Xiaokai Yang & New Classical Economics (1)

Xiaokai Yang, Economics: Neoclassical Framework versus New Classical, Blackwell Publishers (2001)

This is not a review, since I’m not claiming to provide an authoritative critique of the work in question. What follows is my working out what Prof. Yang is trying to say.

Policy initiatives are predictable: the author makes familiar arguments on behalf of the free market (or “invisible hand of the market”) whenever possible, which is generally what economists do. His examples are abstract; there are no detailed examples, such as how to match a system of equations describing an economy to an actual economy or portion thereof. In some of his other works, e.g., Yang 🙵 Sachs (2008) he lists prominent economic historians as being conceptually allied with his opinions.

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Reading Sraffa (3)

(Part 1 | Part 2)

In referring to passages from Sraffa (1960), I will hereafter refer to "paragraph" numbers used by Sraffa (thus: §11). These are numbered 1-96 through the entire book.

THE STANDARD COMMODITY

At this point it probably is a good idea to tell readers that I really want to avoid making any conclusions about Piero Sraffa's Production of Commodities by Means of Commodities (PDF), at least for now. One reason is that if conclusions are needed, I'm not a good source: I'm not an expert, and in intellectual pursuits like this I'm more interested in collaborating with other people to find out what manner of wisdom Sraffa has to offer.

In Chapter III (starting with §13), Sraffa introduces the variation technique he will use to develop the model. Using the standard methods of matrix algebra, he examines the effects on the solution sets of varying selected coefficients.¹ The most important of these is variation in the wage rate in response to the introduction of profits; another is allowing for the effect of time discounting on the value of labor contributions to value (§47).

In §12 (immediately preceding Chapter III) he had dropped the bombshell that there were too many unknowns for the available equations; the system of linear equations has no unique solution, and in fact, no market economy can exist with a unique "market" value of labor or capital. I was led to wonder why Sraffa didn't just select one commodity at random to serve as the unit of value, the way Léon Walras had done. Sraffa could also eliminated another variable by exploiting w's inverse linear relation with r. Both questions are addressed by Sraffa in Chapter III.

First, Sraffa needed a monetary unit that actually said something explicit about value. Supposing he picked commodity a, which happens to be exceptionally labor intensive. If there is some macroeconomic event that leads to a shift in w from 1 to 0.90, then this will have a disproportionate effect on the price of a (p_a).

Second, he is very concerned about the scenario of surplus production in which products as outputs are not in proportion to products as inputs. While it is reasonable to suppose that any complete economic system (i.e., one that imports/exports nothing) with raw inputs a → k will produce outputs (a) → (k), such that each output is greater than or equal to its supply, it is not reasonable to assume the inputs enter the economy in the same proportions that they are released. The a industry might produce a very large surplus (from which a lot of non-basic products are made), while b produces scarcely any at all.

Proportional Relationships of Commodity Inputs 🙵 Outputs By the way, we're interested here in ratios; so, for example, as inputs we have a ratio of b as 1.4 × a, and so on, whereas outputs leave a ratio of b as 1.9 × a (etc.) commodity a b c . . . k input 1 1.4 2.3 . . . 6 output 1 1.9 2.4 . . . 8.2 In terms of outputs, (a) ≥ a, (b) ≥ b, (b) ≥ b, . . . (k) ≥ k; but notice the ratios of (a):(b):(c): . . . :(k) are quite different from the ratios a:b:c: . . . :k (Sraffa 1960, §4). An obvious corollary is that the ratios of a:(a) are not necessarily equal to b:(b), or to c:(c), and so on. The reason (a) has to equal, if not exceed, a, is that if it did not, a would have to be imported from somewhere—which would introduce a feature of the model for which no provision has been made. We could have the economy be changing relative proportions (for example, if Sraffa were interested in modelling the effects of peak oil, and a = petroleum, the annual supply of which is presumed to be declining), but for now we're not interested in that. There could be likewise be substitution (as the supply of a is less than output, somehow (b) → (k) are used to produce more of a for the following production cycle, each year. For example, (b) → (k) might collectively be used to synthesize petroleum products from coal to make up the deficit in (a). But in that case, we would treat the synthetic a as its constituents, a → k.2 It is not a problem for the realism of the model that outputs of commodities are not in the same proportion as inputs. In real life, of course, such proportions would be constantly changing somewhat, in response to depletion of some resources and new discoveries of others. Production functions would also change, and inevitably require different inputs of commodities. More significantly, though, is that all of the surplus of production over inputs is the national income. That "income" takes the form of a mix of commodities that are not required to produce next year's supply; all consumption goods are made from the national income, whether consumer durables, food, public buildings and fixtures, and so on. Plausibly the massively complex arrax of millions of different consumption goods we actually consume is derived, á la Walras, using constrained optimization, utility functions, and so forth. But it is limited by the supply of raw materials available for use.

The two problems are interrelated.

When the wage is reduced to less than 1.00 of the economic surplus (AKA national income), the exchange values of the commodities required to permit the production cycle to begin anew, with the requisite surplus, will shift. This is a built-in problem of the matrix

(A_ap_a + B_ap_b + . . . + K_ap_k)(1 + r) + L_aw = Ap_a
(A_bp_a + B_bp_b + . . . + K_bp_k)(1 + r) + L_bw = Ap_{b
. . . . . .}(A_kp_a + B_kp_b + . . . + K_kp_k)(1 + r) + L_kw = Ap_k
where the capital letters are all given, and the prices/wages are the unknowns. Sraffa laments that it is impossible to to say if changes in the overall price level reflects global change, or merely a peculiarity of the measuring standard. Moreover, any such peculiarities would be multiplied through the multistage production process that actually exists, in which a major non-basic input of most consumer goods consists of semi-finished parts produced in the past—the "residue" of prior production periods:

The relevant peculiarities, as we have just seen, can only consist in the inequality in the proportions of labour to means of production in the successive 'layers' into which a commodity and the aggregate of its means of production can be analyzed; for it is such an inequality that makes it necessary for the commodity to change in value relative to its means of production as the wage changes.
Sraffa (1960), §23

The only way to avoid this problem is if we already know what the average contribution of labor is towards the production of the total national product, and find some commodity for which labor's contribution is the same as the overall average. Commodities for which labor played a larger role would (in the event of a wage reduction³) experience a comparative reduction in the cost of production. If prices in the matrix remained the same, and the values of the total amount of each commodity remained fixed, then the producers of the labor-intensive firms would be running a surplus, while those in the commodity input-intensive industries would be running a deficit (Sraffa 1960, §16).⁴ So we need to find an object that not only straddles the average in its direct consumption of labor, but also in its indirect consumption of labor.

This is not likely to exist, so Sraffa recommends instead a bundle of all the commodities (standard commodity), in proportions determined the following way: imagine a modified version of the matrix economy above, in which each of the production functions for A → K are multiplied by a different constant. These constants are chosen so that the new, modified matrix produces goods (a) → (k) in the exact same ratios to each other as they appear (a → k) as inputs. This new standard commodity, once derived, can be further reduced to the lowest common denominator: it now is guaranteed to remain unchanged in relative price by any change in the labor wage rate.

Sraffa will hereafter use the standard commodity for virtually all econometric observations about his system. For example, when talking about growth of the economy, he is not concerned about reducing the economy to monetary values for the purpose of comparing heterogeneous bundles of goods. Now expansion is applied uniformly to the standard commodity.

NET PRODUCT

The net product is important because it represents the output that is available for consumption or for future investment. Everything else is required in order to keep the system running the way it is. But Sraffa also links it to the idea of the standard commodity, because he is keen to avoid ex post facto exchange values. The reason is that he is not interested in saying the market is good because it does what it does; one of the things the market needs to do is be capable of finding exchange values of goods that meet certain goals. If the market persistently undervalues some commodities relative to others, then the producers of the undervalued commodities will be compelled to underproduce, leading to the system's inability to reproduce itself.

Another point is that eventually he is going to need a uniform concept of value, which will survive technical or macroeconomic changes.⁵ The point of the standard commodity was that, instead of attempting to find out the value-weighted share of labor in the economy at the same time as finding the exchange values of all the various commodities (which would be impossible), or assuring "losers" that, by definition, their loss of income was "fair," it was necessary to restrict economists to unimpeachably homogeneous data.⁶ Instead of begging the question about the value of various components of net national product/economic surplus, Sraffa wants to rely on measures that are irrefragable.

One especially challenging discussion (Sraffa 1960, §26) refers to something called the standard net product (SNP). We suppose that the entire labor force is engaged in the production of the standard product (defined above), i.e., the entire laborforce is engaged in an economy that puts out exactly the same ratio of commodities as are used as inputs in the production cycle. Given whatever the prevailing rate of surplus is (R), a certain percent of this imaginary economy will be surplus of output over input.

That is the standard net product.

