Wednesday, December 26, 2012

Technological Progress and the Capital Labor Ratio

So Krugman is again trying to make sense of his marginalist theory of distribution and the choice of technique. He suggests the following graph, which I slightly modified, to express the possibilities available to the firm.
The profit maximizing firm will choose the 'labor-intensive' technique to the left of the intersection level between the two techniques. For example, at L/Y=0.4, where there is a vertical dotted line, now the firm needs less capital per unit of output (around half, approximately 0.3, rather than 0.6) to produce one unit of output. To the right of the intersection the opposite applies.

Note correctly that, as is well known by Krugman, the neoclassical theory of distribution applies here. The slope of the techniques is given by the negative of the capital to labor ratio, and relative remuneration of capital and labor are associated to the intensity of the use of the factors of production. In his words:
"and if you’re worried, yes, workers and machines are both paid their marginal product."
That is why the firm uses the labor intensive technique as real wages fall, and technical progress is associated with worsening income distribution. There is only one problem. The linear technologies of his example presume that all sectors have the same capital to labor ratio. That is a very peculiar assumption (the same needed for production prices to be determined by the amount of labor directly and indirectly incorporated in production, or what Marx referred to as the same organic composition of capital, by the way). If that proposition is dropped the whole thing is incorrect (see here).

Samuelson (1966; subscription required) was well aware of the problems brought about by the special assumption of his Surrogate Production function. It's time Krugman reads some of Samuelson's classic papers.

PS: Note that Krugman's point here is that maybe worsening income distribution was caused by technical progress, something he has been trying to deny in his recent research, suggesting that the increase in inequality was caused by political factors (conflict) and not skill biased technical change (technology). Make up your mind dude!

PS': Might be good to quote Samuelson directly. According to him (1966, pp. 582-83):
"There often turns out to be no unambiguous way of characterizing different processes as more 'capital-intensive,' more 'mechanized,' more 'roundabout,' except in the ex post tautological sense of being adopted at a lower interest rate and involving a higher real wage. Such a tautological labeling is shown, in the case of reswitching, to lead to inconsistent ranking between pairs of unchanged technologies, depending upon which interest rate happens to prevail in the market."
A tautology that may lead to mistakes. That's what marginalism produces when you want to understand income distribution and technical change.


  1. "and if you’re worried, yes, workers and machines are both paid their marginal product."

    The thing is, this is just an incorrect interpretation of MP theory. In actuality only one of two schedules is *determined* by the *different* marginal products, and the real wage or rental rate will be whichever corresponding MP is intersected by the supply curve.

    1. I think you're concerned with some different, the amount produced. Here it is less about how much the firm produces, but about what technique it uses faced with different wage-profit frontier.

  2. "The linear technologies of his example presume that all sectors have the same capital to labor ratio."


    I think I follow most of your and Krugman's respective arguments.

    The above, however, is not obvious to me, maybe because I am just studying these matters. So, please bear with me.

    If I understand things right, an homogeneous (degree n) production function is one such that

    F(K,L) = (k^n)*F(K,L) (k > 0)

    F is said to be linear iff n = 1, in which case

    Y = diff(F,K)*K + diff(F,L)*L

    (diff(...) are the partial derivatives)

    Could you, please, elaborate why the capital to labour ratio should be the same for all firms?

  3. Magpie,

    I believe you are using the term "linear production function" correctly. But Matias, as I understand it, is using "linear" to modify a different expression. A production function expresses an uncountably infinite number of "technologies", as Krugman is using the word. (I prefer the word "technique".) Krugman, in his diagram, is illustrating two such technologies.

    Maybe this post of mine might help:
    Can you see how capital intensity differs with different rate of profits except in the special case that Matias refers to?

