Saturday, December 31, 2016


I study economics as a hobby. My interests lie in Post Keynesianism, (Old) Institutionalism, and related paradigms. These seem to me to be approaches for understanding actually existing economies.

The emphasis on this blog, however, is mainly critical of neoclassical and mainstream economics. I have been alternating numerical counter-examples with less mathematical posts. In any case, I have been documenting demonstrations of errors in mainstream economics. My chief inspiration here is the Cambridge-Italian economist Piero Sraffa.

In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.

I've also started posting recipes for my own purposes. When I just follow a recipe in a cookbook, I'll only post a reminder that I like the recipe.

Comments Policy: I'm quite lax on enforcing any comments policy. I prefer those who post as anonymous (that is, without logging in) to sign their posts at least with a pseudonym. This will make conversations easier to conduct.

Saturday, October 22, 2016

Multiple And Complex Internal Rates Of Return

Figure 1: One Real and Two Complex Rates of Profit for Alpha Technique (TODO: Update graph for readability)
1.0 Introduction

My intent, in this post, is to refute a few lines in Osborne and Davidson (2016). I want to do this in the spirit of this article, while not denying any valid mathematics. Osborne and Davidson have this to say about the numeric example in Samuelson (1968)1:

In other words, when [the Internal Rate of Return] shifts, affecting the capital cost, the product of the unorthodox rates (the duration of the adjusted labor inputs) also shifts such that the overall interest-rate-cost-relationship is linear. This linearity implies that, in the context of this model at least, switching between techniques can happen but reswitching cannot because two straight lines cross only once. Moreover, the relationship between capital cost and the composite interest rate is positive, implying that the neoclassical 'simple tale' that lower rates promote more roundabout technology, is valid when the interest rate is broadly defined.

Samuelson's example is well-established, and it is incorrect to draw the above conclusion from the Osborne and Davidson model. They derive an equation which, when no pure economic profits exist, relates the price of a consumer good to its cost when a certain composite rate of profits is applied to dated labor inputs. This equation is a tautology; the capital cost on the Right-Hand Side of this equation cannot take on different values without the price on the Left-Hand Side simultaneously varying. Thus, however intriguing this equation cannot be, it cannot support Osborne and Davidson's supposed refutation of reswitching.

2.0 A Model

Consider a flow-input, point-output model of production of, for example, corn. For a given technique of production, let Li, i = 1, ..., n; be the input of labor, measured in person-years, hired i years before the output is produced, for every bushel corn produced. Suppose, for now, that a bushel corn is the numeraire2. Let the wage, w, be given (in units of bushels per person-year), and suppose wages are advanced. Define:

R = 1 + r,

where r is the rate of profits. The cost per bushel produced is:

w L1 R + w L2 R2 + ... + w Ln Rn

Define g(R) as the additive inverse of economic profits per bushel produced:

g(R) = w L1 R + w L2 R2 + ... + w Ln Rn - 1

Divide through by w Ln to obtain a nth degree polynomial, f(r), with a leading coefficient of unity:

f(R) = Rn + (Ln - 1/Ln) Rn - 1 + ... + (L1/Ln) R - 1/(w Ln)

The Internal Rate of Return (IRR), when this technique is adopted for producing corn, is a zero of this polynomial.

3.0 A Composite Rate of Profits

A nth degree polynomial has, in general, n zeros. These zeros need not be positive, non-repeating, or even real. For a polynomial with real coefficients, as above, some of the zeros can be complex conjugate pairs. The IRR is the rate of profits, r1, corresponding to the smallest real zero, R1, exceeding or equal to unity.

r1 = R1 - 1 ≥ 0

The IRR is well-defined only if the wage does not exceed the maximum wage, where the maximum wage is the reciprocal of the sum of dated labor inputs for a bushel corn:

wmax = 1/(L1 + L2 + ... + Ln)

Let r2, r3, ..., rn be the other n - 1 zeros of the above polynomial. As I understand it, these zeros, especially any complex ones, are ignored in financial analysis. Notice that these rates of profits are found, given the quantities of dated labor inputs and the wage. One cannot consider different rates of profits without varying the wage or vice versa.

For any complex number z, one can calculate a corresponding real number, namely, the magnitude (or absolute value):

|z| = |zreal + j zimag| = (zreal)2 + (zimag)2

where j is the square root of negative one. (I have been hanging around electrical engineers, who use this notation all the time.) Consider the magnitude of the product of all rates of profits associated with the zeros of the polynomial f(R):

| r1 r2 ... rn| = r1 |r2| ... |rn|

One can think of this magnitude as a certain composite rate of profits. Michael Osborne's research project, as I understand it, is to explore the meaning and use of this composite rate of profits in a wide variety of models.

