable only at the margins — e.g. constant returns to scale — and using it drive the capital of an entire firm to zero, with the assumption that this linearization of constant returns to scale will continue to hold independently of the size of the firms capital stock, or the number of competing firms. There will be at least one firm in each industry, and so you will never reduce the number of firms to 1 unless all industries but one disappear. But firms will become monopolies long before then, so that CRS will no longer be valid.
In general, the scale-behavior of a firm will change as the number of competitors changes and as the size of that firm’s own capital stock decreases. In the beginning, all firms have increasing returns, then they reach the point of diminishing returns, and then they finally hit the consol rate. If you shrink the capital stock of the firm too much, you will push it back to a different production function, and so you can’t use these linearization to deduce that all firms will disappear but one.
OK. Suppose the two firms are in different industries, producing different goods. Then we have to drop the assumption that both goods have the same price.
And instead of MPL1=W etc., we have to write P1.MPL1=W=P2.MPL2 and P1.MPK1=R=P2.MPK2. Where P1 is the price of the good produced by firm 1, and P2 the price of the good produced by firm 2. And we would also assume, if demand curves for the two goods slope down, that P1/P2 is a decreasing function of Y1/Y2. And what that assumption does is to get rid of the “flat spot”. So the whole model changes. But you still don’t get unemployment, as long as wages are flexible, and there’s no shortage of AD, regardless of the stock of K.
All that assumes perfect competition. If we drop that assumption, then we replace P1 with MR1, and P2 with MR2. But in “normal” cases, MR1/MR2 is still a decreasing function of Y1/Y2. And we have to do this if we want to assume Increasing Returns to Scale, also.
Monopolistic competition by itself will mean lower real wages, and lower capital rentals, in equilibrium. If the labour supply curve slop[es upwards, that will give you lower employment, but still no excess supply of labour. You can get unemployment, however, if demand is inelastic enough.
“Then we have to drop the assumption that both goods have the same price.”
I think you are confusing goods (= firm output) with factors of production. Different industries still draw from the same labor pool, they pay the same price for oil, and electricity, etc. and all capital competes with all other forms of capital for return.
This is the distinction between occupations and industries. An accountant can work for an airplane manufacturer or for Disney. So assume that there is one price for each occupation, or with 2 factors of production, you can assume that there is one common price for each factor.
This is really the key distinction here between assuming that the economy contains a single firm and multiple firms. In the case of a large set of micro-firms, all drawing from a similar (and it need not be exactly the same pool of resources), but selling the output in one of several different markets, you naturally get monopolistic competition as an outcome even if you start with perfect competition. Perfect competition is extremely unstable when you have more than one firm (a requirement) due to the dynamics that you point out.
The requirement that there be a single price for each factor of production, even though the price of the finished goods is different, is a strict (and realistic) transformational constraint that forces a lot of dependencies between the quantities of factors of production used in competitive markets. All of that is missed by assuming there is only one firm.
“I think you are confusing goods (= firm output) with factors of production.”
Nope. I was talking about the prices of firms’ output goods. If firms 1 and 2 produce different output goods, those goods will have different prices. But if they hire the same labour and capital goods, they pay the same wages and rentals (assuming competitive factor markets).
And the first-order conditions for profit maximisation are, as I said, P1.MPL1=W=P2.MPL2 etc.
Look. My guess is that you probably have a graduate degree in math (or similar). But you are largely self-taught in economics. Right? Because you are obviously really intelligent, and know some stuff well, but also have some gaps in mainstream micro theory.
I’m crap at math, and my micro is very rusty. But remember, a load of guys who are very good at math and extremely good at basic micro theory have been ploughing this field for many decades. And they haven’t come up with a theory of unemployment from this stuff. You need to throw something else into the mix. Playing around with Cobb-Douglas production functions alone won’t do it.
Nick
Thanks for encouraging this dialogue. I’m not even going to chime in on the technicalities of this discussion (RSJ is clearly capable and I would in effect simply be saying “What HE said”!) but I think a stark contrast I can already see and Im only third of the way through the comments, is that the mainstream laws, theorems and relationships have been created in so many instances to simply satisfy a small set of “perfect” conditions which almost never exist. They dont do “messy” very well. Often times when its pointed out that such and such a real world situation doesnt meet these relationships described in this particular curve (like the Phillips curve) the response too often is “Well it would if the govt werent in the way” or something like that. Explaining things away is not the same as explaining.
Greg: I look at it a little differently. It is really hard to understand “messy”. We are forced to make simplifying assumptions so we can understand it, define a question clearly, and answer it. So we face a trade-off.
In this case, RSJ made some simplifying assumptions, so the question was well-defined, and it has an answer. (And if my maths were better, or I could draw diagrams in these comments, I could have answered it much more conclusively and clearly.)
But when we go to a messier set of assumptions, it’s not so clear that the question is well-defined, and it’s so easy for everyone to just wave their hands around and BS.
Nick Rowe and RSJ,
the issues of heterogeneity of resources, indivisibilities, the aggregation measurement problem and reswitching of non monotonicity are at the core of your discussion. There is complex math involved beyond the simple assumptions of this discussion but the bottonm line is that there is no stable relation between resources and their renumeration!
Sure. But has anyone ever worked out a coherent model of unemployment based on all that stuff? (Without assuming sticky wages etc. that could generate unemployment without needing all that stuff). Because, I confess, my immediate reaction is to suspect obscurantism.
By the way, let’s take the simple model that RSJ and I have sketched out above. The equations that define the equilibrium are:
P1.MPL1=W=P2.MPL2 and P1.MPK1=R=P2.MPK2
RSJ (in particular): Notice anything peculiar about that system of equations, just looking at it as a mathematician?
