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I don´t have any problem with the MPL concept - but I hope that you don´t think that wages are set in some perfectly competitive market of L demand/supply (rather than through wage bargaining as in e.g. a matching model)

nemi: if I zoomed in, I would want to switch to some sort of matching model. Stuff like matching due to heterogeneity of workers and jobs is much more important than kinks and flat spots.

With linear production functions, you don't need preferences (labor supply, capital supply) to determine the equilibrium wage and return on capital. You can instead make assumptions about the steady-state endowment. For instance, suppose that the economy has one consumption good C and one intermediate good I. The technology is such that (1) you produce 10 units of C by using 7 units of C and 3 units of I, and (2) you produce 5 units of I by using 3 units of C and 2 units of I. The way I wrote these relations, we have a steady state: in each period, we exactly regenerate the endowments of C and I. Then the price vector is determined uniquely, since C producers and I producers must make zero economic profit in the steady state. Hence, 5 units of I must be worth as much as 3 units of C and 2 units of I, and 10 units of C must be worth as much as 7 units of C and 3 units of I. Hence, C and I exchange at parity.

Let's consider a case with a growing economy, as opposed to a steady state. Suppose that (1) 56 units of C and 3 units of I can produce 115 units of C, and (2) 24 units of C and 2 units of I can produce 5 units of I. Now we have a surplus of 35 units of C per period, so the above condition no longer determines the price vector. All we can say is that (115 - 56) = 59 units of C must be worth at least as much as 3 units of I, and (5 - 2) = 3 units of I must be worth at least as much as 24 units of C. However, we can reach a unique solution by supposing that the rate of normal profit is in equilibrium across industries. Then we have: (1 + r) (56 C + 3 I) = 115 C, and (1 + r) (24 C + 2 I) = 5 I. Then r is 1/4 per period, and 36 units of C exchange for 3 units of I.

This analysis can be extended to more complex cases, not least by including labor (thus the real wage) in the model.

anon: agreed.

I used the same simplifying assumption of a linear technology to solve for the wage and rate of interest without specifying preferences in my simple little model of a robot economy.

If you have a linear Production Possibilities Frontier between any two goods, like apples and bananas, or consumption and leisure, or C and I, and if both goods are produced in equilibrium, then you can determine the relative price of those two goods from technology alone. The slope of the PPF is the Marginal Rate of Transformation, and that must equal the relative price of apples and bananas if firms maximise profits in competitive markets. And if the PPF is a straight line, that means it has a constant slope, so the relative price of apples and bananas is determined independently of preferences. Preferences determine the mix of consumption.

If I had drawn a linear production function (which is just a PPF between consumption and leisure) in my diagram, the MPL would be a constant, and MPL would determine the wage, independently of preferences. Preferences would determine employment and output at that wage. But neither the curved nor the kinked production functions are linear, so I need preferences to co-determine the wage, as well as co-determine employment.

My view is that linear PPFs are a very special case. There are lots of different types of labour, land, and capital goods, all with very different comparative advantages at producing different goods, which in turn require different capital/labour/land intensities, so the PPF will almost always be non-linear and bowed out. So we need preferences to determine wages, rents, interest rates, as well as prices of goods.

Even in your examples, if preferences change, so people want more C and less I (or vice versa) we are going to get a change in the relative prices. We need to know time-preferences to determine whether the economy will be growing or not, and how quickly it will be growing.

First off, thanks for the "Money and bond bubbles' post; very clear and readable. I really feel I understand the issues quite a bit better. Admittedly, I started at a pretty low level.

I commented on Profs. Krugman's and Thoma's blogs; I'll try to be clearer here. Goods are produced and delivered, not by unstructured aggregates of workers and capital assets, but by highly-integrated production systems designed to particular scales. This is why, as you grant, production functions may be essentially horizontal. In the short term, increases to output mean overtime or additional shifts.

In the longer term, a single technology typically spans only a fraction of the range of output volume. So increases in output may favor transitioning to an alternative technology; and that drives changes to the design of the product itself (e.g. sheet metal for an equipment enclosure made in low volumes, injection molding for high volumes; reaction-injection molding and die-cast are viable alternatives in between, perhaps). One technology does not supersede another, they exist simultaneously, and they may overlap each other in range. Alternatively, one could stay with an existing technology and set up a second (or third, or fourth...) line. So there is a curve family as there is with indifference curves. Of course, only the leftmost point of any horizontal segment is really relevant. But a family of isolated points and disconnected line segments precludes any unique solution.

As services delivery becomes more integrated, the workflows more closely coupled, production functions will progressively become less well-behaved in that domain also. I don't think manufacturing is special in this regard.