This represents an imaginary version of the economy—a virtual economy inside of a virtual economy. The standard ratio is the ratio of commodities to each other (reduced to a lowest common denominator). Sraffa is interested in the relationship of the SNP to the whole standard gross product (SGP), or R. This is the rate of surplus, and both profits and extra-subsistence wages must come from it.

The point of this exercise is that the ratios of wages and profits to total product always consist of multiples of the standard ratio/standard commodity, and therefore such ratios used to represent the share of wages or profits in the SNP are, in fact, literal ratios of apples to apples, oranges to oranges, and kilos of potash to kilos of potash.

Sraffa proceeds to prove that mathematical inferences made about wages and profits in regards to the SNP actually do apply to the "actual economy"—by which we mean, the virtual economy wherein the standard system is a virtual economy (Sraffa 1960 §§31-34). The end-user products that people actually consume, being non-basic goods not used in the production of other goods, play no role in this calculation (Sraffa 1960 §35).

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NOTES

1. In matrix algebra, one uses a set of n equations to solve for n unknown values. Hence, the solution of a problem in matrix algebra is known as a "solution set." The coefficients are the known values in the linear equations.


2. Notice the use of parentheses to indicate outputs versus inputs. This is introduced in Sraffa (1960) in Chapter VII (§51), where the linear equations that produce a single commodity per each are superseded with equations that produce two.
   
   Supposing in 2012 a deficit occurs in the production of a, such that a < (a). In order to produce the same complement of commodities in 2013, some amount (a') = [ a - (a) ] has to be produced by the surplus of (b) → (k), which are all outputs. In 2013, the output (a') is now input a', which is added to (a) to permit the 2013 input of a to match what was available in 2012, plus a likely surplus. Readers see that a' is nothing more than a fraction of the 2012 surplus of (b) → (k) over b → k. If we treat it as such, then a ≤ (a).


3. The reason why Sraffa is always referring to wage reductions is that he is analyzing what happens to the solution set if you start out assuming w = 1.0, and go downward from there to permit a rate of profit. This progression is necessary because it permits Sraffa's model to capture more and more aspects of a real economy.
   
   For people who are agitated by the suggestion that wages are inversely related to profits (thereby implying the need for class struggle)—remember that in Sraffa's model, economic growth requires profit, and greater profits correspond to higher growth rates—which, in turn, cause higher absolute wages.


4. Here is probably a good time to mention that Sraffa's matrices employ very advanced mathematical methods, and cannot be solved using the methods of linear algebra. See K. Vela Velupillai, "Sraffa's Mathematical Economics—a Constructive Interpretation"(PDF), Discussion Paper, Universita' degli Studi di Trento—Dipartmento di Economica (Feb 2007). However, Velupillai mainly asserts that the systems of equations are good ones and Sraffa has included sufficient mathematical proofs. Another, much more helpful essay is Peter Newman (1962), pp. 58-75, which walks readers through the mathematical analysis used by Sraffa.
   
   That said, I think Newman's criticism of the model is mostly missing the point.


5. By "technical changes," I am including changes in the production functions, whether of the sort that Sraffa can be reasonably expected to have provided for, or the ones he will introduce later in the book. By "macroeconomic changes," I am talking about changes in any of the various conditions that can lead to profits rising relative to wages (including, for example, an economic boom accompanied by rapid economic growth).


6. A recurring theme in conservative political discourse about economics is that the government "shouldn't be picking winners and losers," because that is the job of the market. A logical corollary of this, somewhat less often spelled out, is that the decline in worker share of net national income reflects the irrefutable judgment of the market. As always, we need to ask, "In what sense is the judgment of the market irrefutable?" For an example of someone claiming that secular losses in worker income are immune to appeal, look up the long-simmering debate over the "skill-biased technical change" (SBTC) hypothesis. For a examination of SBTC claims (and debunking of them), see David Card 🙵 John E. DiNardo, "Skill Biased Technological Change and Rising Wage Inequality: Some Problems and Puzzles" (PDF), National Bureau of Economic Research (2002).

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SOURCES 🙵 ADDITIONAL READING

Piero Sraffa, Production of Commodities by Means of Commodities, Cambridge University Press (1960): link goes to complete text online.

Léon Walras (trans. William Jaffé), Elements of Pure Economics, Orion Editions (1984/1954); translation based on 4th Edition (1900).

Peter Newman's "Review of Production of Commodities by Means of Commodities," Schweizerische Zeitschrift für Volkswirtschaft und Statistik, Vol. XCVIII, March 1962, pp. 58-75.

Reading Sraffa (2)

(Part 1)

In referring to passages from Sraffa (1960), I will hereafter refer to "paragraph" numbers used by Sraffa (thus: §11). These are numbered 1-96 through the entire book.

Sraffa's Production with a Surplus Let La, Lb, Lc,... represent the labor used in the production of commodities a, b, c... and w be the wage per unit of labor; the sum of all the labors is 1, representing the total labor available for production. The production equations takes the form (Aapa + Bapb + . . . + Kapk)(1 + r) + Law = Apa (Abpa + Bbpb + . . . + Kbpk)(1 + r) + Lbw = Apb . . . . . .(Akpa + Bkpb + . . . + Kkpk)(1 + r) + Lkw = Apk where r is the share of the total surplus (R) that goes to the non-labor residue—i.e, the profit. The term "total surplus" (which Sraffa calls "net product" or "national income") has a special meaning that will be explained later. Taken from Sraffa (1960), §10-11

As mentioned in Part 1, both Léon Walras and Piero Sraffa (initially) tended to treat production as if it were a single act of exchange. This may seem odd, especially since the outputs are always a different mix of inputs. Steel and coal are exchanged by producers of either to produce... steel and coal. In the case of steel and coal, that's actually fairly realistic, but for the huge variety of other objects produced, it isn't: bookshelves or chairs have no role in the production of bookshelves or chairs; table saws have no role in the production of table saws (but they are essential for bookshelves).

Most production processes are largely uni-directional, leading from a standard class of goods (raw materials, electricity, PNG, clean water) to finished goods. For Sraffa (and Walras) capital goods are represented symbolically, as bundles of heavily processed raw materials. The processing of pig iron into any of a large number of available grades of steel, followed by drop-forging, milling and hogging, assembly, and installation are all subsumed into various retail markups. For them, a marketed product like a piano represents its raw inputs plus many middlemen.

The purpose of this is to explain the process of abstraction: are such matrices inherently unrealistic in representing the economy? Sraffa occasionally refers to non-basics (e.g., §6, §35, §57-61)--commodities that are enter the economy only in the production of themselves, or else as endpoints of production. Because of the irreversibility of "commodity refinements" that transform raw materials into finished goods, one could argue that the vast majority of end-user goods are non-basics. Electricity, flour, and lumber are generic examples of end-user goods that are not confined to end-use; (most) manufactured goods, groceries, and services definitely are.

Non-basics are the dark matter of matrix economies: Sraffa argued that they could be ignored in resolving questions his matrix had been contrived to solve.¹ Since they could be ignored, it was possible that Sraffa's conceptual economy might consist mostly of non-basics whose price was determined by the commodities used to produce everything else. In order for this to be physically possible, of course, there had to be an "economic surplus" consisting of goods or services that were "distributed" in some way to the general population depending on their contribution to production.

All of this relates, ultimately, to the question of whether or not the neoclassical school could actually explain the price of factors. If, according to Joan Robinson and Nicholas Kaldor (and even neoclassical champion Paul Samuelson), it could not, there was a compelling case to be made that no such equilibrium existed. While ideologues might hail the infallibility of the market, the problem here was that "the market" was at risk of being an idea without a coherent explanation. Mainstream economics teaches that "markets" find the socially optimum prices and quantities of goods and inputs spontaneously; but if the most rigorous economic analysis proves that such a thing is inherently unfindable, then "the market" is actually a figleaf for something else.

SOME OBJECTIONS TO SRAFFA'S MATRIX

Several writers have attacked the approach Sraffa used for lack of realism. One that can dispensed with easily is the system ignores such questions as firms, consumer preferences, etc. However, Sraffa doesn't need to include things that make it harder for the market to identify clearing prices for goods. Sraffa's system is also missing technical shocks, such as a sudden fluctuation in the supply of a.

One author objected that his schemata lacked any marginal variation in output (say, as a result of a change in tastes).² This is the result of walking into a conversation without knowing anything about what came before or after. Sraffa's object was to make the system explicitly independent of marginal anything. The marginalists (Walras, W. Stanley Jevons, and Carl Menger) had introduced the matrix economy to demonstrate the concept of general equilibrium; Sraffa's idea was to demonstrate the impossibility of any marginal theory of value determining the correct value of prices. For Sraffa to actually go on to deal with utility functions would be like General Sherman's march through Copenhagen: not terribly helpful for winning the American Civil War.