    1. Hi Magpie, Robert is correct. Note that I also used the term technique. My post on the capital debates, with a link also shows, how only in the case in which both techniques have the same ratio of capital to labor you would have Krugman's results. Quote from that post:

      "Let’s assume that there are two methods of production, associated to the manufacture of capital (iron) and consumption (corn) goods respectively. The prices are determined by:

      (1) pc=wlc+rpkkc
      (2) pk=wlk+rpkkk

      where the subscripts refer to consumption and capital, l and k are the technical coefficients of production, and w and r are the real wage and rate of profit. Using pc as a numeraire and solving for pk we obtain:

      (3) pk=(1-wlc)/rkc

      From (3) into (2) we get:

      (4) [(1-wlc)/rkc]=wlk+rkk[(1-wlc)/rkc]

      Simplifying, and solving for w we find:

      (5) w=(1-rkk)/[lc+(lkkc-lckk)r]

      If the expression in the small parenthesis in the denominator is equalized to zero we obtain a wage-profit frontier that is linear."

      By the way, subscripts don't show here, but do in the original post. But lk is the quantity of labor needed to produce k and kk the quantity of capital needed to produce capital.

    2. @Robert,

      After giving your comment a lot of thought (I am afraid your own post is above my pay grade :-), but worth going back to in a future opportunity) I think I know your meaning. [1]

      Krugman said:

      "No problem: we can just use a mix of the two techniques to achieve any input combination along the blue line in the figure."

      He seems to mean that we can use a convex combination of the production functions. Say, F is the low labour intensity one and G is the low capital intensity one:

      H = a.F + (1 - a).G (0 =< a =< 1),

      H seems to be Krugman's mix.

      So, although both original functions (F and G) have different capital/labour intensities, the combined one (i.e. H) does not. (I might be mistaken, but I think of it as a firm with two plants: one more capital intensive and one less).

      But I think this would be true only for a given a, not necessarily for the whole segment.

      [1] By the way, and although this is not really important, after checking Varian's Microeconomic Analysis, I think that, although a technology includes things like the Input Requirement Set, the Isoquants Set, etc. they are all defined from a production function. Thus, Varian presents a Cobb-Douglas technology, a Leontieff technology, etc.

      Therefore, although strictly speaking not interchangeable, I think one could speak of a technology as a production function. This would seem consistent with the use Krugman makes of the term in his article (he, for instance, speaks of a unit isoquant).

      But I am just guessing. I don't really know how the terms are used by academic economists in every day communication.

      Maybe Matías, who has contact with academic economists, could correct me on this.

    3. @Robert and Matías,

      Incidentally, I do notice the contradiction in Krugman's argument on technology and inequality. At one hand, he attributes the fall in wages on technological improvements, but he has also claimed that other variables are responsible for that:

      Krugman Revisited: Inequality

      PS, at Matías,

      Apologies for not replying to your comment. I need some more time to give it due consideration.

    4. @Robert,

      On further thoughts, the way I think of the problem may be wrong, after all.

      I am thinking of a single firm, with two plants. But Krugman could as well be thinking of a multitude of firms, whose plants are distributed among the two extremes (say, 100*a% firms are low capital intensity, 100*(1-a)% are low labour intensity).

      Individually, each firm has a different capital/labour relation. Collectively, though, they have only one: the average.

      Was that what you meant?

    5. @Matías and Robert,

      I think I see your point. You both are probably right on the shared organic composition of capital (i.e. "same capital to labor ratio") thing.

      I'll have to think about this.

  4. "There is only one problem. The linear technologies of his example presume that all sectors have the same capital to labor ratio."

    Are you sure you're still talking about Krugman's example? The way I understood it, there is just one sector, producing consumption good from labor and capital (I guess he's thinking about static model where inputs are fixed in advance), using a mix of two techniques (implicitly assuming parameters are such that both techniques will operate in equilibrium). The two techniques themselves have different capital-to-labor ratios (4 and 1/4).

    1. Same difference. Use technologies 1 and 2 rather than K-intensive and L-intensive sectors (technologies) and you get the same problem. The only way to show that the techniques are both linear is if you assume that both techniques have the same K/L ratios. The algebra doesn't change if you call them sectors or technologies.