Ken: thanks.

At the micro-micro level, what you say sounds plausible to me. For very low volume production, cars may be handbuilt, using fibreglass. For higher volumes, production will be more mechanised, using steel. But those two cars aren't really the same. So at that level, the production function is very complicated. Labour is a vector of millions of very different workers. So is land. So is output. And capital is a whole set of time lags between vectors of inputs and vectors of outputs, that may or may not be possible to modify mid-stream. But when we are trying to talk about changes in aggregate employment and real wages, where those wages are measured in terms of a bundle of many very different goods, we need to zoom out and try to see the big picture. Which was what Paul Krugman was trying to do, when he asks: what might have caused (average) real wages to stop rising?

I'm not sure it's quite right to think of the linear-PPF analysis as "just a special case". I'd think it could be seen as an alternate way of describing an economy: instead of starting from preferences, initial endowments and a complete PPF in order to determine what goods are produced and what proportions are used in each production, we start with an assumption about what goods are produced in the economy and what fixed-proportions techniques may be used in producing each good.

This seems to be better when you want to describe a long-run steady state. In the traditional analysis, you can never be sure that you will end up characterizing a steady state where all endowments of goods are evenly replaced in each period (except by random chance, obviously), but the alternate description allows you to make this a starting assumption.

Large companies have been known to look the data sets here and adjust merit review amounts that intend to move the statistical spread of wages and salaries to the center value. Thus rewarding, even recruiting, mediocrity rather than effort or competition.

anon: I hope you will keep commenting here, and give yourself a nom-de-plume, so I know it's you.

I think I can see those two different ways of looking at the economy. I happened to be thinking this over for the last few hours, thinking up a post on the subject. Let me say that I prefer to approach it the other way around. We start from a particular state, with a set of preferences, and we work out what happens. We may or may not approach some steady state. Actually, if technology is changing all the time, the economy never does replicate itself, because old durable goods become obsolete before they wear out. They stopped making my 13 year old car 11 years ago, and have made two different model changes since then. Nevertheless, it's still an interesting question to ask: what would an economy look like, and what would preferences have to be, if an economy were replicating itself?

Interesting post....still digesting it. Minor note: "With a kinked production function the labour demand curve is horizontal for levels of employment below the kink, then vertical at the kink, then horizontal again above the kink." That's not what you have shown....you're showing kinks without restricting slopes to be 0 or infinite everywhere else.

Krugman wrote

"... we would expect the labor force to achieve full employment by accepting whatever real wage is consistent with said full employment. And what is that real wage?"

He is clearly taking the labour supply as exogenous and fixed. That's why he has a vertical line in his diagram labelled "full employment" and he looks at equilibria on that line.

Despite what you say, your diagram is not what Krugman drew in the article that you referenced. You are assuming that the labour supply is endogenous and depends on the real wage.

Perhaps you should note that you and PK are considering different problems?

Simon @12.21: Thanks. The production functions PF1a and PF2b are supposed to be horizontal to the right of the kink, so the MPL is b to the left of the kink, and zero to the right of the kink. That's what you get with a production function Y=min{aN,bL}, if N is fixed. But yes, in the more general case, it won't be exactly like that.

12.41. Well, I would just say that Paul is maybe oversimplifying it a bit, so the average NYT reader can get it, but that if he weren't oversimplifying it for the NYT he would probably have done what I did. (I don't want to make a big deal out of that difference.)


I think you are making a big deal out of that difference. You are writing (in boldface) "You still need both the production function and the indifference map to work out the equilibrium level of output, employment, and wages." I think we both agree that, in PK's model, you don't.

I didn't think PK was oversimplifying for the NYT readers. Rather, I think he's working with a framework common in international trade theory, where labour supply is often taken as fixed.

Yes, macroeconomists like to worry about endogeneity of the labour supply. But, as you said, this is really a microeconomic model, not a macro one.