4.0 A Derivation

One can express any polynomial in terms it zeros. For f(R), one obtains:

f(R) = (R - R1)(R - R2)...(R - Rn)


f(R) = (r - r1)(r - r2)...(r - rn)

Two equivalent expressions of the polynomial of interest can be equated:

Rn + (Ln - 1/Ln) Rn - 1 + ... + (L1/Ln) R - 1/(w Ln)
= (r - r1)(r - r2)...(r - rn)

The above equation holds for any rate of profits. In particular, it holds for a rate of profits equal to zero. Thus, one obtains the following identity:

1 + (Ln - 1/Ln) + ... + (L1/Ln) - 1/(w Ln) = (-r1)(-r2)...(-rn)

Some algebraic manipulation yields:

(1/w) = (L1 + L2 + ... + Ln) - Ln(-r1)(-r2)...(-rn)

Take the magnitude of both sides. One gets:

(1/w) = (L1 + L2 + ... + Ln) + Lnr1 |r2| ... |rn|

The above equation, albeit interesting, is a tautology, expressing the absence of pure economic profits. For a given technique (that is, set of dated labor inputs), one cannot consider independent levels of the two sides of the equation. Osborne and Davidson's mistake is to fail to notice that the tautological nature of the above equation invalidates their use of this equation to say something about the (re)switching of techniques.

The Left Hand Side of the above equation is the cost price of a unit output, in terms of person-years. The Right Hand Side is the sum of two terms. The first is the labor embodied in the production of a commodity. The second term is the first labor input, from the most distant time in the past, costed up at the composite rate of profits. Somehow or other, that composite rate of profits, as Osborne and Davidson note, expresses something about the number of time periods over which that first input of labor is accumulated and the distribution of dated labor inputs over those time periods. The number of time periods is expressed in the number of rates of profit that go into forming the composite rate of profits. I find how the distribution of labor inputs affects the composite rate of profits more obscure3. I also wonder how the composite rate of profits appears for a technique in which a first labor input cannot be found.

5.0 Numerical Example

An example might help clarify. Suppose labor inputs, per bushel corn produced, are as in Table 1.

Table 1: The Technology
Labor Hired for Each Technique
133 Person-Years0 Person-Years
20 Person-Years52 Person-Years
320 Person-Years0 Person-Years

5.1 Alpha Technique

The number of time periods, n, for the alpha technique, is three. The polynomial whose zeros are sought is:

fα(R) = R3 + (33/20)R - 1/(20 w)

The maximum wage is (1/53) bushels per person-years. The above polynomial, not having a term for R2, is a particularly simple form of a cubic equation. Nevertheless, I choose not to write explicit algebraic expressions for its zeros. Instead, consider the complex plane, as graphed in Figure 1, above. The traditional rate of profits is on the half of the real axis extending to the right from zero. The other two zeros are on the rays shown extending to the northwest and southeast. When the wage is at its maximum, the traditional rate of profits is zero and the complex rates of profits are at the rightmost points on those rays, as close as they ever come to zero. For wages above zero and below the maximum, the rates of profits are correspondingly further away from the origin. Figure 2, on the other hand, graphs the traditional and composite rates of profits, as functions of the wage.

Figure 2: Rate of Profits and Composite Rate of Profits for Alpha Technique

5.2 Beta Technique

For the beta technique, the number of time periods, n, is two. The polynomial whose zeros are sought is:

fβ(R) = R2 - 1/(52 w)

For wages not exceeding 1/52 bushels per person-year, the traditional rate of profits is:

r1, β = 1/(52 w)1/2 - 1

The other rate of profits is:

r2, β = -1/(52 w)1/2 - 1

The composite rate of profits is:

r1, β | r2, β | = [1/(52 w)] - 1

The dependence of the composite rate of profits on the wage is clearly visible in the beta technique.

5.3 Cost Minimization

Figure 3 graphs the traditional and composite rate of profits, as a function of the wage. In the traditional analysis, the cost-minimizing technique is found by choosing the technique on the outer envelope for the two curves to the left in the figure. Although I do not what meaning to assign to it, one could also form the outer envelope for the two curves on the right, that is, the composite rate of profits. If the (composite) rate of profits is zero, the technique on the outer envelope is the one that intersects the wage axis furthest to the right. This is the technique with the smallest total of dated labor inputs, that is, the beta technique. The outer envelope for both the traditional and composite rate of profits yield the same conclusion.