The absence of individuals, firms, or anything else but unimproved heaps of commodities serves to give the general equilibrium matrix every possible chance to find the prices (including the residue). The whole point of free markets is that they possess vastly more real-time information than anyone could possess. Assuming they are perfect, they will find whatever institutional arrangement is socially ideal, and Sraffa does not presume to limit this with his meager faith.³

Another possible objection I expected to see was that Sraffa had arbitrarily restricted the number of equations to the actual number of commodities, or that the number of commodity inputs (a → k) was necessarily equal to the number of commodity outputs. At the very least, I expected someone to point out that Sraffa could have arbitrarily set the price of a as 1, and made the prices of b → k in units of a.⁴ The reason for the identity of inputs and outputs is that the inputs have to come from someplace (so that Sraffa's economy, simple though it may be, is a global one); and once they have entered the economy, they never leave, they merely change form.

Taking an example, we can suppose a future cloud architecture-based computer that manages to gather all of the data required to simulate the global economy à la Sraffa. It includes tens of thousands of inputs and outputs, and a similar number of industrial procedures. For descriptive purposes, I'll pretend that the "industrial procedures" are linear equations that represent a recipe for the production of any one of those tens of thousands of outputs from a combination of specific quantities of the exact same items—as inputs. But rather than represent the billions of products available for sale (including things like replacement parts or billable services) in a modern global economy, a typical "production genome" is exclusively concerned with explaining how each of the various key inputs are combined to result in each other.

So, for example, an inventory of key industrial ores, energy resources, and farm commodities might actually suffice. The "economy" would consist of estimates of how much was required to recover all the inputs for the following year of production, adjusted to represent indirect inputs by way of imports.⁵ While there would be additional outputs not mentioned, these are subsumed into the "surplus," and since they are contain physical quantities of goods not actually used in reproduction of the original inputs, represent net national income.

So while there have been some objections regarding the realism of Sraffa's stylized economy, they don't prevent it from symbolically representing a vast range of plausible economies.

(Part 3)

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NOTES

wage unit: in Production of Commodities, Sraffa's wage unit has a very special significance. The wage unit is introduced as a share of the surplus produced by the economy (which itself has to be understood in a very special sense). Very loosely speaking, the economy may be described as either (a) operating at a subsistence level, in which case the wage can be ignored, and we can pretend the goods required to produce other goods are partly used to pay workers in kind; or (b) operating at a surplus, in which case we're interested in wages as a share of the excess of output over inputs. Again, very loosely speaking, the wage is expressed as a share of the economic surplus produced, with a potential range of 0 → 1.00.

The "very loosely speaking" qualifier, with red typeface, refers to Sraffa's attempts to address important mathematical quandaries I'll explain in a later part of this series.

1. See Walras (1984), §167 (i.e., pp.211-212). Walras wrote the first edition in 1877; Sraffa wrote his first draft in 1926. I'm guessing both contained essential elements of the texts available to me. So Sraffa was writing about half a century after Walras, and was in part arguing with him over the ability of pre-marginalist Classical economics to explain the prices of factors of production (as, for example, the rate of wages and profits). The concepts of "non-basics" is Sraffa's; Walras had used a totally different approach.
   
   Walras divided physical valuables into income and capital; he used examples of livestock, that could produce eggs and milk (i.e., a stream of wealth), or be slaughtered to produce meat (i.e., objects of immediate consumption); or trees, that could produce fruit, or else be cut down to be burned as fuel. Buildings and machines "by their very nature" are items of capital, not of income; but he goes on to claim that "every kind of social wealth, whether from its intrinsic nature or the use to which it is put, can serve either more than once, or only once," and this determines if it is capital or income. In this sense, Walras simply regards social wealth as commodities that, through abstraction, may be either end-user or "basic" (as Sraffa would put it).


2. Robert P. Murphy, "Sraffa's Production of Fallacies by Means of Fallacies" Mises Daily (Ludwig von Mises Institute) (7 April 2004). Murphy erroneously treats Sraffa as a quasi-Marxist popular amongst all manner of leftists. This is exactly wrong: Sraffa is generally regarded as having debunked the labor theory of value, and therefore burying academic Marxism in economics. See Steve Keen, Debunking Economics, 2nd Edition, Zed Books (2011), Chapter 17: "Nothing to Lose but their Minds." Murphy also jumps to the erroneous conclusion that because Sraffa proved "the market" could never find a unique market-clearing value for all necessary inputs, because there were too many variables, it therefore followed that it made no difference what these values were (with respect to productivity, anyway). See next post, footnote 3.


3. In other words, modeling all possible [free market] economies with the greatest possible simplicity reduces the risk of a failure of imagination on the part of the person doing the modeling. If that is how you actually model the economy, then negative conclusions (statements about what is impossible) are more convincing than if you impose arbitrary restrictions on what sort of values the matrix is allowed to return.


4. The reason why is that Sraffa needs to set the price in units of the standard commodity. The standard commodity and its importance will be explained in a future post in this series, but lest that future post never come, it forms the subject of Chapter IV in Sraffa (1960), p.20.


5. For a list of key industrial ores, see "Commodity Statistics and Information" from the website of the USGS. For statistics on coal, natural gas, petroleum, and uranium, see the US Department of Energy's Energy Information Administration Web site. For farm commodities (including livestock and timber), see the National Agricultural Statistics Service (NASS).

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SOURCES 🙵 ADDITIONAL READING

Piero Sraffa, Production of Commodities by Means of Commodities, Cambridge University Press (1960): link goes to complete text online.

Léon Walras (trans. William Jaffé), Elements of Pure Economics, Orion Editions (1984/1954); translation based on 4th Edition (1900).

Graham Joncas, "Piero Sraffa’s Non-Economics: An Introduction, part I," Linguistic Capital (21 May 2012): analysis of Sraffa's interaction with interlocutors Gramsci, Hayek, and Wittgenstein. Sequel pending as of this writing. Includes a brief discussion of Production of Commodities by Means of Commodities and a recapitulation of the Cambridge Capital Controversy.

Alvaro Cencini, Macroeconomic Foundations of Macroeconomics, Psychology Press (2005)

Reading Sraffa (1)

In his book, Debunking Economics (2nd Ed), Steve Keen mentions Pierro Sraffa in connection to the dilemma of capital formation.

Capital formation is treated in [neoclassical] economics as occurring between static phases of production; capital itself is treated as if the financial meaning of "capital" were identical to the industrial sense of "capital." Economists seldom include any consideration of the industrial steps between financial "capital" (i.e., income diverted from present consumption into production, in the hopes of future income) and industrial capital (i.e., tools, buildings, fixtures, or inventories).¹ They treat the provision of "spare" income as a hydraulic process, translating directly from money to productive instruments in a single step.

In the 1950s, a major preoccupation of economists was modeling simple economies using a set of equations. The idea was to simulate the behavior of a real economy with a virtual one. Obviously, a plausiblely complex economy complete with gas chromatographs and botts dots is not one that can be described with a manageable set of equations; so the economy was simplified immensely. Instead of acknowledging the existence of different modalities of capital, it was treated as a single fluid whole. Sraffa bucked this trend; he insisted on models of the economy that recognized the existence of physical objects used as capital.

The main historical impact of Sraffa was to attack the idea that a market economy can find a unique value of anything. A basic principle of neoclassical economics (and its successor, DGE) is that, in a free market, the share of economic surplus going to capital or labor will reach a socially optimum value.

THE CONCEPT OF GENERAL EQUILIBRIUM

Walrasian General Equilibrium The pool of available commodities is represented as a matrix of n goods and m people. People are indicated as 1, 2, 3... and goods are represented as letters a, b, c... Qa = qa, 1 + qa, 2 + qa, 3 . . . Qb = qb, 1 + qb, 2 + qb, 3 . . . Qc = qc, 1 + qc, 2 + qc, 3 . . . . . . . where Qa represents the total quantity of a, and qa, 2 represents the amount of a held by 1. "Redistribution" takes place to meet the preferences of the original owners (who presumably made them in the first place). In this initial presentation of the problem, the prices of the commodities pb, pc,... are fixed (the price for commodity a is set to 1, and all other prices are in units of a). Person (1) buys x'1, y'1, z'1,... of commodities a, b, c... (this is Walras' notation, and it wasn't thought out very well!) to wind up with q'a, 1 + x'1 = qa, 1 + x'1, q'b, 1 + x'1 = qb, 1 + x'1, q'c, 1 + x'1 = qc, 1 + x'1, . . . . And Walras proceeds to demonstrate that markets will clear. Later he introduces the production process and utility functions, in a comprehensive effort to outline the system of equations the market must "solve." Chart taken from Walras (1984), p.182-184

This is a loaded concept, and it's unlikely I'll explain it in a way that satisfies anyone. But neoclassical economics presupposes there is—at any given moment—an equilibrium value for prices and quantities of output that maximizes public welfare. If one had infinite knowledge of the production processes and utility functions, one could determine the amounts and prices of all the things the economy produced, such that everyone had the most satisfying mix of goods they could afford. An additional benefit is that the prices of goods would stimulate the correct amount of production, and cover the costs of inputs.