Fred Moseley sent me this by email (with permission to post it):

Labor supply and kinked production functions

This is a response to Nick Rowe’s http://worthwhile.typepad.com/worthwhile_canadian_initi/2012/12/on-fred-moseley-on-paul-krugman.html
which was a response to my criticism of Krugman’s explanation of stagnant real wages in the US in recent years: http://economistsview.typepad.com/economistsview/2012/12/krugmans-explanation-of-stagnant-real-wages.html

First of all, the theory of labor supply emphasized by Rowe, based on individuals’ utility functions and their choices between labor and leisure, is unrealistic in the extreme and does not apply to the real capitalist economy. In the real capitalist economy, employers generally decide how many hours their employees work, within limits set by the government, and individuals have little or no choice. “Involuntary part-time” (at a record high in the US today) and “mandatory overtime” obviously contradict this “voluntary choice” theory of labor supply. It is generally not true that more individuals today are choosing to work more part-time jobs because they prefer more leisure!
Individuals sometimes choose to work part-time, but this choice is usually because of personal factors such as child-rearing, attending to school, etc., rather than the wage rate. So again, the labor supply theory does not apply. If anything, the relation between wages and hours is the opposite: part-time jobs have lower wages, rather than workers choosing to work fewer hours because wages are low. And then there is the phenomenon of workers working two part-time jobs because that is all they can get and wages are so low, i.e. an inverse relation between wages and hours, rather than a positive relation (i.e. a “forward-bending” labor supply curve at the low end).

So, Rowe is correct that the marginal productivity theory of wages needs a supply side, but it does not have a credible and realistic one.

Furthermore, Rowe’s kinked (Leontief intensive) production function graph does not address the main point of my criticism – that raw materials and other intermediate goods cannot be held constant as labor and output increases at any level of output. Therefore, the MPL does not exist at any point, not just at a kinked point, and the whole theory falls apart.

Finally, Rowe’s post does not address my empirical criticism: Krugman presents no empirical evidence to support his explanation that stagnant real wages have been caused by “capital-biased technological change”.

So I repeat my conclusion: It is time we stop talking about non-existent marginal products – and individuals’ choices about working hours – and look for other better, logically consistent and empirically supported theories of the distribution of income and the stagnation of real wages in recent decades.

Fred Moseley
Mount Holyoke College

Fred: "Furthermore, Rowe’s kinked (Leontief intensive) production function graph does not address the main point of my criticism – that raw materials and other intermediate goods cannot be held constant as labor and output increases at any level of output. Therefore, the MPL does not exist at any point, not just at a kinked point, and the whole theory falls apart."

That's where I disagree with Fred. I thought my post had addressed this very point. The MPL is the slope of the production functions. And my kinked production functions are what you get when you assume a Leontief technology. And the slope of a straight line is very well-defined. The slope is only undefined at the kink.

Take a very simple version of the Leontief fixed proportions technology: Y=min{L,N}. L is labour and N is land. One worker and one acre of land produce one ton of wheat. Suppose you have N acres of land. If you have less than N workers, every extra worker increases output by one ton of wheat. The Marginal Product of Labour is one ton of wheat. And the Marginal product of land is zero. If L > N, the MPL is zero and the MPN is one. Both are well-defined.

The farmer would of course be daft to pay any positive rent to rent extra land if L < N, and would be equally daft to pay any positive wage to hire extra workers if L > N. If both wages and rents are positive, and if the farmer can choose both the number of workers and the number of acres, we know that L=N for profit-maximisation.

When N=L we can also define the Marginal Product of a team of one worker plus one acre. It equals one. Profit-maximisation in competitive markets implies that wage for one worker plus rent of one acre equals the value of the marginal product of the team.

A little update: See Paul Krugman's latest post

Paul says: "Imagine that there are only two ways to produce output. One is a labor-intensive method – say, armies of scribes equipped only with quill pens. The other is a capital-intensive method – say, a handful of technicians maintaining vast server farms. (I’m thinking in terms of office work, which is the dominant occupation in the modern economy)."

I said in my post above: "Suppose there were two ways of growing wheat: a land-intensive method with big a and small b; and a labour-intensive method with small a and big b. As you add more and more labour to a fixed amount of land, you start off with the land-intensive method, then when all the land is farmed you start switching to the labour intensive method. The production function will have two kinks; it starts steep, then gets less steep, then gets flat."

And, as Paul correctly says "Oh, and if you’re worried, yes, workers and machines are both paid their marginal product."

Paul is talking about an equilibrium that is on the upward-sloping bit between two kinks on my production function.

(I am not saying that PK ripped off "my" idea, because it isn't mine. I am not saying "great minds think alike", because mine isn't great. I am saying "this is just basic micro-theory, so it's not surprising he and I said the same thing". )

Just a short response to Fred's other two points:

1. Empirical support for Paul Krugman's hypothesis? Well, as I said in my earlier post, I'm not too happy myself with the empirical support for Paul's hypothesis, because it would seem to predict rising real interest rates. So I'm not a good person to defend Paul's position empirically.