Figure 3: Wage-Rate of Profits Curves

If one based the choice of technique on the composite rate of profits, one would find the alpha technique preferable for all composite rate of profits above a small rate. This would be a switching example, not a reswitching example. There would only be one switch point, as shown on the diagram. And, by the traditional analysis, it is indeed a reswitching example, with switch points at r1 equal to 10% and 50%. I still see no reason to believe otherwise or to accept a non-equivalent model.

6.0 Conclusion

Although I reject Osborne and Davidson's conclusion about reswitching, I find the concept of the composite rate of profits intriguing. I suspect Osborne is more interested in impacting corporate finance, with the Cambridge Capital Controversy being a by-the-way kind of application. I do not see how the composite rate of profit helps with the analysis of the choice of technique. Osborne (2010) uses the composite rate of profits to clarify the relationship between the Internal Rate of Return and Net Present Value. I like that in my previous exposition of the above example, I applied an algorithm in which both IRRs and NPVs are relevant. I have not yet absorbed Osborne's NPV analysis.

  1. I have an example with reswitching at more reasonable rates of profits.
  2. Osborne and Davidson take a person-year of labor as the numeraire. I do not see anything in this model can depend on which commodity is the numeraire.
  3. Osborne and Davidson state that the composite rate of profits describes the weighted-average timing of labor inputs. Unlike this average, the Austrian average period of production was originally meant to be defined without references to prices.
  • Micheal Osborne (2010). A resolution to the NPV-IRR debate? Quarterly Review of Economics and Finance, V. 50, Iss. 2 (May): pp. 234-239 (working paper).
  • Michael Osborne (2014). Multiple Interest Rate Analysis: Theory and Applications, Palgrave Macmillan [I HAVE NOT READ THIS].
  • Michael Osborne and Ian Davidson (2016). The Cambridge capital controversies: contributions from the complex plane, Review of Political Economy, V. 28, No. 2: pp. 251-269.
  • Paul Samuelson (1968). A summing up, Quarterly Journal of Economics, V. 80, No. 4: pp. 568-583.

Saturday, October 08, 2016

Why Republicans in the USA are "The stupid party"

1.0 Introduction

In 1865, John Stuart Mill, when he was almost 60, was elected to Parliament. He represented the radical wing of the Liberal party. He had been a public intellectual for decades, with lots of books, editorials, and articles for the Tories to draw on in attacking him. Some Tories overreached. This led to the conservative party becoming known as "The stupid party".

2.0 Adventures in Parliament

I find Mill's attitude towards being a Member of Parliament (MP) unusual, albeit consistent with his stated opinions. He was not interested in giving speeches in support of his party's view when many others were willing to do so. He "in general reserved [him]self for work which no others were likely to do." (from his Autobiography. Uncited quotes below are from this book.) He had such opportunities, for few radicals were in Parliament. (Earlier in his life, such a group was known in Britain as the Philosophical Radicals.)

Despite his radicalism, some of his advocacy was in opposition "to what then was, and probably still is, regarded as the advanced liberal opinion". For example, Mill was against abolishing capital punishment and "in favour of seizing enemies' goods in neutral vessels".

But other efforts seem more progressive, when viewed from the standpoint of later times. In a speech on Gladstone's Reform Bill, Mill argued for sufferage of the working class. He also promoted women's sufferage through his parliamentary work. He put out a pamphlet for reforming British rule in Ireland, including "for settling the land question by giving to existing tenants a permanent tenure, at a fixed rent." He joined in an organization that attempted to have British officers in Jamaica prosecuted, in a criminal case. These officers had engaged in killing, flogging, and general brutality, under the pretence of having civilians brought before court-martials.

3.0 Considerations on Representative Government

J. S. Mill had long been what we would call a public intellectual. I want to particularly focus on his book with the above title. He gives a qualitative discussion of particular voting games. Mill was for proportional representation, also known then as "personal representation". And Mill recommended Thomas Hare on the topic. Other issues he considered include:

  • Provide multiple votes (a greater weight) to more highly educated members of the electorate.
  • Giving voters multiple votes for distributing in elections for a district that had multiple members to elect to a council.
  • Working class and women's sufferage.
  • The advantages and disadvantages of a secret ballot (as opposed to an open one).
  • The advantages and disadvantages of having a two-stage election (e.g., the electoral college, Senators being elected by a state's legislature.
  • The advantages and disadvantages of an upper house (e.g., the Senate, the House of Lords), under various assumptions about its composition.
  • Whether or not the chief executive should be independently elected (e.g., the President of the United States) or by the legislature (e.g., the Prime Minister in the United Kingdom).
  • How the central government and localities should interact and what should the authority and responsibility of each be.