Lest anyone get the wrong idea, another crucial point of economics is that of course no one could possibly have remotely enough information to do this, but the free exchange of goods and productive inputs is constantly moving us closer to this equilibrium. Changing objective realities, such as shifting supplies of inputs, or changing preferences, pull us further away from it.

"Covering the costs of inputs" means—in this precise context—enough of a return to the supplier of each input that the supplier is motivated to provide enough to that particular goods producer to produce the market's demand for that particular good. So, for example, if the input is hours of productive labor, then there is a socially optimum wage that is best determined by the market.

The reason why I don't just say "Let the market decide!" is that, if economics is claiming to be a science instead of a religion, it has to claim there is some concrete social optimum out there to be discovered by the market, it has to explain what this social optimum will accomplish, and it has to explain why the market is likely to find it.


SRAFFA 🙵 GENERAL EQUILIBRIUM

Sraffa's System Superficially differs from Walras (above) in representing production rather than redistribution (though commerce), but mathematically the concept is similar: in Sraffa, redistribution occurs because producers of wheat, iron, and pigs are exchanging quantities of those commodities in order to produce another batch of each. 240 qr wheat + 12 ton iron + 18 pigs = 450 qr wheat 90 qr wheat + 6 ton iron + 12 pigs = 21 ton iron 120 qr wheat + 3 ton iron + 30 pigs = 60 ton pigs In this example, the output is equal to the inputs, and exchange must take place again afterwards so that the same amount of everything (as before) can be produced. Subsequently, Sraffa switches to scalable matrix algebra to allow for many different commodities. Here he shows the arrangement of identities required to ensure continuous production. The term Ba prefers to the amount of commodity b used in the production of a. The amounts used in the production of each commodity are fixed by available methods and the amount of each commodity "required" by Sraffa's "economy." Aapa + Bapb + . . . + Kapk = Apa Abpa + Bbpb + . . . + Kbpk = Bpb . . . . . Akpa + Bkpb + . . . + Kkpk = Kpk When solved, the values of p ensure that the producers of each commodity will be able to buy the inputs they need in order to reproduce the output of the previous cycle. Subsequently, Sraffa develops his imaginary economy to include wages and economic surplus. Taken from Sraffa (1960), p.4

When Sraffa published his book, it triggered a major controversy over the question of general equilibrium in economics. With respect to mainstream economics in the USA, neoclassical economics had never really been displaced.² In effect, Sraffa was writing a treatise on general equilibrium in a world that was still Walrasian; he had written a rough draft in 1926, then dusted off the manuscript, and published it in 1960—still in a Walrasian world.

The book is puzzling to most readers. It's not obvious if Sraffa is trying to carry a torch for his beloved David Ricardo, or if he really is making a pointed gesture for the benefit of Walrasians. Rather than construct a model economy with indifference curves and household endowments of capital/labor, Sraffa uses merely production functions and abstract commodities. The capital contribution to production represents all inputs besides labor. The commodities must be end products of production as well as inputs of production. Over successive chapters, Sraffa builds up his analysis to include a richer and more detailed model of the economy. But to what purpose?

In subsequent posts, I will attempt to explain what this purpose could be.

(Part 2)

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Notes

1. Classical economists, such as David Ricardo, made a major distinction between "circulating capital" and "fixed capital." Inventories, whether of merchandise or raw materials, are "circulating capital"; firm revenue is directly related to inventory moving out the door. Machinery, buildings, and fixtures are "fixed capital"; firm revenue may increase if the firm buys more of it, but the object is to keep it in working condition as long as possible before it is used up.
   
   Neoclassicals like Lionel Robbins noted that this was actually an ambiguous distinction, since the difference between f- and c-capital was really the time period of production one was looking at. Inventory of finished goods one hopes to sell is clearly circulating, but what about drill bits used in the process of recovering petroleum? The exploration company would like to conserve them, but over the long run, the more bits used, the more petroleum recovered (and sold to refineries).


2. I actually discovered this when I was compelled to read Paul Samuelson (Foundations of Economic Analysis, Harvard University Press (1947). Subsequently, I noticed the persistence of neoclassical assumptions and methods all through the 1960s and 1970s. For readers shocked by my claim, this will seem like a weak defense. However, my posts on Steve Keen and Hyman Minsky will treat this in greater detail.
   
   For an account of the Cambridge Capital Controversy "provoked" by Sraffa's book, please see Eckhard Hein 🙵 Engelbert Stockhammer (2011), pp.3-4.

economic surplus: excess of output over input; not to be confused with "surplus value," a concept from Marxian economics.

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SOURCES 🙵 ADDITIONAL READING

Piero Sraffa, Production of Commodities by Means of Commodities, Cambridge University Press (1960): link goes to complete text online.

Léon Walras (trans. William Jaffé), Elements of Pure Economics, Orion Editions (1984/1954); translation based on 4th Edition (1900).

"Debunking Economics, Part V: The Holy War Over Capital," Unlearning Economics (26 July 2012): for a summary of Steve Keen's account of the Cambridge Capital Controversy

Eckhard Hein 🙵 Engelbert Stockhammer, A Modern Guide to Keynesian Macroeconomics and Economic Policies, Edward Elgar Publishing (2011). Hein 🙵 Stockhammer's account of the Cambridge Capital Controversy includes a plausible explanation of why Joan Robinson's victory over Samuelson and Solow was ignored by the economics profession.

KLEMS-1

Productive Factors in DGE Economics

In economics classes and the vast majority of monographs I've read on recent economic theory, it's common to refer to two factors of production: capital (K) and labor (L). In the late 19th century, this was very important since there was a lot of polemics over the correct theory of value, with the conservatives of the day insisting on the utility theory of value. Much more recently, we have seen the Solow-Swan Classical Growth Theory, in which the Keynesian theories of the business cycle were supplemented with, then replaced by, a comprehensive model of capital and technology (A) accumulation.

This has typically been disappointing to me precisely because it seemed to me that a model of the economy in which there was a single universal production function Y = f(A, K, L) would yield only certain results. Bear in mind that we're always interested in output per worker (Y/L, or y), which is always assumed to be a function of capital per worker (K/L, or k). Some modern theories of economic growth are described as exogenous, such as Solow's; they are "exogenous" in the sense that they believe the main determinant of economic growth, A, is something outside of the economic model. "Endogenous growth theory," in contrast, focuses on the tendency of capital accumulation to cause A directly. Both theories ran into serious problems with respect to international comparisons. Exogenous growth implied that the difference between countries was the result of capital accumulation, but capital accumulation in the richest nations of the OECD is actually not much larger than that of low-income countries such as the Philippines or India. Nor was this an anomaly of the present day; today, of course, saving and investment in the least-developed countries (LDC's) surpasses that of, say, the USA (where net saving is negative).

Conversely, endogenous growth theory merely consisted of assuring us that there were increasing returns to scale of capital investment; by allowing any exponent on capital that would fit the data, the endogenous growth theorists came up with the most Panglossian view imaginable of economic development. They suffered from the logical dilemma that small, island nations like Singapore often responded better to capital accumulation than large, integrated regions (like Western Europe). While Western Europe, taken as a unit, is very affluent, and K is huge, its network spillovers ought to be larger than Singapore's. As everyone knows, the opposite is true; not only that, but investors in Europe and Japan are keen on exporting capital as if they—i.e., "the market" for investment opportunities—knew better.

Also, I was well aware that some dissident economists had objected to the obsession with human inputs to the industrial economy. Labor is superabundant; economists are usually expected to make sure the supply of that is utilized. Capital is always treated as a component of prior output, or, K = ∑(Y_t-i - C_t-i)(1 - δ)ⁱ, for i = 1 --> ∞.In the equation above, δ is the rate of capital depreciation; in each year of the past t - i, output was Y_t-i and consumption was C_t-i; so ∆K_t-i is always the difference between the two, and it will hereafter depreciate at (1 - δ)ⁱ where i is how many years ago ∆K_t-i occurred. While i may be as large as ∞, (1 - δ)ⁱ would make any capital accumulated before 1968 worth about 1% of its real value in '68.

A problem, though, is that this implies that nothing bad can happen to the economy, provided the immense stock of capital survives (and the world population doesn't shrink). What about peak oil? What about significant changes in climate that reduce farm output? And, for workers, what about corner solutions in which the full employment of all non-labor resources (renewables, energy, and capital) leaves much labor unemployed?

(Part 2)

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ADDITIONAL READING & SOURCES: Susan Houseman "Outsourcing, Offshoring, and Productivity Measurement in Manufacturing" (PDF), Upjohn Institute Staff Working Paper No. 06-130 (June 2006); Harold Cole & Lee Ohanian, "The Great Depression in the United States from a Neoclassical Perspective" (PDF), Federal Reserve Bank of Minneapolis (1999);

On the labor theory of value: Albert C. Whitaker, "History and Criticism of the Labor Theory of Value in English Political Economy" (PDF) Stanford University (1904); for an introduction to the utility theory of value, I have to recommend William Stanley Jevons, The Theory of Political Economy (5th edition), IV (complete text). As you can see, Whitaker's work of 1904 was a piece of historical research; Jevons introduced the alternative "utility theory" in 1870. The whole issue of value theory is hence pretty antiquarian. Or so you'd think; but Joseph A. Schumpeter, in his History of Economic Analysis, III.6.2, seems to think it's the basis of any system of economic analysis. Polemically, it means a lot to him, although he argues that one should not draw any polemical conclusions from any theory of value!

