Wednesday, December 31, 2014

Welcome

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.

Tuesday, September 02, 2014

Failing to Empirically Render Visible What Was Hidden

Figure 1: Wage Share versus Ratio of Rate of Profits
1.0 Introduction

Consider the theory that Sraffa's standard system can be used to empirically predict distribution and prices in existing economies. Although individual commodities might be produced with extremely labor-intensive or capital-intensive (at a given rate of profits?) processes, large bundles of commodities chosen for technical characteristics, such as net output or wage goods, would be expected to be of average labor intensity. And the standard commodity formalizes the idea of a commodity of average capital intensity.

The data I looked at rejected this theory as a universal description of economies around the world.

2.0 Theory

The standard system is here defined for a model of an economy in which all commodities are produced from labor and previously produced commodities. The technique in use is characterized by the Leontief input-output matrix A and the vector a0 of direct labor coefficients. The gross output, q, of the standard system is a (right hand) eigenvector of the Leontief input-output matrix, corresponding to the maximum eigenvalue of the matrix:

(1 + R) A q = q,

where R is the maximum rate of growth (also known as the maximum rate of profits). The maximum rate of profits is related to the maximum eigenvalue, λm, by the following equation:

R = (1λm) - 1

From previous empirical work, I know that the maximum rate of profits is positive for all countries or regions in my data. The standard system is defined to operate on a scale such that the labor employed in the standard system is a unit quantity of labor:

a0 q = 1

The standard commodity, y, is the net output of the standard system:

y = q - A q

In the standard system, such aggregates as gross output, the flow of capital goods consumed in producing the gross output, the net output, the commodities paid in wages, and the commodities consumed out of profits all consist of different amounts of a single commodity basket, fixed in relative proportions. Those proportions spring out of the technical conditions of production in the actual economy.

Prices of production represent a self-reproducing system in which tendencies for capitalists to disinvest in some industries and disproportionally invest in other industries do not exist. In some sense, they arise in an economy in which all industries are expanding so as to maintain the same proportions. Such prices can be represented by a row vector, p, satisfying the following equation:

p A(1 + r) + a0 w = p,
where r is the rate of profits and w is the wage paid out of the net product. The adoption of the standard commodity as numeraire yields the following equation:
p y = 1

One can derive an affine function for the wage-rate of profits. (Hint: multiply both sides of the first equation above for prices of production above on the right by the standard commodity.) This relationship is:

w = 1 - (r/R)

Prices of production in the standard system can easily be found for a known rate of profits.

p = a0 [I - (1 + r) A]-1 [1 - (r/R)]

If wages were zero, the rate of profits would be equal to its maximum in the standard system. If the rate of profits were zero, the wage would be equal to unity. The wage represents a proportion of the net output of the standard system. It declines linearly with an increased rate of profits.

The gross and net outputs of any actually existing capitalist economy cannot be expected to be in standard proportions, particularly since some (non-basic) commodities are produced that do not enter into the standard commodity. But do conclusions that follow from the standard system hold empirically? in particular, the average rate of profits, the proportion of the net output paid out in wages, and market prices are observable. Given the average rate of profits for the economy as a whole, the proportion of the standard commodity paid out in wages can be calculated. Is this proportion approximately equal to the observed proportion of wages? Do the corresponding relative prices of production calculated with the standard commodity closely resemble actual relative market prices? This post answers the question about wages. The empirical adequacy of prices of production is left to a later post.

3.0 Results and Discussion

I looked at data on 87 countries or regions, derived from the GTAP 6 Data Base, compiled by the Global Trade Analysis Project at Purdue. (I had help extracting the database and putting it in a format that I can use.) GTAP 6 data is meant to cover the year 2001. The data covers up to 57 industries. (Not all industries exist in each country.)

For each country or region, I calculated:

  • The observed proportion of the net output paid out on wages.
  • The observed rate of profits, as the proportion of the difference between net output and wages to the total prices of intermediate inputs.
  • The maximum rate of profits for the standard system.
  • The ratio of the observed rate of profits to the maximum rate.

