Hahnel ABCs of Political Economy Modern Primer

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We call equations (1) and (2) the “price equations” for the economy. They are 2 equations with 4 unknowns: w, r, p(1), and p(2). (The a(ij) and L(j) are technological “givens.”) But we are only interested in relative prices, i.e. how many units of one good trade for how many units of another good. If we set the price of good 2 equal to 1, p(2) = 1, then p(1) tells us how many units of good 2 a unit of good 1 exchanges for, and w tells us how many units of good 2 a worker can buy with her hourly wage. So we now have 2 equations in 3 unknowns: w, r, and p(1), the price of good 1 relative to the price of good 2. We proceed to discover: (1) that the wage rate and profit rate must be negatively related, (2) that the relative prices of goods can change even when there are no changes in consumer preferences, productive technologies, or the relative scarcities of resources, (3) which new technologies will be adopted and which will not be, (4) when the adoption or rejection of a new technology will be socially productive or counterproductive, and (5) how the adoption of new technologies will affect the rate of profit in the economy.

(1) What would the wage rate be in this economy if the rate of profit were zero? We simply substitute r = 0, p(2) = 1, and the values representing our technologies (or recipes) for producing the two goods, the a(ij)’s and L(j)’s, into the two price equations and solve for p(1) and w:

(1+0)[0.3p(1) + 0.2(1)] + 0.1w = p(1);	0.3p(1) + 0.2	+ 0.1w	= p(1)
(1+0)[0.2p(1) + 0.4(1)] + 0.2w = 1;	0.2p(1) + 0.4	+ 0.2w	= 1
0.1w = 0.7p(1) – 0.2;	w = 7p(1) – 2
0.2w = 0.6 – 0.2p(1);	w = 3 – p(1)
7p(1) – 2 = w = 3 – p(1); 8p(1) = 5;	p(1) = 5/8; p(1) = 0.625
w = 3 – p(1) = 3 – 0.625;	w = 2.375.

(2) Suppose the actual conditions of class struggle are such that capitalists receive a 10% rate of profit. Again, with p(2) = 1, what will the wage rate be under these socio-economic conditions?

(1 + 0.10)[0.3p(1) + 0.2(1)] + 0.1w = p(1) (1 + 0.10)[0.2p(1) + 0.4(1)] + 0.2w = 1

Solving these two equations as we did above yields: p(1) = 0.649 and w = 2.086

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(3) Suppose the actual conditions of class struggle are such that capitalists receive a 20% rate of profit. Again, with p(2) = 1, what will the wage rate be under these socio-economic conditions?

(1 + 0.20)[0.3p(1) + 0.2(1)] + 0.1w = p(1) (1 + 0.20)[0.2p(1) + 0.4(1)] + 0.2w = 1

Solving these two equations as we did above yields: p(1) = 0.658 and w = 1.811

The answers to the first three questions reveal an interesting relationship between the rate of profit and the wage rate in a capitalist economy. As the rate of profit rises from 0% to 10% to 20% the wage rate falls from 2.375 to 2.086 to 1.811 units of good 2 per hour.6 Moreover, the change in r and w is not due to changes in the productivity of either “factor of production” since productive technology did not change in either industry. It is possible the fall in w (and consequent rise in r) was caused by an increase in the supply of labor making it less scarce relative to capital – which mainstream micro economic models do recognize as a reason there would be a change in returns to the two “factors.” But this is by no means the only reason wage rates fall and profit rates rise in capitalist economies. A decline in union membership, a decrease in worker solidarity, a change in workers’ attitudes about how much they “deserve,” or an increase in capitalist “monopoly power” leading to a higher “mark up” over costs of production on goods workers buy are also reasons real wages fall and profit rates rise in capitalist economies. Political economy theories like the “conflict theory of the firm” explore how changes in the human characteristics of employees affect wage rates (and consequently profit rates), and how employer choices regarding technologies and reward structures affect their employees’ characteristics. Political economy theories like “monopoly capital theory” explore factors that influence the size of mark ups in different industries and the economy as a whole.

