Karl Marx’s invisible hand

I was recently invited to give a talk by the Communist Corresponding Society, in Oxford, on the labour theory of value. Much academic discussion of the labour theory of value is unfortunately cast entirely in terms of equilibrium models. Yet Marx’s theory of value concerned with identifying causal laws, and therefore is irreducibly dynamic. So I decided to talk about the dynamics of the labour theory of value, especially the relationship between out-of-equilibrium market adjustment and the allocation of the total labour of society. I also wanted to emphasise the intimate relation between Marx’s economic theories and the theoretical contributions of Adam Smith and David Ricardo, since this connection isn’t fully appreciated by all Marxists.

Here is the transcript of the talk.

Introduction: prices are, and are not, related to labour time

In 1868 Marx wrote a short letter to his friend Ludwig Kugelmann which contains some of the most significant paragraphs written in the history of economics. Marx begins his letter by stating:

Every child knows a nation which ceased to work, I will not say for a year, but even for a few weeks, would perish. Every child knows, too, that the masses of products corresponding to the different needs required different and quantitatively determined masses of the total labour of society. That this necessity of the distribution of social labour in definite proportions cannot possibly be done away with by a particular form of social production but can only change the mode of its appearance , is self-evident. No natural laws can be done away with.

So, in case you missed it, Marx states that — for any economic system to reproduce itself through time — it must have a mechanism that allocates the total labour of society to different useful tasks. He then continues:

What can change in historically different circumstances is only the form in which these laws assert themselves. And the form in which this proportional distribution of labour asserts itself, in the state of society where the interconnection of social labor is manifested in the private exchange of the individual products of labour, is precisely the exchange value of these products.

Marx therefore states that prices are the mechanism through which labour is allocated in a market-based economy.

More precisely, Marx thinks that an economic “law of value” is ultimately responsible for allocating labour. The law of value governs the movement of prices. If the labour-time required for the production of a commodity reduces, then its price falls; and if the labour-time increases, then its price will rise. So prices vary in lockstep with labour time.

But Marx then states that:

The vulgar economist has not the faintest idea that the actual everyday exchange relations can not be directly identical with the magnitudes of value.

And by value Marx means the labour time required to produce a commodity.

So Marx’s letter raises a puzzle. First, he claims that prices vary with labour time. On the other hand, he boldly asserts that prices are always different from labour time.

So in this talk, I want to examine what the “law of value” actually is, and how it coordinates the division of labour in market economies. And I also want to dig deeper into the reasons why Marx proposed a seemingly paradoxical labour theory of value where prices always differ from labour time.

The invisible hand in classical economics

The coordination of millions of independent production activities in a large-scale market economy is not perfect nor equitable but nonetheless we should be far more surprised by coordination than by disorder.

Marx is most associated with explaining the crises of capitalism and its eventual collapse. But he was just as much concerned in explaining how capitalism persists and reproduces itself over time.

In this respect, he was heavily influenced by the British economists, Adam Smith and David Ricardo. They developed a theoretical framework in which this surprising fact could be understood.

Adam Smith published his “Inquiry into the Nature and Causes of the Wealth of Nations” about 100 years before the publication of Capital. In that book, Smith introduces his famous metaphor of the invisible hand.

Smith argues that the uncoordinated and selfish actions of individuals, who pursue private profit, has the unintended side-effect of increasing the wealth of all. In other words, capitalist competition has emergent properties that are materially progressive for the whole of society.

Marx knew Smith’s argument, and borrows the metaphor of the invisible hand in his own writings, although he’s quick to emphasise the destabilising effects of market relations.

So how is the invisible hand supposed to work? According to Smith and Ricardo, there are two basic processes that coordinate a capitalist economy.

First, whenever the quantity brought to market is larger than the demand then the market price of that commodity will fall. And conversely, if too little is brought to market, the price will rise.

Second, the quantities brought to market increase in sectors where profits are above average and decreases in those sectors with below average profits.

So owners of firms adjust their prices according to supply and demand, and reallocate their capital away from loss-making and towards profit-making activities. This scramble for profit, according to the classical authors, has the unintended consequence of stabilising the economic system.

Smith used the term “gravitation” to describe the process of stabilisation, and to explicitly connect with Newton’s theory of gravity.

During gravitation the prices of commodities follow a special trajectory. According to Smith, market prices gravitate towards, converge to, or oscillate around, their natural prices.

This distinction between a market price and its natural price is very important in classical economics.

Market prices are subject to fluctuations due to supply and demand, and other accidental causes, such as what Smith describes as the “higgling and haggling” of the marketplace.

The natural price of a commodity, in contrast, is the relatively stable price that manifests when supply equals demand.