Figure 2 shows the distributions of the observed and maximum rate of profits.

Figure 2: Distribution of Actual Rate of Profits and Maximum in Standard System

Four countries or regions in the data had an actual rate of profits exceeding the theoretical maximum rate of profits: The rest of North America, Uruguay, Belgium, and Cyprus. The rest of North America is a region consisting of Bermuda, Greenland, and Saint Pierre and Miquelon. The four countries and regions are excluded from the linear regression and statistics given below.

Figure 1 shows the results of a linear regression of the wage on the ratio of the rate of profits. If, for each country or region, the standard system were empirically applicable to that country or region the intercept of the regression line would be near one, and the slope would be approximately negative one. But the 99% confidence intervals of the intercept and slope do not include these values. In this sense, the theory is rejected by the data.

Figure 1 points out the twelve countries with the wage furthest away from the prediction from the standard system. Why might the theory be off for these countries and the four excluded from the regression? Perhaps the net output is not near standard proportions. This possible variation of between the proportions of the standard commodity and the actual net output is abstracted from when plugs the observed rate of profits into the wage-rate of profits function for the standard system. I have looked at wage-rate of profits curves, drawn with the observed technique in use and the observed net output as numeraire. And countries far from the theory generally stick out as having wage-rate of profits curves with extreme curvatures.

Another possibility is that the industries in an economy are not earning nearly the same rate of profits, not merely because of barriers to entry but because of the economy not being in equilibrium. Prices of production, for any numeraire do not prevail.

Another possibility is that the Leontief matrix and the vector of direct labor coefficients do not capture the economic potential of the country or region. For example, the calculation of the rate of profits abstracts from the existence of land and fixed capital. Most interestingly, suppose the country or region does not characterize an isolated economic system. A region in the data combines several countries for which data is difficult to get. And the above analysis highlights several of these regions: the rest of North America, Central America, and the rest of Middle East (which consist of all of the Middle East besides Turkey). Or the country under consideration might be small and heavily dependent on imports and exports. You might notice Hong Kong and Singapore, which are important international ports. Think also of small countries that provide off-shore banking facilities. Recent events have alerted me to Cyprus serving this purpose for the countries that were formerly in the Soviet Union. I do not know much about Ireland, but recent discussion of how Apple shields its profits makes me wonder about the reported profits for its economy.

I do not know what to fully make of this analysis. The empirical use of the standard commodity seems to be more of a heuristic than the application of a claimed international law. And the failure of its application seems to point out aspects of the deviating countries that seem of economic interest.

Appendix: Data Tables
Table 1: Descriptive Statistics for Rate of Profits (Four Countries Removed)
StatisticMaximum
Rate of
Profits
Observed
Rate of
Profits
Ratio of
Observed Rate
To Maximum
Sample Size838383
Mean84.85248.6230.591
Std. Dev.26.08814.8980.138
Coeff. of Var.0.3070.3060.234
Skewness-0.374-0.0440.623
Kurtosis0.3260.5910.134
Minimum8.6235.4950.356
1st Quartile66.19539.9470.476
Median86.24247.3850.575
3rd Quartile104.13958.1240.662
Maximum144.81884.8220.967
Interquartile Range/Median0.4400.3840.323

Table 2: Descriptive Statistics for Wages (Four Countries Removed)
StatisticWage in
Standard
System
Observed
Wage
Sample Size8383
Mean0.4090.431
Std. Dev.0.1380.085
Coeff. of Var.0.3380.198
Skewness-0.623-0.397
Kurtosis0.134-0.597
Minimum0.0330.246
1st Quartile0.3380.360
Median0.4250.453
3rd Quartile0.5240.491
Maximum0.6440.597
Interquartile Range/Median0.4380.289

Thursday, August 28, 2014

The Temporal Single System Interpretation and Marx's History of Political Economy

I associate the Temporal Single System Interpretation (TSSI) of Marx's Capital most notably with Alan Freeman and Andrew Kliman. The TSSI must be addressed today by those grappling with the mathematics of the Transformation Problem, with how prices and labor values are related. But I think the TSSI makes much of Marx's work incomprehensible.