The answers to the first three questions also reveal something interesting about relative prices in a capitalist economy. As we

6.This negative relationship between w and r holds in more sophisticated versions of the model and appears again in our long run political economy macro model in chapter 9.

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changed from one possible combination of (r,w) to another – from (0, 2.375) to (0.10, 2.086) to (0.20, 1.8106) – p(1), the price of good 1 relative to good 2, changed from 0.625 to 0.649 to 0.658 even though there were no changes in productive technologies (or consumer preferences for that matter). In other words, the relative prices of goods are not determined solely by preferences, technologies, and “factor” supplies. Relative prices are also the product of power relationships between capitalists and workers (and owners of natural resources in an extended version of the model).

Technical change in the Sraffa model

One of the conveniences of a Sraffian model is that it allows us to determine when capitalists will implement new technologies and when they will not, and what the long run effects of their decisions on the economy will be.

(4) Under the conditions of question one, [r = 0%, w = 2.375, p(1) = 0.625, and p(2) = 1], suppose capitalists in sector 1 discover the following new capital-using but labor-saving technique:

a'(11) = 0.3 a'(21) = 0.3 L'(1) = 0.05

Will capitalists in sector 1 replace their old technique with this new one?

The new technique is capital-using since a'(21) = 0.3 > 0.2 = a(21). But it is labor-saving since L'(1) = 0.05 < 0.10 = L(1). The extra capital raises the private cost of making a unit of good 1 by: (0.3 – 0.2)p(2), or (0.3 – 0.2)(1) = 0.1. The labor saving lowers the private cost of making a unit of good 1 by: (0.1 – 0.05)w, or (0.1 – 0.05)(2.375) = 0.119. Which means that when the rate of profit in the economy is zero and therefore w = 2.375, this new capital-using, labor-saving technology lowers the private cost of producing good 1 and would be adopted by profit maximizing capitalists in sector 1.

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(5) Under the conditions in question three, [r = 20%, w = 1.811, p(1) = 0.658, and p(2) = 1], suppose capitalists in sector 1 discover the same new technique: Will they replace their old technique with this new one?

As before the extra capital raises the private cost of making a unit of good 1 by: (0.3 – 0.2)p(2), or (0.3 – 0.2)(1) = 0.1. But now the labor savings lowers the private cost of making a unit of good 1 by: (0.1 – 0.05)w, or (0.1 – 0.05)(1.8106) = 0.091. Which means the new technique now raises rather than lowers the private cost of making a unit of good 1, and would not be adopted by profit maximizing capitalists.

The model permits us to easily deduce what new technologies would be adopted by profit maximizing capitalists. And if a new technology is adopted we can use the model to calculate how the new technology will affect wages, profits and prices in a very straightforward way – as we do below. But the answers to questions four and five reveal a surprising conundrum worth considering before we proceed. The new technique either improves economic efficiency, and is therefore socially productive, or it is not. If it improves economic efficiency, capitalists in industry 1 serve the social interest by adopting it, as we discovered they would under the conditions stipulated in question four. But then, capitalists will obstruct the social interest by not adopting the new, more efficient technique, as we discovered they will not under the conditions stipulated in question five. On the other hand, if the new technique reduces economic efficiency, capitalists will serve the social interest by not adopting it, as we discovered they will not under the conditions stipulated in question 5, but will obstruct the social interest by adopting it, as we discovered they will under the conditions stipulated in question 4. In other words, no matter whether the new technique is, or is not more efficient, capitalists will act contrary to the social interest in one of the two sets of socio-economic circumstances above!