    2. Sorry, I don't follow. Is there a difference between technique and technology? In Krugman's example, given the numbers he chose, we will have K/L ratio 4 in the first firm/sector/technology/technique and 1/4 in the second one, both before and after technical change.

    3. Let me clarify. I used technologies and techniques as analogous (and you can think of a sector as using a different technique). At any rate, note that in order to choose a technology (or a sector that they might want to enter, after all a new technique may imply as hard a choice as entering a new sector) capitalists will try to maximize profits. With free entry (which in the case of the technique it means its available to all) it implies that a uniform rate of profit will be generated. We also assume for simplicity homogenous labor and a uniform real wage. Hence in order to choose a technique based on which one would lead to a more profitable outcome, meaning in Krugman's (marginalist) case that the firm uses the labor intensive technique when the real wage is low, you must solve the system (in this case 2 techniques) for a uniform rate of profit (and a given real wage). You compare the techniques in equilibrium. In that case, you find that it is only possible to show that the technique will be the chosen one at low levels of wages, but not at high, if the capital labor ratio (for a uniform rate of profit and a given real wage) is the same for both techniques. If not you may have that one technique might be more profitable at low and high levels of real wages, which also make it impossible to say it is labor-intensive. Note that the idea of a uniform rate of profit was central not just for classical authors (Ricardo, Marx), but also the old marginalist authors (Marshall, but also Walras). This is an idea that has been abandoned by the intertemporal General Equilibrium authors.

    4. It would be interesting to have an update to this paper:

      Flaschel et al: "The Classical Growth Cycle
      after fifteen years of new observations" more recent than (PDF)

      Some of the hand-wringing over low employment ratios in the population were agreeing with a Classical, not neo-Classical approach (Goodwinian with some modifications from Rose, in the paper), as of 2007.

      It would be interesting to add 5 years more data and see if anything has changed besides the 50 year 'long wave' version of the cycle.

    5. @Matias:

      the equilibrium conjectured by Krugman will be such that both techniques will be operational. Since both techniques have constant returns to scale, in competitive equilibrium they must earn zero profit (after paying for labor and capital inputs). Thus, given the technical coefficients and price of output normalized to one, zero-profit conditions will uniquely determine wage and rental rate of capital. With math, if a_ij is amount of input j (=L,K) needed in technique i (=1,2) to produce one unit of output, prices W, R are determined by:

      a_1L * W + a_1K * R = 1
      a_2L * W + a_2K * R = 1

      In comparison to your post on CCC, there is no interest rate, since the model is essentialy static (there is some amount of capital K, which is fixed, perhaps more like land).

      With these equations in hand, we can compute comparative statics, i.e. response of prices to technological coefficients. The coefficients must obviously satisfy some restrictions so that both techniques will be utilized in equilibrium (my guess is that sufficient conditions will be: a_1L > a_2L, a_1K < a_2K, so no technique is strictly dominated by the other, and that aggregate endowments of capital and labor are not too unbalanced). Other than that, there is no need to assume that "that all sectors have the same capital to labor ratio".

      About intertemporal general equilibrium: in what sense does it abandon uniform rate of profit? Competitive equilibrium rules out arbitrage, so one-period rates of returns on all assets (financial or physical) must be equalized (in deterministic case, with uncertainty they would be equalized after adjusting for risk). If the economy converges to a steady state, this surely implies also equality of steady-state rates of return.

    6. Ivan, think of production taking a year (for example, in a harvest cycle) to complete in some steady state. And think of capital goods as being paid at the start of the cycle and labor being paid out of the harvest. (One can redo the argument with wages advanced.)

      Your "rental rate of capital" embodies an interest rate (also known, in some sense, in classical terminology as the "rate of profits". In the circulating capital case, one can write R = p[i]*(1 + pi). In Matias' post on the CCC, his r is equal to my (1 + pi). These equations are consistent with the assumption of the nonexistence of pure economic profits. In classical terminology, what you are calling "profits" are known as "supernormal profits".

      Perhaps this post of mine (where I have changed the notation) might help you understand the point:


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