Whatever else Marx was, he was very well read. And he had many comments on the political economy of his predecessors and contemporaries. You can see this most obviously in Theories of Surplus Value, the so-called fourth volume of Capital. But, really, you can find such comments throughout Marx's work, extending back even to the Economic and Philosophical Manuscripts of 1844.

Arguably, Marx was not trying to create a scientific theory of capitalist economies1, although he did extend classical political economy along these lines. Rather Marx thought that even the best work of British political economy - that is, David Ricardo - took too much for granted. How does capitalism create the illusion that labor is a commodity, freely bought and sold on the market like any other commodity? Why do so many come to believe that profits are a return to capitalists for the contribution of capital to production? How did the institutions of capitalist economies emerge from a feudal past? These are central questions for Marx. He addressed them through a process of immanent criticism.

I am not sure that Marx was always fair to Smith and Ricardo. He often castigates them for not recognizing distinctions that Marx himself created. (On the other hand, I can see the point of arguing that Ricardo was not clear on the difference between relative natural prices and a notion of absolute value that he was struggling to develop.) Marx's unfairness, if that is what it is, strengthens my point. Does he argue that Ricardo should have been developing the sort of supposedly dynamic concepts essential to the TSSI? Or does he accept that Ricardo has adopted an approach consistent with TSSI, with his difficulties being located elsewhere? On the other hand, a dual system interpretation, in some formulation or other, has no problem with understanding the differences between market and natural prices and Smith's idea, for example, that natural prices act as centers of gravitational attraction for market prices.

One can find many proponents of the TSSI writing in a style drawing on Hegel, whether on his head or right-side up. But I am not aware of any detailed work by such proponents exploring Marx's comments on, say, William Petty, Francois Quesnay, Adam Smith, Ricardo, with an emphasis on if or how they disagreed with the TSSI.

Footnotes
  1. I recognize a tension here with the empirical work I have been presenting in the last couple of weeks.

Monday, August 25, 2014

Estimates Based On Labor Values More Precise Than Those Based On Direct Labor Coefficients

Table 1: Variations Across Countries
1.0 Introduction

This post is an empirical exploration of a simple labor theory of value as a theory of price. The precision of estimates of labor values is compared with the precision of estimates based on direct labor coefficients. The question of the accuracy of the labor theory of value is left to later posts.

I think of precision and accuracy in terms of darts. Suppose all your dart throws cluster together. Then they are precise, even if that cluster is not near the bulls eye. But if they are also in the bulls eye, then your throws are accurate, as well.

2.0 Direct Labor Coefficients and Labor Values

Labor values are calculated in the manner I find most straightforward, from a pure circulating capital model. Each industry in a modeled country, in the year in which the country is observed, produces a flow of a single commodity. Inputs for each industry consist of labor power and a flow of commodity inputs. The quantity of labor directly used, per unit output of the industry, constitutes the direct labor coefficient for that industry.

The labor value embodied in a commodity consists of all labor directly or indirectly used as an input for producing it. In the model, all inputs into production can be reduced to an infinitely long, dated stream of labor inputs. For example, the input into the industry for wearing apparel includes labor directly employed in the given year, as well as some labor directly employed in the textile industry in the previous year. (In calculating such dated labor inputs, one abstracts from changes from technology, at least in the approach that I am using. The same technique is assumed to have been used forever in the past.) Inputs directly used in the textile industry include outputs of the industry for wool and silk worm cocoons. Thus, the labor inputs into the industry for wearing apparel include some labor directly employed in that industry two years ago, as well as some labor employed three years ago in the industry for bovine cattle, sheep and goats, and horses. Given that the technique for the economy is viable, the sum of the infinite sequence of labor inputs constructed in the way outlined converges to a finite sum. I know that the techniques for all countries that I am considering are viable, based on previous empirical work.