Adam Smith actually envisioned two, not one, invisible hands at work in capitalist economies: One invisible hand promoted static efficiency, and the other one promoted dynamic efficiency. He not

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only hypothesized that the micro law of supply and demand would lead us to allocate scarce productive resources to the production of different goods and services efficiently at any point in time, he also believed that competition would drive capitalists to search for and implement new, socially productive technologies thereby raising economic efficiency over time. Smith assumed that all new technology that reduced capitalists’ costs of production – and only technologies that reduced capitalists’ production costs – improved the economy’s efficiency. We have just discovered that apparently Smith’s second “invisible hand” is imperfect, just like his first! In some circumstances capitalists will serve the social interest by adopting new, more productive technologies that lower their costs of production, but in some circumstances they will not. And in some circumstances capitalists will serve the social interest by rejecting new, less efficient technologies that lower their costs of production, but in some circumstances they will not.

To sort out the logic of when the first invisible hand works, and when it does not, we needed to be able to identify the socially efficient level of output for any good. We used the “efficiency criterion” to do that: The socially efficient amount of anything to produce is the amount where the marginal social benefit of the last unit consumed is equal to the marginal social cost of the last unit produced. To sort out the logic of when the second invisible hand works, and when it does not, we need to be able to identify when a new production technology is more efficient, or socially productive. The surplus approach proves remarkably adept at helping us identify when a new technology improves economic efficiency and is therefore socially productive, and when it reduces economic efficiency, and is therefore socially counterproductive. The only thing we care about in the simple economy in this model is how many hours of labor it takes to get a unit of a good. There is only one primary input to “economize on” in the simple version of the model – labor. Moreover, as long as labor is less pleasurable than leisure, being able to get a unit of a good with less work is socially productive. Whereas any new technology that meant we had to work more hours to get a unit of a good would be socially counterproductive.

It may seem that we have the answers ready made in L(1) and L(2). Since L'(1) < L(1) it may appear that the new technique is obviously socially productive. But unfortunately L(1) is not the amount of labor

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it takes us to get a unit of good 1. L(1) is the number of hours of labor it takes to make a unit of good 1 once you already have a(11) units of good 1 and a(21) units of good 2. But since it takes some labor to get a(11) units of good 1 and a(21) units of good 2, it takes more labor than L(1) to produce a unit of good 1. We call L(1) the amount of labor it takes “directly” to get a unit of good 1 – once we have a(11) units of 1 and a(21) units of 2 for L(1) to work with. The amount of labor it took to get a(11) units of 1 and a(21) units of 2 is called the amount of labor needed “indirectly” to produce a unit of good 1. The total amount of labor it takes society to produce a unit of good 1 is the amount of labor necessary directly and indirectly. And while the new technique in question reduces direct labor needed to make a unit of good 1, i.e. is “labor-saving,” it unfortunately increases the amount of indirect labor it takes to make a unit of good 1, i.e. is “capital-using.”

Fortunately it is not terribly complicated to calculate the amount of labor, directly and indirectly necessary to produce a unit of good 1 and a unit of good 2 in our simple model. Let v(1) represent the total amount of labor needed directly and indirectly to make a unit of good 1, and v(2) represent the total amount of labor needed directly and indirectly to make a unit of good 2. Since v(i)a(ij) represents the amount of labor it takes to produce a(ij) units of good i we can write the following equations for the total amount of labor needed both directly and indirectly to make each good:

(3)v(1) = v(1)a(11) + v(2)a(21) + L(1)

(4)v(2) = v(1)a(12) + v(2)a(22) + L(2)

These are two equations in two unknowns, so v(1) and v(2) can be solved for as soon as we know the technology, or “recipe” for production in each industry. All we have to do is solve for the original values for the initial technologies – v(1) and v(2) – solve for the new values with the new technologies – v'(1) and v'(2) – and compare them. If v'(1) < v(1) and v'(2) < v(2) the new technology is socially productive. If v'(1) > v(1) and v'(2) > v(2) the new technology is socially counterproductive.7

7.It is obvious why the new technology for industry 1 will change v(1) since it changes L(1) and a(21). But even though there is no change in technology in industry 2, since good 1 is an input used to produce good 2 and since v(1) will change, v(2) will also change. This also resolves another potential concern.