In short, Mill seems to write about concerns often of interest today in analytical political science, albeit in a qualitative way and grounded in concrete practices in his time.

4.0 Attention and the Aftermath

The Tories in Parliament took advantage of Mill's long paper trail. In debates, they would ask if he wanted to defend some of his previous written statements. Because of Mill's forthrightness, this strategy backfired:

"My position in the House was further improved... by an ironical reply to some Tory leaders who had quoted against me certain passages of my writings, and called me to account for others, especially for one in 'Considerations on Representative Government,' which said that the Conservative party was, by the law of its composition, the stupidest party. They gained nothing by drawing attention to the passage, which up to that time had not excited any notice, but the sobriquet of 'the stupid party' stuck to them for a considerable time afterwards."

Considerations on Representative Government contains this passage:

"...It is an essential part of democracy that minorities should be adequately represented. No real democracy, nothing but a false show of democracy, is possible without it.

Those who have seen and felt, in some degree, the force of these considerations, have proposed various expedients by which the evil may be, in greater or lesser degree, mitigated. Lord John Russell, in one of his Reform Bills, introduced a provision that certain constituencies should return three members, and that in these each elector should be allowed to vote only for two; and Mr. Disraeli, in the recent debates, revived the memory of the fact by reproaching him for it, being of opinion, apparently, that it befits a Conservative statesman to regard only means, and to disown scornfully all fellow-feeling with any one who is betrayed, even once, into thinking of ends."

And that passage has this footnote (which I read as noting the existence of negative partisanship):

"his blunder of Mr. Disraeli (from which, greatly to his credit, Sir John Pakington took an opportunity soon after of separating himself) is a speaking instance, among many, how little the Conservative leaders understand Conservative principles. Without presuming to require from political parties such an amount of virtue and discernment as they that they should comprehend, and know when to apply, the principles of their opponents, we may yet say that it would be a great improvement if each party understood and acted upon its own. Well would it be for England if Conservatives voted consistently for every thing conservative, and Liberals for every thing liberal. We should not then have to wait long for things which, like the present and many other great measures, are eminently both the one and the other. The Conservatives, as being by the law of their existence the stupidest part, have much the greatest sins of this description to answer for; and it is a melancholy truth, that if any measure were proposed on any subject truly, largely, and far-sightedly conservative, even if Liberals were willing to vote for it, the great bulk of the Conservative party would rush blindly in and present it from being carried." (emphasis added.)

I assume Mill's refers to the following statement, in parliamentary debates, as his "ironical reply":

"I did not mean that Conservatives are generally stupid; I meant, that stupid persons are generally Conservative. I believe that to be so obvious and undeniable a fact that I hardly think any honourable Gentleman will question it."
5.0 Conclusion

And so, to this day, the more conservative party in some countries, such as the United States, is sometimes called "The stupid party".

  • J. S. Mill (1861). Considerations on Representative Government
  • J. S. Mill (1873). Autobiography of John Stuart Mill

Saturday, September 24, 2016

Parliamentary Parties In A Presidential System and the Failure of the Principle of Subsidiarity

1.0 Introduction

Some have argued that the Republican Party, in the United States of American, has been acting, since Newt Gringrich's speakership of the house, more like a parliamentary party1. And that this creates tensions in a presidential system2, like the USA. I think I have located another tension that, so far as I know, had not been previously identified when I started this post, months ago3.

People line up in local elections often for local reasons, to pursue local interests. In mass publics, even the political leaders in town, district, city, county, and municipal systems cannot be expected be knowledgeable about national issues and political ideology. In big-tent parties, the aggregate of such local movements need not form coherent ideologies. But when at least one national party is dominated by ideological beliefs, local politics might tend to be seen through an ideological lens. Not only might local political bickering become more bitter and rancorous, local politicians might become less responsive to specific characteristics of their areas. Federalism will work worse. Delegating decisions to the lowest authority possible among municipalities, states, and nations does not necessarily lead to more democratically responsive decisions, in some sense.

2.0 Local Politics

County-level splits in big-tent political parties do not result in ideological shifts. Suppose there are both right and left wings in two dominant political parties in a country, and these ideological spectra overlap. One party might be more dominant in one region than another. How urban and rural populations; ethnic groups; landholders, financiers, industrialists, professionals, small business owners, and workers line up might vary among regions. Once, say, in the 1950s, the Democrats were the party in the USA of southern whites and urban ethnic immigrants from southeastern Europe. And Republicans were simultaneously the party of African-Americans and big business4. Supporting a party at a local level, switching sides, and so not need not reflect strong ideological view in such circumstances. It could be a matter of simply seeking more resources for an interest group.