3.0 Source of the Data

Labor values are found, for each of one of 87 countries or regions, as calculated from a Leontief matrix and vector of direct labor coefficients for a country. Each Leontief matrix was derived from a transaction table. The transactions tables, in turn, are derived from the GTAP 6 Data Base, compiled by the Global Trade Analysis Project at Purdue. (I had help extracting the database and putting it in a format that I can use.) GTAP 6 data is meant to cover the year 2001. The data covers up to 57 industries. (Not all industries exist in each country.)

Quantities of each commodities, including labor power, are measured such that a unit of each commodity can be purchased with one billion dollars at prices observed when the data was taken. With this choice of units, and the adoption of one billion dollars as the numeraire, observed market prices are unity for each produced commodity.

4.0 Results and Discussion

Figures 2 and 3 show direct labor coefficients and labor values, as calculated from the data. Each point in, say, Figure 2, represents the direct labor coefficient in a specific country for the industry with the label on the X axis. Many points are plotted for each industry, since that industry exists in many countries.

Table 2: Direct Labor Coefficients By Industry
Table 3: Labor Values By Industry

The labor value for each industry, in a given country, exceeds the corresponding direct labor coefficient. I was surprised to see that any direct labor coefficients or labor values exceed unity. The largest labor coefficient and labor value is for the industry producing oil seeds in Greece. Looking at the transactions tables, I see value added includes rows for a value-added tax, as well as income for labor, returns to capital, and rents on land. In Greece, the value-added tax for oil seeds is negative. Perhaps the government of Greece has decided that, for example, the olive oil industry is important to them for cultural reasons. And they subsidize it. So this most extreme point on my graph points to something of economic interest.

The labor values, for example, for a specific industry constitute a sample, with each country contributing a sample point. For the labor values for that industry, one can calculate various statistics, including the sample size, the mean, the standard deviation, skewness, and kurtosis. The sample size will never exceed 87, since Leontief matrices were calculated, in the analysis reported here, for 87 countries.

The coefficient of variation is a dimensionless number. It is defined as the quotient of the standard deviation to the mean. Since the coefficient of variation is dimensionless, it does not depend on the choice of physical units in which to measure the quantities of the various commodities.

Figure 1, at the top of this post, shows the distributions of the coefficient of variation, for labor values and direct labor coefficients, across countries. The variation in labor values tends to be smaller and more clustered than the variation in direct labor coefficients. Consider two theories, where one states that prices in a country tend to be proportional to labor values. The other theory is that prices tend to be proportional to direct labor coefficients. This post is an empirical demonstration that the first theory is more precise.

Monday, August 18, 2014

Even If The Workers Could Live On Air

The Maximum Rate Of Growth Around The World

Consider a model of an economy in which all commodities are produced from inputs of labor and previously produced commodities. And suppose the commodities needed as inputs in the production of commodities are described through a Leontief input-output matrix in which no commodity can be produced with (unassisted) direct labor alone. Consider the special case in which wages are zero. In a sense, this special case can be seen as a description of a futuristic economy in which all production is automated, and robots are used to produce robots.

In the theory, the input-output relations determine a finite maximum rate of profits, corresponding to the maximum eigenvalue of the Leontief matrix. This maximum rate of profits is also the maximum rate of growth that arises in the Von Neumann growth model. A composite commodity, proportional to the associated eigenvector, arises from the Leontief matrix. Along the Von Neumann ray, the output of the economy each year consists of an evenly expanding output of this standard commodity, as Piero Sraffa called it. The standard commodity, in some sense, is a generalization of "corn" in David Ricardo's corn model (which was expounded in his 1815 Essay on the Influence of a Low Price of Corn on the Profits of Stock). The commodities with positive quantities in the standard commodity are known as basic commodities, once again in Sraffa's terminology.