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For the old technologies we write:

v(1) = 0.3v(1) + 0.2v(2) + 0.1

v(2) = 0.2v(1) + 0.4v(2) + 0.2

Which can be solved to give: v(1) = 0.2632 and v(2) = 0.4211 For the new technologies we write:

v'(1) = 0.3v'(1) + 0.3v'(2) + 0.05 v'(2) = 0.2v'(1) + 0.4v'(2) + 0.2

Which can be solved to give: v'(1) = 0.2500 and v'(2) = 0.4167 – revealing that the new technology is truly more efficient, or socially productive, because it lowers the amount we have to work to get a unit of either good to consume. Why is it capitalists will serve the social interest by adopting the new, more efficient technology when w = 2.375 and r = 0%, but obstruct the social interest by rejecting this technology that would make the economy more efficient when w = 1.811 and r = 20%?

To solve this puzzle we start with what we know: We know that the new technology made the economy more efficient. We know that the new technology was capital-using and labor-saving. And we know capitalists in industry 1 embraced it when the wage rate was 2.375 (and the rate of profit was zero), but rejected it when the wage rate was 1.811 (and the rate of profit was 20%). The reason for the capitalists’ seemingly contradictory behavior is clear: When the wage rate was higher the savings in labor costs because the new technology is labor-saving was greater – and great enough to outweigh the increase in non-labor costs because the new technology was capital-using. But when the wage rate was lower the savings in labor costs were less and no longer outweighed the increase in non-labor costs. Apparently the price signals [p(1), p(2), w, and r] in the economy in the first case led capitalists to make the socially productive choice to adopt the technology, whereas different

If the new technology lowers v(1) then it necessarily lowers v(2), whereas if it raises v(1) it necessarily raises v(2). We will never face the dilemma that a new technology in one industry will lower v in one industry but raise v in others – and thereby make it impossible for us to conclude whether or not the technology was socially productive or counterproductive.

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price signals in the second case led capitalists to make the socially counterproductive choice to reject the technology.

No matter how efficient, or socially productive a new capitalusing, labor-saving technology may be, it is clear that if the wage rate gets low enough (because the rate of profit gets high enough) the efficient technology will become cost-increasing, rather than cost-reducing, and capitalists will reject it. Similarly, no matter how inefficient, or socially counterproductive a new capital-saving, laborusing technology may be, if the wage rate gets low enough (because the rate of profit gets high enough) the inefficient technology will become cost-reducing, rather than cost-increasing, and capitalists will embrace it.8 In other words, Adam Smith’s second invisible hand works perfectly when the rate of profit is zero but cannot be relied on when the rate of profit is greater than zero. Moreover, as the rate of profit rises from zero (and consequently the wage rate falls), the likelihood that socially efficient capital-using, labor-saving technologies will be rejected, and the likelihood that socially counterproductive capital-saving, labor-using technologies will be adopted by profit maximizing capitalists increases.

Technical change and the rate of profit

In any case, clearly it is cost-reducing technological changes that a capitalist will adopt – whether they be capital-using and laborsaving or capital-saving and labor-saving, and whether they be socially productive or counterproductive. Can we conclude anything definitive about the effect of any cost-reducing technical change on the rate of profit, prices, and the wage rate in the economy? Marx hypothesized that capitalist development would entail capital-using, labor-saving changes more often than capitalsaving, labor-using changes, and that this would eventually produce a tendency for the rate of profit to fall in capitalist economies in the long run since Marx’s labor theory of value led him to believe that profits came only from exploiting “living labor,” not “dead labor.” For over a hundred years some Marxist political economists

8.For proof that in a simple, static Sraffa model if and only if the rate of profit is zero will there be a one-to-one correspondence between efficient, or socially productive, and cost-reducing technological changes see theorem 4.9 in John Roemer, Analytical Foundations of Marxian Economic Theory (Cambridge University Press, 1981).