Schumpeter is correct when he says one ought not to draw any polemical conclusions from one's theory of value; but the logical corollary is that one therefore ought to use theories of values like domain-specific programming languages, a proposition that would no doubt cause Schumpeter to turn several different shades of magenta.

Solow-Swan Classical Growth Theory

This is an economic model of development conceived by Robert Solow and Trevor Swan separately and concurrently. In the late 1940's and early 1950's, the Keynesian Revolution in economics was working its way through the entire field, and in pursuit of a theory of economic growth. The initial model had the logical shortcoming that, if investment fell below a certain level, the capital-output ratio would fall, causing the economy to contract further, and so on. This was incompatible with the obvious fact of the business cycle; it was referred to as a "knife-edged equilibrium," in the sense that the economy was supposed to be balanced precariously on the precise equilibrium. Another problem was that economic growth and development are confused; they are not the same.

Solow's initial project was to model the economy as consisting of two factors, labor and capital; after deducting expansion of capital growth and labor growth, Solow identified that the residual accounted for virtually all economic growth; the residual was variously identified as "total factor productivity," "technology," or "knowledge." It ought to be noted that estimating what the residual is, is a controversial business since capital (machinery, fixtures, infrastructure) is heterogeneous mix of things, and labor varies in value or payment, and one can't just add "labor + capital" and divide the gross domestic product (GDP) by that. We have an estimate of the total national pool of capital, and another for the total national supply of labor hours worked, and economists do compare the growth in the two with the growth in GDP; since GDP grows faster that the total supply of either, the difference can be attributed to better use of both labor and equipment.

Solow's model, like Harrod-Domar's, focused on capital and labor inputs. In fact, the model is always presented intensively, so that we divide the volume of capital in the entire economy (K) by the total volume of labor (L), in order to get k. Capital k depreciates as a function of the total amount, and if the population is growing, then investment is required to keep the capital stock growing as fast as the population. The blue diagonal line in the chart indicates the correlation between k and required investment i. According to Solow's model, increasing capital produces an increase in output, but the relationship is not linear; past a certain point, increasing the capital stock will not actually increase the amount of output that each worker can consume, since it will have to be replaced so fast.


Click for larger image
While the heavy black line represents total output per worker, the red line indicates saving as a constant share of income. In the illustration above, it is true that if saving as a share of output were increased, then the economy would indeed settle at a higher rate of equilibrium consumption. (I just wanted experts to know I knew that). But with s/y at its present level, capital will necessarily settle at k*. A temporary increase in i (or ∆k/y) would increase k momentarily, but the economy would soon return to its former level of output and capital as the new capital resumed its usual rate of depreciation.

Now, the interesting thing about this model is that it was developed partly in response to, and in conformity with, the Keynesian Revolution. But it had the advantage of adapting efficiently to Neoclassical Growth Theory (PDF), and became absolutely fundamental to it. To this day, the concept of the Real Business Cycle (RBC) has remained the prevailing view in research economics; business cycles are the result of actual cycles in capital and labor inputs, accompanied by "technology shocks (abrupt changes in the Solow residual). Robert Solow was not an early supporter of RBC theory, but the concept of an economy consisting of just two inputs, plus technology, was the crucial concept. In the great majority of literature on economic research since that time, the (unattributed) Swan-Solow model is paramount.

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ADDITIONAL SOURCES & READING: CEPA New School, "Neoclassical Growth" and "Keynesian Growth: the Cambridge Version" [Harrod-Domar Model]; Manfred Gärtner, "The Solow model" EurMacro; Bennett T. McCallum, "Neoclassical vs. Endogenous Growth Analysis: An Overview" (PDF), Carnegie Mellon University (1996)

On the Solow Residual: Yasser Abdih & Frederick Joutz, "Relating the Knowledge Production Function to Total Factor Productivity: An Endogenous Growth Puzzle" (PDF-2006) + Shekhar Aiyar & Carl-Johan Dalgaard, "Total Factor Productivity Revisited: A Dual Approach to Development Accounting" (PDF-2005), International Monetary Fund; Charlotta Groth, Maria Gutierrez-Domenech, & Sylaja Srinivasan, "Measuring total factor productivity for the United Kingdom" (PDF), Bank of England Structural Research (Spring 2004); Scott L. Baier, Gerald P. Dwyer Jr., & Robert Tamura, "How Important Are Capital and Total Factor Productivity for Economic Growth?" (PDF) Federal Reserve Bank of Atlanta (2002); Charles Steindel & Kevin J. Stiroh, "Productivity: What Is It, and Why Do We Care About It?" (PDF), Federal Reserve Bank of New York (2001);

On Real Business Cycle Theory: for a favorable treatment of RBC theory, see Sergio Rebelo, "Real Business Cycle Models: Past, Present, and Future" (PDF) Northwestern University & NBER (March 2005); a good article explaining the connection between the Solow-Swan Classical Growth Theory and RBC theory is Mark Rush's "A primer on real business cycles or the ABCs of RBCs," Business Economics (July 1990).

In order to conduct empirical work with their models, many real cycle researchers often use so-called "Solow residuals" as a proxy for technology shocks. [...] Basically, a production function relationship for aggregate GNP is assumed, say a Cobb-Douglas production function for GNP with labor and capital as inputs. Assuming that factors are paid their marginal products, data on the total shares of output going to an input can be used to infer the coefficients of the production function. Now, any change in output can be viewed as resulting from either a change in inputs or a change in technology. Using the estimated production function coefficients, it is easy to deduce what would be the change in GNP from one year to the next if only the inputs changed. Then, any difference between the actual change in GNP and this calculated change must be attributed to changes in technology.

The estimated Solow residual shocks play a crucial role in calibrating real cycle models. [...] If the basic work adequately captures the impact from changes in inputs, then any additional change in output must result from a shift in technology.

Constant Relative Risk Aversion

There is a game of chance called "St Petersburg," which is the simplest thing possible. Take a fair coin and flip it. You have bet 2 rubles on the outcome. If the coin comes up heads then you win 2 rubles, but if it comes up tails you play again, this time for 4 rubles. Each time, the stake is doubled, so n plays yields a prize of 2ⁿ rubles. Each flip of the coin is called a "trial" and the string of trials with their outcomes that concludes the game, is called a "consequence."

The probability of a consequence of n flips ('P(n)') is 1 divided by 2n, and the "expected payoff" of each consequence is the prize times its probability. The ‘expected value’ of the game is the sum of the expected payoffs of all the consequences. Since the expected payoff of each possible consequence is 1 ruble, and there are an infinite number of them, this sum is an infinite number of rubles. This became known as the St. Petersburg Paradox.

Bernoulli, the Swiss philosopher and mathematician, suggested the problem lay in rewarding people with money rather than utility. At the back of the paradox is the assumption that (2ⁿ)(2^-n) = 1 for all values of n; as n becomes (or could become) infinitely large, the sum of probable outcomes reaches ∞. In fact, that's not true for utility, and Bernoulli proposed that the expected utility--as opposed to expected payout in rubles--was necessarily finite.¹

Economists were increasingly interested in risk² because it applies to virtually all decisions, particularly those related to savings. Suppose you have a temporary employee working at a longterm assignment. The temp could be dismissed at any second; temps are almost never given any notice, and an abrupt dismissal typically causes the temp a lot of hardship. Because of this, the temp faces a risk if she accrues any debt; savings are vital to surviving periods between assignments. Yet there is also potential benefit in taking night school courses--say, in accounting. She must therefore weigh the risk (weighted for consequences) of dismissal, against the probability of getting a permanent job with benefits (weighted for the benefits of doing so).

Putting this another way, let U be the utility experienced by the temp. U is a function of consumption C, which of course varies over time; U = U(C(t)). The temp may prefer to take risks in order to enhance her estimated future consumption: an increase in income caused by risky investment of scarce money in tuition. For small values of θ, marginal utility diminishes more slowly--i.e., U˝(C) is smaller--than for larger values. That's the crucial significance of θ.

In the equation above, a high value of θ signifies that the consumer is quickly sated by increasing consumption. Hence, both high values of consumption all at one time, and a high payoff from a high risk bear less gratification, than would be the case if θ were low. Hence, another term for "constant relative risk aversion" is "constant intertemporal elasticity of substitution" (CIES). On average, the tendency to accept risk (in exchange for a payoff) and the tendency to accept a major belt-tightening (in exchange for a future payout) are comparable.