Once upon a time in Chicago, the Democratic party was extremely dominant, and the party was run like many another big city machine. Harold Washington was a successful reform candidate who became major. The old-time machine politicians had to go somewhere, and they became Republicans. Whatever local tensions were involved in it, this kind of local party split and reforming of one party did need not align with any national movement.

The county I live in has two urban centers. As I understand it, the Democrats are traditionally dominant in the larger city, and the Republicans are dominant in mine. We have had in both cities, in my memory, mayors that were either independent - in the sense, that they ran on neither party line - or bipartisan, in that he ran on both.

So there are two examples of alignments in local politics that might be said to be more about interest groups, and less about ideological movements. Politics in the USA has been becoming more ideological and falling along a one-dimensional continuum (Hare & Poole, 2013). And I think that has affected local politics.


Rogers (2016) does show that state legislative elections are dominated by national politics. But he does not show any break in such trends with national politics becoming more partisan. But I have stumbled upon Abramowitz and Webster (2015), which support my thesis. I do not know of any literature investigating national effects on local elections, as I postulate.

  1. For this post, I am more interested in the first paragraph in the following quotation. The second paragraph is probably the most widely quoted passage from this book, partly because of Ornstein's standing among right-leaning think tanks and partly because of an accompanying Washington Post editorial:
  2. "...we identify two overriding sources of dysfunction. The first is the serious mismatch between the political parties, which have become as vehemently adversarial as parliamentary parties, and a governing system that, unlike a parliamentary democracy, makes it extremely difficult for majorities to act. Parliamentary-style parties in a separation-of-powers government are a formula for willful obstruction and policy irresolution. Sixty years ago, Austin Ranney, an eminent political scientist, wrote a prophetic dissent to a famous report by an American Political Science Association committee entitled 'Toward a More Responsible Two-Party System.' The report, by prominent political scientists frustrated with the role of conservative Southern Democrats in blocking civil rights and other social policy, issued a clarion call for more ideologically coherent, internally unified, and adversarial parties in the fashion of a Westminister-style parliamentary democracy like Britain or Canada. Ranney powerfully argued that such parties would be a disaster within the American constitutional system, given our separation of powers, separately elected institutions, and constraints on majority rule that favor cross-party coalitions and compromise. Time has proven Ranney dead right - we now have the kinds of parties the report desired, and it is disastrous.

    The second is the fact that, however awkward it may be for the traditional press and nonpartisan analysts to acknowledge, one of the two major parties, the Republican Party, has become an insurgent outlier - ideologically extreme; contemptuous of the inherited social and economic policy regime; scornful of compromise; unpersuaded by conventional understanding of facts, evidence, and science; and dismissive of the legitimacy of its political opposition. When one party moves this far from the center of American politics, it is extremely difficult to enact policies responsive to the country's most pressing challenges." -- Thomas Mann and Norman Ornstein (2012).

  3. From an agenda-setting paper on the differences between presidential and parliamentary systems:
  4. "...the president's strong claim to democratic, even plebiscitarian, legitimacy [stands out]... Following ...Walter Bagehot, ... a presidential system endows the incumbent with both the 'ceremonial' functions of a head of state and the 'effective' functions of a chief executive, thus creating an aura, a self-image, and a set of popular expectations which are all quite different from those associated with a prime minister, no matter how popular he may be.

    But what is most striking is that in a presidential system, the legislators, especially when they represent cohesive, disciplined parties that offer clear ideological and political alternatives, can also claim democratic legitimacy... [W]hen a majority of the legislature represents a political option opposed to the... president...[,] who has the stronger claim to speak on behalf of the people: the president or the legislative majority that opposes his policies? ... One might argue that the United States has successfully rendered such conflicts 'normal' and thus defused them... [T]he uniquely diffuse character of American political parties - which ironically, exasperates many American political scientists and leads them to call for responsible, ideologically disciplined parties - has something to do with it... [T]he development of modern political parties, particular in socially and ideologically polarized countries, generally exacerbates, rather than moderates, conflicts between the legislative and the executive." -- Juan Linz (1990): pp. 53-54.'