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explored this area looking for explanations of crises in real world capitalist economies. But in 1961 a Japanese political economist, Nobuo Okishio, published a theorem proving that if the wage rate did not fall, no cost-reducing technical change could lower the rate of profit in the Sraffa model. Instead, cost-reducing changes, including capital-using, labor-saving changes, would raise the rate of profit, or leave it unchanged – contrary to the expectations of generations of Marxist theorists. We can see these results even in our simple numerical example.

Let the economy be in the “equilibrium” described in question two, i.e. the rate of profit is 10%, and consequently the wage rate is 2.086, and p(1) is 0.649 if p(2) = 1 – as we calculated. Under these conditions the capital-using, labor-saving technical change in industry 1 we have been analyzing is cost-reducing, and will be adopted. Non-labor costs increase by: (0.3–0.2)(1) = 0.1 as before, while labor costs decrease by (0.1–0.05)(2.086) = 0.104, which is greater, making the technology cost-reducing. The question is not if the capitalist in industry 1 who discovers the new technique will get a higher rate of profit than before right after she adopts it. Clearly she will since she was previously getting 10% and now will have lower costs than all her competitors, yet still receive the same price for her output as they and she did before, p(1) = 0.649. Nor is the question if all capitalists in industry 1 will receive a higher rate of profit if they copy the innovator as long as p(1) holds steady at 0.649. Clearly, as long as prices and the wage rate stay the same, all those who implement the change will have lower costs per unit than before and therefore a higher rate of profit than before. Instead, the question is what will happen to the rate of profit in the economy after capitalists from industry 2 move their investments to industry 1 because the profit rate is temporarily higher there, until the profit rates are once again the same in both industries? As long as r(1) > r(2) capitalists will move from industry 2 to industry 1, thereby decreasing the supply of good 2 and driving p(2) up, and increasing the supply of good 1 and driving p(1) down until r(1) = r(2) = r', the new, uniform rate of profit in the economy. We want to know if the new uniform rate of profit in the economy with the new equilibrium prices will be higher or lower than the old rate of profit, r – assuming the real wage rate stays the same. To answer this question we simply substitute in the new technology for industry 1, set the wage rate equal to the old wage rate, w = 2.086, set p(2) = 1, as always, and

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solve for the new equilibrium price of good 1, p'(1), and the new uniform rate of profit in the economy, r'.

(1 + r')[0.3p'(1) + 0.3(1)] + (0.05)(2.086) = p'(1) (1 + r')[0.2p'(1) + 0.4(1)] + (0.2)(2.086) = 1

Solving these two equations in two unknowns yields p'(1) = 0.644 and r' = 0.102. So when the economy reaches its new equilibrium after the introduction of the cost-reducing new technology in industry 1, the price of good 1 relative to the price of good 2 is slightly lower (0.644 < 0.649) as we would expect since the costreducing change took place in industry 1, and the uniform rate of profit in the economy is slightly higher (10.2% > 10%.) Since the change was capital-using and labor-saving this is contrary to Marx’s prediction but consistent with what Okishio proved would always be the case for any cost-reducing technical change as long as (1) the real wage stayed constant, and (2) good 1 entered into the production of both itself and good 2.

A note of caution

Micro economic models are notorious for implicitly assuming all macro economic problems away. This means conclusions drawn from micro economic models can be misleading when macro economic problems exist – which is the case with Sraffa models of wage, price, and profit determination as well. Just because the rate of profit cannot go up unless the wage rate goes down in the simple Sraffa model does not mean this is always true in the real world. If an economy is in a recession an increase in the wage rate, by increasing the demand for goods, often leads to an increase in the rate of profit in the short run. We study how increasing wages can increase the demand for goods and services, and thereby lead to increases in business production, sales, and profits in the short run in chapter 6. If an economy has a long run tendency to produce at less than full capacity, increasing the real wage may move the economy closer to full capacity utilization by shifting income from capitalists who save more to workers who save less and consume more. Because the redistribution of income from capitalists to workers increases demand for goods, and therefore capacity utilization, it can increase the rate of profit for capitalists even in the long run. We study the possibility of “wage-led growth” in a long run political economy macro model in chapter 9. The reason the Sraffa