As this post demonstrates, this is an operational model. The graph above is based on an eigenvector decomposition of Leontief matrices. Each Leontief matrix was derived from a transaction table for a country or region. The transactions tables, in turn, are derived from the GTAP 6 Data Base, compiled by the Global Trade Analysis Project at Purdue. (I had help extracting the database and putting it in a format that I can use.) GTAP 6 data is meant to cover the year 2001. Quantities of each commodities are measured such that a unit of each commodity can be purchased with one billion dollars at prices observed when the data was taken.

The graph above and the table below show the maximum rate of profits or growth for each country or region for the snapshot yielding the data. The actual rate of profits for prices that allow for the smooth reproduction of the economy falls below the maximum, sometimes considerably, because the workers do not live on air. The larger the proportion of the net output of the economy paid out in wages, the lower the corresponding rate of profits. At any rate, prices of production fall out, given some information on the distribution of income and production conditions.

Along with calculating the maximum rate of profits, I found the standard commodity and identified which commodities are basic for each country or region. For example, the commodities produced by the following industries are basic commodities in the United States: Cereal Grains; Vegetables, Fruits, Nuts; Crops; Bovine Cattle, Sheep and Goats, Horses; Animal Products; Raw Milk; Coal; Oil; Minerals; Bovine Meat Products; Meat Products; Dairy Products; Sugar; Food Products; Beverages and Tobacco Products; Textiles; Wearing Apparel; Wood Products; Paper Products; Publishing; Petroleum, Coal Products; Chemical, Rubber, Plastic Products; Mineral Products; Ferrous Metals; Metals; Metal Products; Motor Vehicles and Parts; Transport Equipment; Electronic Equipment; Machinery and Equipment; Manufactures; Electricity; Gas Manufacture, Distribution; Water; Construction; Trade; Transport; Water Transport; Air Transport; Communication; Financial Services; Insurance; Business Services; Recreational and Other Services; and Public Administration, Defense, Education, Health. Which commodities are basic varies among countries, and I typically found a few non-basic commodities in each country.

I think this data is fairly comprehensive, and I hope that I can do further believable analyses with it.

Maximum Rate Of Growth By Country
CountryRate of Growth
(Percent)
Peru144.8
Turkey132.5
Rest of Southeast Asia127.4
Albania125.3
Uganda122.5
Zambia121.6
Rest of Southern Africa Development Community120.7
Mozambique119.3
Greece117.8
Mexico117.3
Argentina116.9
Columbia115.0
France109.8
Sri Lanka109.6
Chile108.8
United States107.1
Bangladesh106.4
Zimbabwe106.3
Rest of Sub-Saharan Africa105.6
Spain104.4
Madagascar104.1
Rest of South Asia104.1
Venezula104.1
Japan103.4
Switzerland102.5
Italy101.0
Botswana98.2
United Kingdom97.7
India93.8
Indonesia93.0
Rest of Free Trade Area of the Americas92.8
Rest of EFTA92.0
Portugal91.4
Canada90.7
Malta89.3
Rest of the Caribean88.9
Denmark88.7
Rest of Central America88.4
Tanzania87.7
Rest of South America87.1
Australia87.1
Rest of Europe86.2
Brazil85.6
Sweden85.2
Rest of North Africa84.3
South Africa83.5
New Zealand82.7
Rest of Middle East82.5
Tunisia82.3
Taiwan82.1
Netherlands81.7
Finland80.0
Poland76.8
Germany76.6
Latvia75.8
Rest of South African Customs Union75.0
Malawi72.9
South Korea72.9
Hungary72.4
Austria70.3
Luxembourg67.6
Romania66.6
Russia66.2
Lithuania66.0
Rest of East Asia64.4
Philippines64.2
Estonia62.3
Thailand61.6
Malaysia60.5
Vietnam60.0
Rest of Oceania57.6
Ireland55.2
Central America54.3
Slovakia54.0
China53.8
Slovenia53.7
Croatia51.1
Czech Republic47.1
Belguim46.9
Morocco45.9
Hong Kong40.6
Singapore35.0
Cypress24.6
Rest of Former Soviet Union12.7
Uruguay12.4
Bulgaria8.6
Rest of North America4.7

Friday, August 15, 2014

Political Intervention in Faculty Selection at the UIUC

This is a post about the University of Illinois at Urbana Champaign (UIUC)1. It is not about current events.