  5. But see Steven Rogers' study, highlighted by a Jeff Stein article at Vox.
  6. These are tendencies. It is part of my point that such tendencies might be violated, at some time in some specific locality.
Selected References
  • Alan Abrsmowitz and Steven Webster (2015). All politics is national: The rise of negative partisanship and nationalization of U.S. House and Senate elections in the 21st century.
  • Christopher Hare and Keith T. Poole (2013). The Polarization of Contemporary American Politics.
  • Matt Grossmann and David A. Hopkins (2015). Ideological Republicans and Group Interest Democrats: The Asymmetry of American Politics, Perspectives on Politics, V. 13, No. 1 (Mar.): pp. 119-139.
  • Juan J. Linz (1990). The Perils of Presidentialism, Journal of Democracy, V. 1, No. 1 (Winter): pp. 51-69.
  • Thomas E. Mann and Norman J. Ornstein (2012). It's Even Worse than It Looks: How the American Constitutional System Collided with the New Politics of Extremism, Basic Books.
  • Steven Rogers (2016). National Forces in State Legislative Elections, AAPSS (Sep.): pp. 207-225.

Friday, September 09, 2016

Tim Lewins: "Economics, Intelligent-Design Theory, And Homeopathy"

Tim Lewens has written a popular introduction to the philosophy of science, The Meaning of Science: An Introduction to the Philosophy of Science. In his first substantial chapter, he writes about what distinguishes science from non-science. Karl Popper and the demarcation problem arise here. He needs examples of near sciences:

Consider the trio of economics, intelligent-design theory, and homeopathy. The only thing that unites these three endeavors is that their scientific status is regularly questioned in ways that provoke stormy debate. Is economics a science? On the one hand, like many sciences, it oozes both mathematics and authority. On the other hand it is poor at making predictions, and many of its practitioners are surprisingly blaseé when it comes to finding out about how real people think and behave. They would rather build models that tell us what would happen, under simplified circumstances, if people were perfectly rational. So perhaps economics is less like science, and more akin to The Lord of the Rings with equations: it is a mathematically sophisticated exploration of an invented world not much like our own.

In a later chapter, Lewens recognize that economics is a diverse discipline. He writes about some interesting analyses in economics. And then we get:

In contrast to these empirically rich forms of economic inquiry [associated with Sen and Kahneman], much work in neoclassical economics is concerned with the largely theoretical analysis of how markets would work if they were populated with individuals endowed with perfect rationality - in other words, creatures of fantasy. We might be tempted to classify these areas of economics as science fiction. Alternatively, we might think that this brand of economics tells us not how the world is but how the world ought to be, if only people would think straight...

I think Lewens is more complimentary to homeopathy than he is to economics. (He does have a bit more to say about economics than I have quoted.) Controlled experiments in medicine, I gather, consider one intervention as applied to a population. Advocates of homeopathic medicine claim to be treating a whole, particular person in a way which cannot be easily analyzed such reductionist experiments. This, no matter how hostile you may be to it, is an interesting claim for a philosopher to consider. Maybe what they advocate are placebos. Suppose you have a patient that is skeptical of big medicine. Would he react better to a placebo if it is administered in an alternative setting? What, ethically, could such a practitioner say when prescribing extremely diluted "medicine"?

I still am of the opinion that labelling a claim in economics as "science" or "non-science" should neither add nor subtract to its plausibility, over and above whatever empirical evidence and disciplinary arguments already do.

Friday, July 29, 2016

Emmanuelle Benicourt Influenced By Steve Keen?

I am thinking of absurdity number 3 below. I go a little further because I am amused by the well-established point with which I end this quotation.

"ABSURDITY No3 'For a price-taking firm, the demand curve for its own output is a horizontal line at the market price' (Unit 8.3)

This is false: the demand curve of a price-taking firm is not, and cannot be, horizontal: a firm supply, even if it is 'tiny', affects the price and then the demand of the good it produces.

The correct assumption should be that the firm believes that the demand curve is horizontal - an erroneous belief, but that is another story...

In their seminal article, Existence of an Equilibrium for a Competitive Economy, Kenneth Arrow and Gérard Debreu don't mention agents' beliefs but they,

'...instruct each production and consumption unit to behave as if the announcement of price p were the equilibrium value' (point 1.4.1, [Benicourt's] italics)

ABSURDITY No4 All agents are price-takers (competitive equilibrium)

...Now, any reasonable person will immediately ask: if all agents are price-takers, who set[s] prices? The e-Book answers (implicitly) this question with a circular reasoning...

Conclusion: 'A competitive market', as defined in the CORE e-Book, is not 'an approximation' of any existing market. It is not:

'...hard to find evidence of perfect competition' (Unit 8.3).

It is impossible.