In the graph below, the horizontal axis C(z) refers to a random outcome; the probability that z₁ happens is p, and the probability that z₂ happens is (1-p). In other words, either z₁ or z₂ can happen. So the expected outcome E(z) is pz₁ + (1-p)z₂. Now, please notice someone has drawn a chord between points A and B. Notice that the expected utility E(U) is substantially lower than the utility of the expected outcome u[E(z)]; or just notice D and E. The position of E on the chord is dependent on the ratio of p:(p-1).

The behavioral inference drawn from this chart is that the utility of expected income U[E(z)] is greater than the expected utiliy E(U), i.e.,

U[pz₁ + (1-p)z₂] > Upz₁ + U(1-p)z₂
This is just a complicated way of saying that risk aversion inflicts a severe hit on the utility of bundle of benefits.

The function above was developed by Milton Friedman and Leonard Savage in 1948. Friedman & Savage also speculated on other shapes of the risk-utility function, but the curve above has a certain usefulness for the economics profession. You see, if a person has a curve very much unlike the one shown above, then one can be presented with a series of risks, each of which one finds acceptable, that lead one into any position; the other party--say, the casino management--can always make a profit, and essentially "pump" money out of players. While some people undoubtedly are like that, the population in the aggregate cannot be, or the economy would grind to a halt forever.

If we are looking at the function as a CIES graph, then the horizontal access merely represents increasing values of consumption. If, however, we are looking at the function as a CRRA graph, then it makes sense to regard the horizontal axis as a series of equally likely payouts. A segment between z_i and z_j with a length of 1% of the entire horizontal axis, would have a 1% possibility of happening.
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NOTES

1 For those of you unfamiliar with calculus: some algebraic functions, like f(x) = x^-2 can be graphed from 0 to infinity, and the total area under their curve is finite. This seems impossible, but it's true.

2 Risk and uncertainty are (usually) regarded as distinct topics in economics. Risk is quantifiable; uncertainty is not. Or, in the words of Frank L. Knight,

The essential fact is that "risk" means in some cases a quantity susceptible of measurement, while at other times it is something distinctly not of this character; and there are far-reaching and crucial differences in the bearings of the phenomenon depending on which of the two is really present and operating. There are other ambiguities in the term "risk" as well, which will be pointed out; but this is the most important. It will appear that a measurable uncertainty, or "risk" proper, as we shall use the term, is so far different from an unmeasurable one that it is not in effect an uncertainty at all. We shall accordingly restrict the term "uncertainty" to cases of the non-quantitative type. It is this "true" uncertainty, and not risk, as has been argued, which forms the basis of a valid theory of profit and accounts for the divergence between actual and theoretical competition.
[Risk, Uncertainty, and Profit, 1921]

The Dynamics of Industrial Choice (2)

(Part 1)

Industries make decisions about the implementation of technologies according to expected returns of that implementation: that's the standard position of orthodox economic theory. The reality is more complex, but it's interesting to see how even the very simple, reductionist mathematical models used to simulate the behavior of a simple economy lead to complex systems.

Since the basic rules of economic explanation are relatively simple, efforts have been made many times to reduce these to mathematical formulas that can describe the workings of a system. One idea has been to use these to re-create the laws of motion that prevail in an economy, so that the rules can be refined based on their predictive powers. The best-known attempt to do this has been the Ramsey-Cass-Koopmans Model, which simplifies the job by treating the economy as if it consisted of a single, average household.

The RAMSEY-CASS-KOOPMANS MODEL
Attributes of the household are inherent in the economic system. Households have endowments of labor (l) and capital (k). Labor is paid at wage w and capital commands an interest rate r. Hence, the income (y) of the household will be

y = wl + rk. However, while the endowment k accrues interest without labor, it also depreciates at rate δ; presumably r < δ, or else there would be little point in holding k. Likewise, L is accumulated from y - c; in the RCK model, all income that is not immediately consumed is saved, and therefore invested in the form of more k.

In economics, K always represents the total stock of capital; k (small) represents the supply available to our sample household at any moment in time. The symbol ρ stands for the discounting of future consumption; economists assume that consumers value future consumption less than present consumption.¹ Stocks of capital depreciate at a rate of δ, so they must be replaced at a rate of δk. There must also be a rate of return on capital, which is r; it's common to assume that r = ρ + δ, since (by definition) if r < ρ + δ, people would be irrationally postponing consumption, and if r < ρ + δ, people are irrationally improvident. Each household is intuitively driven to maximize the equation below.
This is not as far fetched as it might seem. That's because the "average" path taken by millions of households groping towards an optimal allocation may well fit this description. Groping comes in the form of endless brushes with frustration and lost opportunities. Errors or eccentric decisions made by this or that individual may be expected to average out over very large numbers and over great lengths of time.

The household stock of capital increases at rate , which will be
= (r – δ)k + wl - c
There is a consumption function U(C) is assumed to take the form below:

This is the function for constant relative risk aversion (CRRA); it is also known as the continuous intertemporal elasticity of substitution (CIES) function. Since we do not impose a time horizon, there's a risk of what is called a "corner solution," which is where the maximum point of a function lies at one limit or the other of its domain. The danger here is that the solution would be "c = 0" for all t < ∞, since ∞ is the biggest number we have. At the end of time, k_∞ would be extremely large, but the who affair would be utterly pointless since our whole effort to simulate the economy with an average household would lead to that household acting in accordance with totally arbitrary equations. Such a scenario is unreasonable; people have to consume something even when their incomes are so low they can save scarcely anything, so we have limits to the value of infinitely postponed consumption.

The economy also incorporates an average firm, which transforms l and k into y. Beyond this, however, the firm does not appear; it does not have an objective function to maximize, for instance; it is not in conflict with other firms or the representative household. The RCK is an extremely adaptive model, however, and a very large number of variations on it exist. Here, we'll be sticking to the plain vanilla version.



Click for larger image

This chart shows the two phase diagrams in the RCK model. On the left, the blue line represents constant, stable rates of consumption; c-dot represents the 1st derivative of c(t) with respect to time. Let's say that k* represents the point on the horizontal axis where c-dot = 0 (where the blue line touches the bottom edge). Then levels of capital endowment k* leads to a decrease in consumption.

(Please note that c-dot is instantaneous. I point this out because, if one occupies a position {k0, c0} , then one will presently move to another point on the phase diagram.)

On the right, the red curve indicates all the positions where k-dot is 0; if one occupies positions along that line, one's net growth in capital endowments is zero. For values of c above the red line, one's rate of capital accumulation is negative (one is spending out of one's substance!). For levels of c below the red line, one's rate of capital accumulation is positive, because one is consuming so little.



Click for larger image
Here, the two maximization functions are combined. Where the red and blue lines intersect, there is steady state consumption and capital endowment. At points along the violet line passing through the intersection, points are not in equilibrium, but are "gravitating" towards it.

The economy (in the avatar of its representative households) is has a peculiar version of the knife-edged equilibrium. The saddle equilibrium might appear to suggest that the economy, if perturbed from perfect order, would plunge into wreckage, like a locomotive on a tightrope. Over the short-run, as during recessions, this would appear to be the case; and over very long periods of history, flourishing economies do eventually enter periods of decline. However, the RCK model pertains to medium-run trends; the model is not, nor ever could be, rich enough to capture the multifarious forces of the short-run, and over the long-run things such as civil wars, obsolete institutions, demographic changes, and so forth are simply out of the model.

The other important distinction is that when an economy is not on the violet line in the graph above, the optimization preferences of its population will tend to push it toward convergence with the steady-state, balanced growth economy. In fact, the model incorporates projections of how this occurs.


In the graph above, the economy responds with a reduction in consumption, and concomitant increase in saving. Richer models incorporate a "floor" of consumption that causes it to start low, and rise to overshoot the higher balanced growth path (BGP) savings rate.


Here, the economy's stock of capital is converging. Notice that the high saving rate is accompanied by a steep slope for k; high values of s amount to exactly the same thing as high values of . Likewise, in our simplified model, the efforts of households to maximize consumption over infinite time horizons leads to rapid accumulation.


This is the big picture: the balanced growth curve, in which the endogenous factors are growing at a constant pace (dotted line) while the equilibrium growth path gradually catches up.

(Part 3)
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Notes

1 Time discounting: it is usually assumed that humans generally prefer consumption in the present to consumption in the future. As a result, we assume humans have to pay more to consume now than they do if they wait.  Discounting is a way of expressing this preference in the form of a low price for future consumption.

As always, special exceptions may apply but remember, economists tend to be interested in average or median behavior. Even if exceptions are very common, therefore, people with unusual time-of-consumption preferences can demand the same discount as everyone else.

Typically, for purposes of government accounting the Office of Management and Budget (OMB) uses a rate of 7%, and tests for rates of 5%-9%.
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Resources and Additional Reading:

The Ramsey-Cass-Koopmans model is explained formally here (and here), for those of you interested in a backup source. The first link is to the site of Prof. Thomas M. Steger in Zurich; in my opinion, his explanation is not only the best I've seen online (and the most reliable), it's also better than the one in David Romer's textbook Advanced Macroeconomics (1st edition), which is the one I was initially using. Juan Ruiz, a Spanish economist, posted lecture notes on the RCK model that are also very easy to follow.