2. Labour supply. The US economy is currently in recession. In a recession, economies are more "off" their labour supply curves than they usually are. But we are talking about a longer run phenomenon here. If employers could ignore the constraint of labour supply when choosing wages and working conditions, it would make me wonder why all jobs don't pay the legal minimum wage and minimum holidays and maximum hours. Any employer that needs to compete with other employers to hire and keep good-quality workers will need to take their preferences into account when choosing wages, hours, holidays, and working conditions. If you offer the bare legal minimum on all dimensions you will only get lower quality workers with high turnover. That may work OK for some employers, but not for all.

Regardless: there is no such thing as the "marginal product theory of wages". There is the marginal product theory of the *labour demand function*. MPL is a function (or at least a curve), not a number. You need some additional other function, or curve, like a labour supply function, to co-determine wages.

Taking things very literally, it's probably true that it's rarely possible to hold all else constant and increase output by increasing just one input. If you add another worker to your bakery, you'll need more flour etc if they are to achieve anything. This, again, strikes me as the sort of detail that is sensibly ignored.

Luis Enrique: loaves of bread = min{bakers, flour}. Nobody says you have to use all the bakers or all the flour. For many periods in history many natural resources have been left idle because their marginal products were zero. Nobody ever said "We can only use land and farmers in fixed proportions, and we don't have enough farmers to farm all this land. What are we going to do???!!!" They just left some land idle.

More realistically, as in Paul Krugman's example, even if techniques were Leontief, there is more than one technique. You grow wheat on some land (labour intensive) and do ranching on other land (land intensive), and as the aggregate labour/land supply ratio changes, you just do more wheat and less ranching.

Another email from Fred Moseley, which he has asked me to post:

1. Labor supply
Long-run / short-run makes no essential difference with respect to the supply of labor. In either case, workers have little or no choice about their working hours; and what little choice they have does not depend much if at all on the wage rate (positively).
Therefore, all the efforts to derive the demand for labor with fixed proportions (or with variable proportions for that matter), even if successful (which they are not), would still not provide a theory of wages, because there is no credible and realistic the theory of labor supply. To paraphrase Marshall, it is like a “scissors with one blade”.
I am not saying that employers can ignore labor supply and determine wages unilaterally. I am saying that employers pretty much unilaterally determine the hours of labor, within limits set by the government (and these limits are the outcome of decades of class conflict, not individual choices).
There is a further problem here as well. The demand for labor is in units of workers and the supply of labor is in units of hours, so the two are logically incoherent, and cannot mutually determine wages.

2. Land and labor example
In your land and labor example, you count the quantity of land input as the stock of land available (N acres). But the quantity of land input in a production function is the flow of land actually used, as is true of all inputs in a production function, which is a flow-flow function. Land not utilized is not an input to production.
Thus as you add one worker, you are also adding an acre of land input; i.e. you are not holding land constant and thus the MPL is not defined.

3. Krugman
Krugman’s example of office work does not include raw materials and thus does not address my criticism that raw materials render the concept of the MPL (or the MPK) impossible. Raw materials cannot be held constant while increasing labor and output.
With raw materials, there is only one factor proportion possible (i.e. only one “technique”), so Krugman’s example of two techniques and extrapolation to a “bunch” of techniques does not apply. There is no isoquant (kinked or otherwise) between labor and raw materials. One cannot increase labor and decrease raw materials to produce the same quantity of output. A car still needs four tires. There is only one point for each quantity of output.
Krugman just asserts (and you endorse) that “labor and machines are paid their marginal products”. But Krugman does not define marginal products and does not demonstrate the equality between marginal products and factor payments. I argue that this can’t be done in cases that include raw materials, since marginal products do not exist.
You say that “Paul is talking about an equilibrium that is on the upward-sloping bit between two kinks on my production function.” How is this equilibrium determined? What is the other line that is tangent to your total product line? It can’t be the factor price line (which is usually combined with isoquants to determine factor proportions), since output is on the vertical axis. For reasons explained above, I hope you are not going to say individuals’ indifference curves between labor and leisure.
You say that you and Krugman are just presenting “basic micro theory”. I agree, but in this case basic micro theory is invalid, because marginal productivity theory cannot incorporate raw materials and it has no credible theory of labor supply. Plus the capital market is even more problematic than the labor market, but I will save that big topic for another occasion.

Fred Moseley

Fred: let is assume a fixed proportions technology. 1 hour of labour and 1 kilo of wheat produce 1 loaf of bread. The land produces a steady flow of 10 million kilos of wheat per day, all by itself. But nobody wants the wheat, until a worker has made it into bread.

What does the labour demand curve look like? Assume perfect competition, and measure wages in loaves of bread.