In the late 1940s, UIUC attempted to revamp their economics department. They hired many new economists, including, for example, Jacob Marschak and Franco Modigliani2. A bunch of economists previously at UIUC resisted these modernizing changes. They ended up calling for political support in the press, complaining about New Deal politics. And the department was purged, in a violation of academic freedom, of these new-fangled economists3.

I thought I knew about this incident originally from reading an Esther Merjam Sent article about why both rational expectations and bounded rationality could have emerged from research at Carnegie Mellon during the 1950s - maybe, "Sargent versus Simon: Bounded Rationality Unbound" (Cambridge Journal of Economics, V. 21, No. 3 (1996): pp. 323-338). Or maybe I am recalling Fred Lee's 2009 book, A History of Heterodox Economics: Challenging the mainstream in the twentieth century. Googling, I find a draft of a paper from Antonella Rancan, who I have not otherwise read.

Footnotes
  1. I have many positive impressions of UIUC. As I recall, the first graphical web browser was made there.
  2. Modigliani, in a 1944 paper, extended the Hicksian IS/LM interpretation of Keynesianism to include a labor market with sticky wages. This was a critical contribution towards a politically powerful approach that Post Keynesians quarrel with on theoretical grounds (while agreeing, mostly, on short term political implications).
  3. Modigliani ended up at Carnegie Mellon, which I guess was once not called that.

Friday, August 08, 2014

Labor Demand In A Fog

Figure 1: Labor Demanded Per Unit Output in a Stationary State
1.0 Introduction

As a Sraffian, I have no problem with open models in which room exists for exogenous political forces to determine distribution. The example here, though, has more indeterminancy than I expect.

2.0 Technology

Consider a simple capitalist economy, composed of workers and capitalists. After replacing (circulating) capital goods, output consists of a single consumption good, corn. The workers are paid a wage, w (in units of bushels corn per person year) out of the harvest. Capitalists obtain the rate of profits, r. The technology1 consists of an infinite number of Constant-Returns-to-Scale (CRS) techniques. In each technique, a bushel of corn is produced from inputs of:

  • l0 person-years of labor performed in the year of the harvest.
  • l1 person-years of labor performed one year before the harvest-year.
  • l2 person-years of (unassisted) labor performed two years before the harvest-year.

Each technique is determined, given the values of the two index variables s and t. s is a non-negative real number less than or equal to the parameter c. t is a non-negative real number.

l0(t, s) = A - B + (t + 1)(B - s)/2
l1(t, s) = s
l2(t, s) = (B - s)/[2 (t + 1)]

where A, B, and c are positive constants and

cBA

In effect, the above has traced out isoquants for a production function, where the quantity of output is a function of dated labor inputs2. For a given value of the index variable s, labor inputs in the harvest year and two years before can be traded off. That is, if the amount of labor two years before is lower, then more labor must be expended in the harvest year. Likewise, for a given value of the index variable t, more labor being expended one year before the harvest mandates less labor being expended in the harvest year and two years before. So this specification of technology allows for substitution among inputs, at least in comparing steady states3, 4.