For a detailed introduction to production functions, see Egwald Economics;

The Dynamics of Industrial Choice (1)

Often, economic models seek to explain business decisions based on a snapshot of conditions. Hence, we have the case of the indifference curve and the production curve. Both are closely analogous:

INDIFFERENCE CURVE

In economics, one speaks of "utility" as a state that cannot be measured, but can be compared; so, for example, in the chart below, the blue line (U1) represents a lower lever level of utility than the red (U2). It is not valid to say U2 represents 1.5x as much utility, but we can include a very large number of intermediate levels of utility between the two points.



Utility is always described as a function of two sources, such as "wages" and "leisure" (from the POV of the worker). Of course, if you increase wages without reducing leisure, or increase both, then the person obviously has a higher level of utility. But what about when one must trade one for the other?



In the graph above, the economic actor's utility is a function of A and B. The red line is the result of a sudden decline in the price of B. When that happened, the "budget line"—the straight dashed lines slicing diagonally across the graph—moved outward, to the right. That diagonal line intersects the A-axis at the point where the consumer spends 100% of her income on A, and the B-axis where she spends 100% of her income on B. So when the price of B fell, the budget line moved outward to intersect with a new, higher, level of utility.

When the consumer had the lower (blue) budget line, she consumed A1 and B1. When the price of B fell, her consumption of both increased, to A2 and B2. But economists make a distinction between (A2, B2) and (A1.5, B1.5). While some of the change in consumption (ΔA,ΔB) can be explained by the increased income—i.e., the new, "purple" flashpoint on the red curve above—some of (ΔA,ΔB) is the result of substitution. So, for example, the increased real income caused by a decline in the price of B actually caused the consumption of A to decline in absolute terms. An income effect will always cause both to increase, but a substitution effect will always cause consumption of one to fall relative to the other.


OUTPUT CURVE

This is closely analogous to the indifference curve, and so I used a similar graphic with different labels (the original graphic is here).



Here, the tradeoff is between labor (L) and capital (K), or any other combination of inputs. While I've shown only two inputs in the diagram, it's possible to set up optimization equations involving as many inputs as you like... such as different capital structures (bonds versus bank loans versus equity), energy inputs, and so forth. One element that is new to the production curve here is the idea of technology: the possibility that output (X) can increase without an increase in L or K. In fact, economists simply treat technology as another input (A), and have long debated the role it plays (here's a formal treatment).

The concept of the indifference curve in economics dates back to the 1870's; some of the first economists to use it were the "Marginalists," such as William Stanley Jevons (1871) and Leon Walras (1874). A formal explanation of these concepts may be found here.


PARETO OPTIMIZATION

Pareto Optimization is illustrated by the Edgeworth Box shown below. It's really just a pair of indifference curves. One thing to remember is that, while the vertical axis shows rising wages, the direction of the horizontal axis is reversed. That's because the "zero" axis for the utility of the owner of capital is in the extreme upper right-hand corner of the graph.



According to this chart, the rising rate of wages is one contributor to the utility of the worker; another contribution is lower interest rates (or capital rental rates). The latter effectively increases the purchasing power of the worker.

(Incidentally, this is not a radical or leftist conception of labor-capital relations. It's from the work of Francis Ysidro Edgeworth and Vilfredo Pareto, two of the most conservative, orthodox economists who ever lived. Anyone who is seriously disturbed by my dichotomy can relax. Everyone agrees that there's a missing dimension here, which is that of time. A reasonably high value of r leads to an increase in the accumulation of capital, allowing for greater total output.)

The object of this diagram was to illustrate how the market, under optimal conditions, resolves the controversy of the correct distribution of the total economic output between labor and capital. This same dichotomy of interest also exists between producers and suppliers, or taxpayers and the state.

However, the chart also illustrates something else: one can see here the idea of demand reaching a convergence with available output. The optimal solution is one where the rate of indifference is the same as the comparative cost, which is (in turn) determined by the output function.

(Part 2)

Some thoughts on Game Theory & Economics

Readers will probably be familiar already with the fundamental economic notions of supply and demand, of diminishing marginal utility/productivity, and so on. I'm hoping readers understand the mutual compatibility of free will and the "laws" of economics, also. John Ruskin, in Unto this Last (1862) complained that economics had no room for meritorious or selfless motives; there was no room for solidarity between employers and laborers, for example.

Among the delusions which at different periods have possessed themselves of the minds of large masses of the human race, perhaps the most curious—certainly the least creditable—is the modern soi-disant science of political economy, based on the idea that an advantageous code of social action may be determined irrespectively of the influence of social affection.

However, Ruskin's understanding was imperfect: non-economic, or counter-economic, motives (such as emotional affinity for one's employees) are not incompatible with the concept of labor markets per se; rather, such motives tend to average out. For example, if someone boycotts Starbucks, others may just as likely develop "consumer loyalty" to the same firm, or a "social affection" to the staff of a particular franchise. Nor is Ruskin's objection convincing in regards to the labor market: if some firm out there, such as Old Fezziwig's, pays its workers more than it absolutely must to prevent their starvation or flight, then this reduces the number of persons Old Fezziwig may employ, thus pushing the general level of wages down by a very slight amount.

(There are other problems with classical economics, but Ruskin doesn't really know what they are.)

Economics needs to be understood not as a deterministic prediction of how persons will behave given the prices of stuff they like, but rather, as an explanation of the tradeoffs they make. Assuming a population has a reliable set of moral virtues, for instance (like a desire to avoid polluting the environment) at a reliable concentration, a decline in the price of gasoline will still result in an increase in gasoline consumption. That's because people, however much they may dislike consuming petrol, will find a dollar spent on it to provide more utility than it did before. And even if a few diehards are determined to buy absolutely positively no more than x liters of the stuff per week, no matter what, this means that enterprises using it will still be able to buy more; indeed, if a choice exists, they might increase, say, passengers transported, by leasing more cars as cabs or express vans.

However, at the level of business institutions (including, for example, labor unions, banks, and governing bodies in the national economy), economics can make few meaningful predictions. True, the average firm will (over the greatest possible time horizon) expand production until marginal costs equal marginal revenue; but there are many reasons why a particular firm, during a particular period of operation, will not. One reason is that the firm may be a specialty producer of things like ocean-going vessels or large buildings; its production levels may be controlled entirely by the local demand for a highly-specialized product. There is no meaningful demand curve for such a business, since, even if it could cut costs and prices by two-thirds, the industry for which it produces could only consume the same number of hydrocracker units or tower cranes.

Another reason is strategic-temporal. Most firms cannot just increase output by whatever amount they need to reach MC=MR. If you are the manager of a semiconductor fabrication firm, for example, you many want to increase output by 31.5%, but a new facility will increase output by 65%. If you stay where you are, marginal costs may be quite different from marginal revenues; but if if you get the new facility, you'll be working with an entirely new set of cost and revenue curves. Even if you have reached a point where the advantages of such an expansion are totally certain, you may wish to wait until the industrial union has negotiated a new contract before announcing the new plant.

Other reasons—presumably the best-known reason—has to do with solutions of the duopoly problem. If you have only two big firms that sell a product, both will want to reach MC=MR; MR varies with the sales of the firm, so both duopolies and monopolies enjoy higher marginal revenue if sales are held artificially low. Unfortunately for the manager of the duopoly, the other firm wants you to reduce sales so it can reap the higher marginal revenue, while you will want to do the same.


A mathematician, Antoine Augustin Cournot (1801-1877), proposed that game theory offered the solution to what duopolies would sell. He used this to extrapolate the linkage between game theory (which supplied the optimal choices for individual actors) and economics (which anticipated the behavior of vast numbers of actors).

Game theory was an exotic concept for economists until the 1950's. In 1951, mathematician John Nash introduced the Nash Equilibrium (actually, the mathematical explication); then John Harsyani introduced ('58) the concept of Bayesian approaches to a Nash Equilibrium. The tremendous analytical opportunities opened by game theory and the digital computer transformed economics, as theorists sought to meld the study of individual behavior and the behavior of entire economies, to achieve something that had far more predictive power. By the 1960's, Carnegie Mellon University professor John Muth had introduced the concept of "rational expectations," which sought to represent the economy as a scalar multiple of average actors; this idea was examined in far greater detail by Edward Prescott, who also tried to restore neoclassical economics. In effect, the new economists of the 1970's expanded microeconomic analysis to cover the behavior of the entire national economy.

The evolution of game theory and economics continued to drive most research, ultimately superceding "rational expectations." Adherents of the new classical economics, after the mid-90's, tended to downplay "rational expectations" in favor of "stochasticity" and "dynamic general equilibrium" (DGE) analysis [*]. At the same time, the new Keynesian model has incorporated both DGE and stochasticity, although since it does not believe policy is ineffective, it retains the ideological rift with the likes of Prescott, et al. However, like the new classical models, there is an element of expectation. As it happens, the role of expectations applies to prices and wages, as well as to expectations of future interest rates and inflation, so the New Keynesians could retain the old concept of sticky prices (and hence, adjustment of the economy to equilibrium via quantities, rather than fluid prices).