The so-called 'competitive economy' model doesn't 'describe an idealised market structure' (Unit 8, p 44). It is not 'unrealistic' - any model is, by definition - it is irrelevant. In fact, it has nothing to do with capitalism. It can be considered, at most, as a variant of market-socialism models, with a benevolent planner setting prices, adding supplies and demands, etc." -- Emmanuelle Benicourt (2016). Is the CORE e-Book a possible solution to our problems? Real-World Economics Review, iss. no. 75, p. 135-142.

Tuesday, June 28, 2016

Getting Greater Weight For Your Vote May Not Give You Relatively More Power

1.0 Introduction

This post presents a perhaps surprising example of results from measuring political power in a system with weighted voting. I provide examples in which the weight of a person's vote is increased. Yet that voter, in some cases, gains no additional power, in some sense. In one case, by the measures of voting power considered here, the additional weight has no effect on the power of any voter. In another case, another player, with unchanged weight to his vote, is elevated in power with the voter whose weight is increased.

I find these results to be an interesting consequence of power measures. I have not yet found a simple example where the effect on the ranking of voting power is different for the three indices considered here. Nor have I found an example where a voter declines in power with an increase in the weight of his vote.

2.0 An Example of a Voting Game

A voting game is specified as a set of players, the number of votes needed to enact a bill into law (also referred to as passing a proposition), and the weights for the votes of each player. In considering voting games with a small number of players and weighted, unequal votes, one might think of such a game as describing a council or board of directors, where members represent blocs or geographic districts of varying sizes.

As example, consider a set, P, of four players, indexed from 0 through 3:

P = The set of players = {0, 1, 2, 3}

A common way to indicate the remaining parameters for a voting game is a tuple in which the first element is followed by a colon and the remaining elements are separated by commas:

(6: 4, 3, 2, 1)

The positive integer before the colon indicates the number of votes - six, in this case - needed to pass a proposition. The remaining integers are the weights of players' votes. In this case, the weight of Player 0's vote is 4, the weight of Player 1's vote is 3, and so on.

3.0 Two Power Indices

Consider all 16 possible subsets of the four players. These subsets are listed in the first column of Table 1. A subset of players is labeled a coalition. The second column indicates whether or not the coalition for that row has enough weighted votes to pass a proposition. If so, the characteristic function for that coalition is assigned the value unity. Otherwise, it gets the value zero. A player is decisive for a coalition if the player leaving the coalition will convert it from a winning to a losing coalition. The last four columns in Table 1 have entries of unity for each player that is decisive for each coalition. The last row in Table 1 provides a count, for each player, of the number of coalitions in which that player is decisive. The Penrose-Banzhaf power index, for each player, is the ratio of this total to the number of coalitions.

Table 1: Calculations for Penrose-Banzhaf Power Index
{}v( {} ) = 00000
{0}v( {0} ) = 00000
{1}v( {1} ) = 00000
{2}v( {2} ) = 00000
{3}v( {3} ) = 00000
{0, 1}v( {0, 1} ) = 11100
{0, 2}v( {0, 2} ) = 11010
{0, 3}v( {0, 3} ) = 00000
{1, 2}v( {1, 2} ) = 00000
{1, 3}v( {1, 3} ) = 00000
{2, 3}v( {2, 3} ) = 00000
{0, 1, 2}v( {0, 1, 2} ) = 11000
{0, 1, 3}v( {0, 1, 3} ) = 11100
{0, 2, 3}v( {0, 2, 3} ) = 11010
{1, 2, 3}v( {1, 2, 3} ) = 10111
{0, 1, 2, 3}v( {0, 1, 2, 3} ) = 10000

The Shapley-Shubik power index considers the order in which players enter a coalition. For the example, one considers all 24 permutations for the players. The first column in Table 2 lists these permutation. For each row, a player gets an entry of unity in the appropriate one of the last four columns if including that player in a coalition, reading the entries in a permutation from left to right, creates a winning coalition. The Shapley-Shubik power index, for each player, is the ratio of the totals of each of the last four columns to the number of permutations.

Table 2: Calculations for the Shapley-Shubik Power Index
(0, 1, 2, 3)0100
(0, 1, 3, 2)0100
(0, 2, 1, 3)0010
(0, 2, 3, 1)0010
(0, 3, 1, 2)0100
(0, 3, 2, 1)0010
(1, 0, 2, 3)1000
(1, 0, 3, 2)1000
(1, 2, 0, 3)1000
(1, 2, 3, 0)0001
(1, 3, 0, 2)1000
(1, 3, 2, 0)0010
(2, 0, 1, 3)1000
(2, 0, 3, 1)1000
(2, 1, 0, 3)1000
(2, 1, 3, 0)0001
(2, 3, 0, 1)1000
(2, 3, 1, 0)0100
(3, 0, 1, 2)0100
(3, 0, 2, 1)0010
(3, 1, 0, 2)1000
(3, 1, 2, 0)0010
(3, 2, 0, 1)1000
(3, 2, 1, 0)0100

4.0 Three Power Indices for Three Voting Games

Table 3 summarizes and expands on the above calculations. The Penrose-Banzhaf power index need not sum over the players to unity. Accordingly, I break this index down into two indices, where the second index is normalized. The Shapley-Shubik power index is guaranteed to sum to unity. I introduce two other voting games, with corresponding power indices, presented in Tables 4 and 5.