3.0 Choice of Technique

As usual, I consider a competitive, steady state economy in which capitalists have chosen the cost-minimizing technique, at an exogenously specified wage or rate of profits. Consider the function v(r, t, s):

v(r, t, s) = (1 + r)2 l2(t, s) + (1 + r) l1(t, s) + l0(t, s)

Take a bushel of corn as numeraire. The condition that all income be paid out to workers and capitalists leads to a wage-rate of profits curve, as a function of the rate of profits and the technique (specified by the values of the two index variables):

w(r, t, s) = 1/v(r, t, s)

A wage-rate of profits curve can be drawn for each technique. The wage-rate of profits frontier, consistent with a competitive steady-state, is the outer envelope (Figure 2) of all these curves. That is, for a given wage, one finds the values of the index variables that maximizes the wage among all techniques. This maximization does not fix s. But, for each value of s, the maximum is found by setting the index variable t equal to the rate of profits r. The equation for the frontier is:

w(r) = 1/(A + Br)

Notice the frontier is independent of the labor input, s, in the first year before the harvest. In this case, each point on the frontier is consistent with a continuum of profit-maximizing techniques. And these techniques vary continuously along the frontier. None of this indeterminancy is apparent by looking at the frontier5.

Figure 2: The Wage-Rate of Profits Frontier
4.0 Labor Inputs

The analysis of the choice of technique allows one to plot labor inputs versus selected variables from the price system. In any year in a stationary state, some workers will be gathering the harvest, some will be working on preparing for the harvest one year out, and some will be working on preparing for the harvest two years out. So employment, per the unvarying net output, is the sum of l0, l1, and l2. And these labor inputs can be found from a given rate of profits and a choice of s. From the wage-rate of profits frontier, one can calculate the wage for any given rate of profits. Thus, one has the two dimensions needed to draw the curves in Figure 1. One sees that, for any given wage in an interval from zero to a maximum, the quantity of labor demanded by the firms per unit output is a relation, not a function of the wage. If the relation shown were considered to be a labor demand curve, the curve would have a certain (varying) thickness.

5.0 Capital Inputs

The analysis of the choice of technique also allows one to plot the value of capital goods6 versus selected variables from the price system. I define the value of capital per unit output, given the rates of profit and the technique like so:

k(r, t, s) = (1 + r) l2(t, s) w + l1(t, s) w

This definition is such that the value of capital advanced, discounted to harvest time, and the wages paid out of the harvest add up to unity:

k(r, t, s) (1 + r) + l0(t, s) w = 1

Impose the condition here, too, that only cost-minimizing techniques are considered for a given rate of profits. Then one obtains the curves shown in Figure 3. Here, too, the analysis yields an obvious indeterminancy.

Figure 3: Capital Demanded Per Unit Output in a Stationary State
6.0 Conclusion

Does this example undermine Sraffian analysis, as well as introductory textbook labor economics?

Footnotes
  1. Notation and numerical values are chosen to be consistent with a past post.
  2. I am unsure how to explicitly represent such a production function.
  3. With three or more inputs, some complementarity among inputs is possible. I am not sure how to express this formally.
  4. I suppose the production function consistent with the data exhibits non-negative marginal returns. I am not sure it would exhibit non-increasing marginal returns. If not, I would like to see either a proof, in the general case with n dated labor inputs, that the shaded violet regions cannot arise, given such conventional properties for a production function. Or, I would like to see a concrete numerical illustration like mine, but with such conventional properties shown to hold.
  5. Also, notice the analysis of the choice of technique leads to simpler equations than those in the specification of the technology. This is not an accident.
  6. I gather that, for any given value of s, unassisted labor two years before the harvest can be used to produce one of a continuum of capital goods, depending on the value of t. And once one of these capital goods is selected, the minimum dated labor inputs in each of the three years are fixed. Maybe this way of thinking about capital goods makes issues of convexity raised in Footnote 4 of little interest.
References
  • Enrico Bellino (1993). Continuous Switching in Linear Production Models, Manchester School, V. 61, Iss. 2 (June): pp. 185-201.
  • Christian Bidard (2014). The Wage Curve in Austrian Models, Centro Sraffa Working Papers n. 3 (June).