The labour demand curve is horizontal at 1 loaf of bread per hour, until it hits 10 million hours of labour per day. Then it is vertical. Then it is horizontal at zero. It is perfectly possible to define the labour demand curve with a Leontief technology.

What does the production function look like, with labour on the horizontal axis?

If we start where L=0, it is an upward-sloping straight line, with a slope of 1, until it hits 10 million hours of labor. Then it goes horizontal.

The height of the labour demand curve is equal to the slope of that production function.

I want to say that the slope of that production function (and the height of that labour demand curve) is "the marginal product of labour". You object to my saying that, because when we increase the amount of labour we are not holding constant the amount of wheat actually used.

OK, let me define a new concept: the marginal product(NR) of labour is the change in quantity of bread produced with one extra hour of labour holding constant the amount of wheat avaliable to be used. If 10 million kilos of wheat are available to be used, you can either use it, or leave it lying on the ground to rot, if it's not a scarce good. (Or you can redefine "used" so that leaving it lying on the ground to rot is just another form of "use").

My redefinition of MPL would, IIRC, be perfectly acceptable to Ricardo. When population expands, Ricardo has the extra labour being used both at the intensive margin (higher labour/land ratio on the good land already being farmed) and at the extensive margin (more bad land gets used). In my example, there is fixed proportions between land and labour, so there is no intensive margin. All the action is taking place at the extensive margin. And all my land is the same, so all land earns zero rent until employment equals 10 million hours per day. Didn't Ricardo talk about the "marginal product of labour" at the extensive margin too?

So, it is perfectly possible to define the labour demand curve under Leontief technology. And see its relation to the marginal product(NR) of labour. It just has a slightly weird shape, is all. It's not the usual downward-sloping thing. It's got steps in it. Maybe one big step, if there is only one technique and one quality of land. Or two steps if there are two techniques or two types of land. There are lots of different types of land in reality, just as in Ricardo's model.

In economics we want to explain both prices and quantities. (Wage and employment in this example). One single curve (the demand curve in this case, the supply curve in other cases) won't do that. If the demand curve is horizontal it explains price but not quantity. If the demand curve is vertical it explains quantity but not price. If it's downward-sloping it explains neither. Same thing for the supply curve alone. Technology (and resources) alone cannot explain both P and Q. Under very special assumptions it can explain P but not Q, or Q but not P.

Yes. There is no such thing as the "marginal productivity (NR or not) theory of wages" except under very special assumptions. Technology (and resources) alone does not explain prices (except under special assumptions). That's why economists in 1871 brought preferences into the model, to explain how technology (and resources) and preferences co-determined both prices and quantities. And have faced massive resistance from some post-Ricardians and post-Marxists ever since. Only under very special assumptions can their models explain prices, without preferences. And even then they cannot explain quantities, which are equally important.

So I'm going to turn the question around: if not preferences, what do you want to add to your model, to explain both prices and quantities?

Gotta go New Years Day skiing! (Fred said he'd had a lovely couple of days skiing.)


Just in case you're still reading this thread (and it's commendable that you read and respond to your commenters) -

"...micro-micro..." Yep, that's just the way my brain works.

"...at that level, the production function is very complicated..." Indeed. My argument is that "zooming out" doesn't help. If there are multiple solutions for a given micro case, then aggregating just adds in many more solution points, till the aggregate case becomes a cloud of potential solutions. Granted, in the aggregate there are constraints which can't be ignored, as they may be at the macro level. But would they prune the solution space faster than it expands? Or maybe the constraints appear as 'attractors' in the solution space?

institutionalists have also attacked the use of preferences (which can't be measured and ignores prevalent habits and instincts, and hence power, and hence has no meaning outside of models in understanding the real world), there has never been any (good) critique of veblen's characterization of institutions as prevalent habits of thoughts which extends into two areas: the ceremonial side, and the technological (later called the instrumental) side. Veblen actually talks about the real world when he refers to the leisure class, when he refers to money being the end goal and hence having a monetary theory of production meaning that money is never neutral. So if I want to talk about preferences, we have Veblen, why do we need to make ad hoc assumptions other than so we can have models that simultaneously determine prices and quantities? and there is so much wrong with our WANTING a model that does so that we sacrifice everything (that is, everything that matters: understanding how the real world actually operates) in order to get a set of equations that put the system into equilibrium.

Ken: I'm still reading. Here's my way of thinking: zoom in close, and Earth has a very complicated surface, with valleys and hills, ridges and cliffs. Zoom out, and Earth is round.

DarkJaws: sorry. Your comment got stuck in the spam filter.


On Round Earth, everyone needs a boat and nobody needs skis. Global constraint precludes local solution.


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