Table 3: Power Indices for (6: 4, 3, 2, 1)
PlayerPenrose-Banzhaf Power IndexShapley-Shubik
Power Index
05/165/1210/24 = 5/12
13/163/12 = 1/46/24 = 1/4
23/163/12 = 1/46/24 = 1/4
31/161/122/24 = 1/12

Table 4: Power Indices for (6: 4, 2, 2, 1)
PlayerPenrose-Banzhaf Power IndexShapley-Shubik
Power Index
06/16 = 3/86/10 = 3/516/24 = 2/3
12/16 = 1/82/10 = 1/54/24 = 1/6
22/16 = 1/82/10 = 1/54/24 = 1/6

Table 5: Power Indices for (5: 4, 2, 2, 1)
PlayerPenrose-Banzhaf Power IndexShapley-Shubik
Power Index
06/16 = 3/86/12 = 1/212/24 = 1/2
12/16 = 1/82/12 = 1/64/24 = 1/6
22/16 = 1/82/12 = 1/64/24 = 1/6
32/16 = 1/82/12 = 1/64/24 = 1/6

5.0 Constitutional Changes

Consider a change in the constitution, from one of the three voting games with tables in the previous section to another such game. The calculations allow one to measure the impact on voting power for any such change. To simplify matters, I consider only rankings of voting power. And, for these three voting games, the three power indices consider here happen to yield the same ranks, for any given voting game out of these three.

Accordingly, Table 6 shows changes in the rules (the "constitution") for these cases. The change to the rules on the right superficially strengthens Player 1, either by increasing the weight of Player 1's vote or requiring less votes to pass a resolution. As noted below, I am unsure what naive intuition might be for the second row. For the third vote, the number of votes needed to pass a proposition is altered such that a simple majority is needed before and after the change in weight.

Table 6: Changing the Rules to Strengthen the Players?
Starting GamePlayer RanksEnding GamePlayer Ranks
(6: 4, 2, 2, 1)0 > 1 = 2 > 3(6: 4, 3, 2, 1)0 > 1 = 2 > 3
(6: 4, 2, 2, 1)(5: 4, 2, 2, 1)0 > 1 = 2 = 3
(5: 4, 2, 2, 1)0 > 1 = 2 = 3(6: 4, 3, 2, 1)0 > 1 = 2 > 3

The first row shows a case where the weight of Player 1's vote increases, which might intuitively give him more power with respect to the apparently weaker Players 2 and 3. Yet this increase in weight also increases the power of Players 2 and 3, even though the weight of their votes does not change. And Player 1 remains equal in power to Player 2, both before and after the change. In fact, the change has no effect on the ranking of the players' voting power.

The second row shows a case where the votes needed to pass a measure declines, after the change in rules, from a super-majority to a simple majority, given the total of weighted votes. Would one expect such a constitutional amendment to strengthen the most powerful, or moderately powerful voters before the change? I find that this change raises the power of the weakest voter to the power of the middling voters. I am not sure this is counter-intuitive, unlike the other two rows.

The third row shows a case in which, like the first row, the weight of Player 1's vote increases. Both before and after the change, a simple majority, given the total of weighted votes, is needed to pass a proposition. This change makes Player 1 more powerful than the weakest player, as one might intuitively expect. But Player 2 is also made more powerful than the weakest player, despite the weight of his vote not varying. And Player 1 ends up no more powerful than Player 2. These effects on Player 2 seem counter-intuitive to me.

6.0 Conclusions

So my examples above have presented somewhat counter-intuitive results in voting games.

I gather that the Deegan-Packel and Holler-Packel are some other power indices I might find of interest. And Straffin (1994) is one paper that explains axioms that characterize some power index or other.

  • Donald P. Green and Ian Shapiro (1996). Pathologies of Rational Choice Theory: A Critique of Applications in Political Science, Yale University Press
  • P. Straffin (1994). Power and stability in politics. Handbook of Game Theory with Economic Applications, V. 2, Elsevier.