WHO WINS THE "GAME"? NEW CLASSICAL? OR NEW KEYNESIAN?
("New classical" is an economic analysis that emerged after 1961 and became prevalent in the USA after 1978; "neoclassical" is a synthesis of "marginalism" and emerging understanding of monopoloid markets, etc.; it became prevalent from 1890 to 1936)
The impact of game theory on economics can push it in different directions. As understood by the likes of Oskar Morgenstern, Robert Barro, and Vernon Smith, game theory was employed to make the case that monetary policy would be thwarted by financial arbitrage, while social welfare programs tended to expose economically virtuous actors to blackmail by economically weak ones.
(The "economic loser" theory, in which a socially-optimal technology is blocked by an economic class whose rents are threatened by it, is discussed and refuted in this PDF file by Acemoglu & Robinson)

From the other direction, Matthew McCartey points out that the interconnectedness of behavior under game theory assures multiple equilibria—leaving the state with the obligation to ensure the one that prevails is one corresponding to social goals. Likewise, Anatol Rapoport, noted game theorist, repeatedly treated confrontation as an institution in and of itself, which guided the formation of confrontation-promoting institutions. Cooper & John, in "Coordinating Coordination Failures in Keynesian Models" (PDF-1988), which introduced the concept of "strategic complementarity" to explain involuntary unemployment. As McCartney notes, these theories place an extraordinary burden of rationality on the individual, as well as on the individuals' self-knowledge. In the face of life experience demonstrating that most individuals lack either rationality or the opportunity to make predictions that average to accuracy, both justify their assumptions by anticipating that some economic agent will identify arbitrage opportunities and capitalize on them.

But even the notion of arbitrage does not save the rational expectation. That's because a prediction only becomes an arbitrage opportunity when discovery (i.e., the event happening) proves the prediction true or false. An example is the commodity options market. We can assume the options market for crude oil captures all available information for predicting the spot price of oil on 1 June 2006. But when 6/01/06 rolls around, the inferred price (i.e., the price for which the 6-month option price-spread was most advantageous) will probably be wrong, and the fact that a lucky few will have got it right contributes nothing to technical efficiency in the oil industry. A more compelling example is the impact of Volcker's monetary supply growth targeting in 1980, which could easily have been foreseen, but was not.

However, game theory is valuable when we are trying to assess reaction to uncertainty. Moreover, as with economic theory generally, game theory is far more useful examining why institutions fail to achieve desired results. Both are effective as instruments of analysis; they are dubious as elements in positive science. Hence, the typical confusion of critics: physics allows people to make moon landings, while economics offers an ingenious tool for defending absolutely any course of action. Neither it nor game theory can be falsified by observation, whereas physics can be. The flaw in this reasoning is, of course, that while social sciences are necessary, they are quasi-legalistic: whereas a physicist or chemist is productive regardless of anyone else understanding physics or chemistry, an economist is useful only in a population of other economists. The same may be said of lawyers; a single lawyer, entrusted with unilateral power to make decisions, is not particularly likely to make good ones, and lawyers are constantly disagreeing with one another, yet (I speak in compete seriousness), a nation without lawyers is virtually ungovernable. Again, one requires a structure run by legally-trained people to administer law; but there is no such thing as a controlled experiment for ensuring a legal decision is correct.

The ability of an economist to supply useful information about system failure stems from the techniques of analysis and scrutiny that economics has developed; the ability to compare rival historical forces (such as, rising supply and rising demand for a good, which have contradictory effects on price) and establish which one is more important, the ability to render disimilar situations into comparable ones for purposes of relevant comparison, and the ability to use mathematical models to test assumptions for absurdity, are the implements that economics brings to the social sciences. Game theory does the same thing, albeit more starkly.

New-classical economists will not take well to my previous two paragraphs. A few might nod in agreement, then insist that economics is good at identifying costly frictions (such as rent control or minimum wage rates) that can prevent full employment; but beyond that, they'd have a hard time explaining why economics had anything to say after Jean-Baptiste Say and Frederic Bastiat died. The Keynesian, in contrast, would wonder where the IS-LM curve fits into "supply[ing] useful information about system failure." Economists observing the economic implosion of the late 1920's, from '37 onward largely agreed the failure of sustaining adequate demand was responsible for the calamity, and devised various proposals to defend future demand. I would observe that, when economists began conceiving social welfare programs to ensure minimal levels of demand and institutions to prevent global financial illiquidity, they were doing their jobs successfully. But when, having developed institutions to prevent fiscal and monetary failure, they assumed those institutions would always work, they made profound errors in their own field that led to another, albeit much smaller, economic crisis in the 1970's.

The crisis was caused by the economics furnishing, as they evidently surmised, an instrument to solve the problem of the business cycle. It was so powerful its use became addictive. Yet the economists largely overlook the dangers of overuse. They failed to see how the tool could be insidious. And when at last its restorative powers were exhausted, the entire profession simply recanted fiscal and monetary policy, en bloc. Since economists did not offer alternative tools to political leaders (who demanded them), the result was that quasi-automatic fiscal policy and discretionary monetary policy continued to operate.

Today game theory is moving towards offering a meta-theory, or theory about theories. The question is no longer, what is the most appropriate judgment in any particular scenario (from the perspective of an omnipotent authority), but rather, what determines how a decision will be made? Instead of announcing the optimal decision under uncertainty, there is more interest in understanding what sorts approaches will be used, and if they are strategically different. The new field of neuroeconomics seeks to adapt to utility theory without rationality. Such a version of economics, in which actors operate in a system/institution, constrained neurologically, and limited to strategic behavior under uncertainty, would leave us about as far from classical economics as it is possible to get.

UPDATE (27 January 2006): Luka Crnic (Cherishing the Mundane) posts on the "hot topic" of neuro-economics. He begins with an article in The Economist:

No longer will economists rely on crude statistical models of how people behave in response to a policy change, such as an interest-rate rise or a tax increase. Instead, they will be able to peer directly into the brain to predict behaviour.

Here, neuro-economics is used as a research tool in "behavioral economics," i.e., the study of economic decisions made by individual actors. The original study of behavioral economics employed the familiar scheme of translating the researcher's individual hypothesis about economic behavior into a constrained optimization problem, then "modeling" economic events as if the nation consisted of 300 million identical humans. It was obviously a simplification, but it could be used to screen out categorically absurd scenarios. It was Quasi-Rational Economics, by Richard Thaler and Hersh Shefrin, and it introduced such compelling ideas as the split personality ("planner-doer"). Those interested in an introduction to behavioral economics can read "Behavioral Economics" (PDF) by Sendhil Mullainathan & Richard Thaler.
My impression is that further refinements in understanding of economic behavior at the personal level would only lead to a tweaking of the constant relative risk aversion component of utility maximization functions.

In "Social Neuroscience and Neuroeconomics," Crnic outlines some of the actual research in neuro-economics, which does indeed include fMRI scans of people playing a game.

First of all, what role does caudate nucleus play?

[Nature Neuroscience paper]: The human striatum has been implicated as a critical structure in trial-and-error feedback processing and reward learning. In particular, the caudate nucleus, a structure linked to learning and memory in both animals and humans has been shown to have a role in processing affective feedback with responses in this region varying according to properties such as valence and magnitude. It has been shown that activation in the human caudate nucleus is modulated as a function of trial-and-error learning with feedback.

This clearly indicates that the perception of the moral character of the partner in the trust game directly influences the neural mechanisms connected with feedback processing in trial-and-error learning

The next day, in "The Consilience of Brain and Decision," Crnic reviews a paper of the same title; he outlines some of the practical principles, without spelling out how this might affect economic modeling or analysis. I really need to return to these posts and others of Crnic to understand them properly, especially since there are so many links I will need to read.

(cross-posted at Hobson's Choice)

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NOTES:
Rationality: strictly defined in economics:

1. [Transitivity] If a decision maker prefers A to B and prefers B to C, then she should prefer A to C;


2. [Completeness] A decision maker is rational if she can rank all bundles of goods. If two bundles have equal rank, then that is a valid rank and the decision maker is "indifferent" to them

Some add some other attributes, such as monotonicity (i.e., if you like A more than B, then 2xA will be better than 2xB), local non-satiation (i.e., if A is a thing you like, then 2xA is better than 1xA), continuity, convexity (meaning, the marginal rate of substitution of A for B will increase as one's supply of A increases, however slightly) etc.

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ADDITIONAL READING: Marco Antonio Guimarães Dias, "Asymmetrical Duopoly under Uncertainty: the Extended Joaquin & Buttler Model," Pontifica Universidade Católica, Rio de Janeiro, Brazil