P1, P2, W, and R all have $ in the units. (We call them “nominal variables”). And that system of equations is Homogenous (of degree one, IIRC?) in those nominal variables. That means, if you start in any equilibrium defined by those equations, whether or not there is or is not unemployment in that equilibrium, if you double all those nominal variables you remain in that same equilibrium. For any solution to that system of simultaneous equations, there exists a whole range of solutions where all nominal variables are multiplied by an number you care to think of.
And that, dear readers, is the fundamental insight that lies behind the vertical Long Run Phillips Curve.
Which brings us back to the topic of this post.
hmmm
If you are using CRS, then
P1.MPL1=W=P2.MPL2 and P1.MPK1=R=P2.MPK2
implies that both production functions are exactly the same.
Ugh.
I want to look at an economy with more than one firm, and with multiple production functions that are not all identical, or homogenous of degree anything — it seems that statements like “I can always tweak the wage rate to achieve full employment independent of the MPK” should not require production functions that are of a highly specific form, as production is a fluid and changing thing — you should be able to deform the production function a bit and still end up with the same general conclusions.
But I still think that what I am saying is fairly simple, and not obscurantism. When there are two production functions, unless you are in a degenerate situation, then in general the equations:
MPK1/MPK2 = MRTS(K1,K2) = MR1/MR2 (1)
MPL1/MPL2 = MRTS(L1,L2) = MRTS(K1,K2) (2)
Should take away 2 degrees of freedom from (L1, L2, K1, K2), in which case they would leave you with a constraint of the form total labor = f(total capital). That is a pretty simple argument.
The reason for believing that these two constraints are non-degenerate is that in general, you wont have constant elasticity of substitution for micro-level firms.
For the MR1/MR2 non-degeneracy condition, the right hand side is going to be a function of the PED1 and XPED12 of the two goods, which will depend on consumer preferences rather than the technical factors of production. You can argue that, over the long run, capital will be invented with the exact substitution characteristics to match the (changing) consumer preferences — but it’s hard to argue this is a general case at all periods of time.
Btw, I did go through some micro-texts but suffered with a lot of boredom. It’s my own laziness, so thanks for being a gracious host. I will force myself to go look at them again. Feel free to call any BS as and when you see it!
I think it is more about what you count as important than any fancy math.
Most MMTers think that ‘full employment’ – with the simple meaning of ‘everyone who wants a job can get one at a living wage’ – is more important than inflation. This is a political question, not an economic one.
From this position they aren’t really interested in Phillips curves. They see the primary economic job of government to achieve full employment. If a government can do that cleverly with labor market policies – training etc – great. If they can’t, then just do it anyway with aggregate demand and wear whatever rise in prices you get while the economy adjusts.
If anything, they think the most important aspect of a ‘long run phillips curve’ is that it is actually U shaped. High unemployment leads in the long run to an unskilled labor force that actually increases production costs – a sort of ‘supply shock’. We are seeing this in Australia right now, where twenty years of not employing trainees and apprentices has led to a real lack of skilled workers that must be obtained from overseas at great cost.
RSJ: “But I still think that what I am saying is fairly simple, and not obscurantism.”
I fully agree. Sorry, I wasn’t accusing you of obscurantism. In this thread, you are (almost always) being the exact opposite. (What is the opposite of “obscurantism”?) You made a clear claim, and laid out your assumptions clearly. (Or, as clearly as you could). And you gave a simple (or fairly simple) example to back it up. Just like a good economist should. And that’s why I wanted to fully engage you on it.
Any obscurity in our argument here is an accident of my bad math, your bad micro, and the difficulty of writing equations and drawing diagrams in TypePad.
RSJ: Let’s switch back to perfect competition, as it makes the model simpler, and I don’t think the results should depend on it, in this case.
Let me try to lay out the model more clearly.
There are two firms (or two industries each comprised of identical firms). The two firms produce different output goods. Each firm has a different CRS technology.
Your 2 equations:
Replace “MR” with “P” to reflect perfect competition. The P1/P2 should be a function of Y1 and Y2 (the outputs of goods 1 and 2), as well as consumer preferences. A simple assumption (sort of the equivalent to CRS in preferences) would be that P1/P2 is a (decreasing) function of Y1/Y2.
“For the MR1/MR2 non-degeneracy condition, the right hand side is going to be a function of the PED1 and XPED12 of the two goods, which will depend on consumer preferences rather than the technical factors of production.”
Nope. The ratio MR1/MR2 (or P1/P2 under perfect competition) will depend both on consumer preferences (the shape of the indifference curves) and on the relative supplies of the two factors of production. Intuition: as capital becomes more abundant relative to labour, that will increase the supply of the capital intensive good relative to the labour intensive good, and will reduce the marginal utility of the capital intensive good relative to the labour intensive good (change the MRS), and lower the price of the former relative to the latter.
If the ratio P1/P2 were fixed, determined by preferences independently of Y1/Y2, then your equations would indeed do as you say (I think), and let you derive an equation “total labor = f(total capital)”.
Here’s the economic intuition for that case: your indifference curves would be straight lines. The two goods would be perfect substitutes. Formally, this is mathematically equivalent to the one good model (except that 1 unit of good 1 might be equivalent (say) to 3 units of good 2.) So if the aggregate K/L ratio were outside those bounds I mentioned above, one of the firms would stop producing.
begruntled: “From this position they [MMTers. NR] aren’t really interested in Phillips curves.”
That would be a totally incoherent position.
Suppose (for example) the Long Run Phillips Curve sloped the “wrong” way, so that an increase in inflation caused an increase in unemployment. Don’t you think (if they cared about unemployment) they would then recommend very different policies?