This vision of the homeostatic properties of capitalist competition is shared by Smith, Ricardo and Marx. For example, Marx, in Volume 3, states:

The competition between capitalists — which is itself this movement toward equilibrium — consists here of their gradually withdrawing capital from spheres in which profit [is] below average, and gradually investing capital into spheres in which profit is above average.

However, the classical authors didn’t develop formal models of capitalist competition, and so their accounts remain sketched in natural language. Marx went furthest by embarking on a close and extensive study of the calculus. He believed that differential equations held the promise of “determining the main laws of capitalist crisis”.

Yet this work didn’t impact Capital, which contains only small-scale numerical examples of simultaneous equations in the reproduction schemes of Volume 2.

The marginal revolution in the 19th Century shifted economics away from analysing changes in market prices over time. Focus instead turned to the logical determination of prices by the conjectural intersection of supply and demand schedules. So the theory of classical gravitation was essentially forgotten.

Even to this day, a typical neoclassical model assumes instantaneous adjustment of both prices and quantities to their equilibrium or balanced values. In other words, the market is simply assumed to be a perfect coordinator of economic activity.

A formal model

OK, so we’ve noted that economic systems must have a mechanism for distributing labour to different activities. And in market economies this is supposedly achieved by an invisible hand, which involves the uncoordinated adjustment of prices charged and quantities produced.

But is this true? Is a market economy really self-stabilising in this way? Or is it merely a just-so story?

We’re going to decide this question by constructing a formal model of classical competition.

Formal models are particularly important when trying to understand complex systems that have nonlinear relationships between their parts.

I’ll spend a few minutes describing the overall features of the model. Then we’ll take a look at its dynamic behaviour.

In the handout I’ve included equations only for completeness. You don’t need to look at them. But I would like you to get a feel for the different parts of the model and how they interact with each other.

PDF of handout

Worker households

So let’s begin with worker households, which is described in section 1.1 of the handout.

Workers consume a real wage, which consists of a bundle of different commodities. How many bundles they purchase depends on how much money they have to spend, and the price of the bundle. And the real wage consumed by workers forms part of the total demand in the economy.

Workers hold a stock of money. The change in that stock depends on the difference between what they spend and what they earn.

Their total wage earnings depends on the demand for labour, from the different sectors of production, and the current wage rate.

So what causes the wage to change? Marx stated, quite conventionally, that the wage rate, within certain bounds, is determined like the price of any other commodity: it depends on the supply and demand for labour.

So as more workers are drawn into employment, the labour market tightens, and wages rise. And, if overall output falls, then the demand for labour falls, and wages also fall.

Capitalist households

Let’s now turn to capitalist households, described in section 1.2 of the handout. They consume in the same way as workers, but they have a very different relationship to production.

Following Marx we consider two main types of capitalist.

Finance capitalists advance money-capital to production and earn interest from these loans. And industrial capitalists, as the owners of firms, and therefore liable for industrial profits (or losses).

For simplicity, I assume that firms are not self-financing, and instead borrow to finance their purchase of means of production and labour.

So the demand for money-capital therefore depends on how much firms want to produce, current market prices and the wage rate.

The level of interest income, received by finance capitalists, fluctuates with the demand for money-capital and the interest rate. For example, a firm that wants to increase production will borrow additional money-capital to finance the cost of purchasing more means of production and labour.

So that’s finance capitalists. Industrial capitalists, as owners of firms, receive profits from profit-making firms, and have to cover the losses of any loss-making firms.

A firm’s profit is the difference between its revenue and costs.

The firm’s costs consist of means of production and labour, plus interest payments on their loans.

The firm’s revenue depends on the selling price of their goods and the quantity they sell. The quantity they sell depends on demand from firms in other sectors of production, and consumption demand from worker and capitalists households.

Capitalist households consist of both financiers and industrialists. So the total stock of money held by capitalists is augmented by an inflow of profit — consisting of interest income and industrial profit — and reduced by an outflow of consumption spending and industrial losses.

The interest rate

How is the interest rate set? We want to keep things simple, and we follow Marx, who adopted a loanable funds theory.

Let’s assume that both workers and capitalists money stocks are deposited in financial institutions, and then finance capitalists loan out these savings. The total stock of loanable funds is constant since money is conserved in exchange. So, in this simple model, the interest rate is a fixed parameter.


Let’s now turn to the firms in each sector of production, described in section 1.4 of the handout. This is where we specify the uncoordinated adjustment processes that are supposed to give rise to the invisible hand.

In this model, firms don’t hold their own stocks of money. Money just circulates in and out of them.

But they do hold stocks of inventories. For example, a corn-producing firm will hold a stock of corn available for sale in the market.

Inventories fluctuate depending on the difference between the supply and demand for goods. If a firm is selling more than it produces then its stock of commodities will decrease.

Firms raise their prices when inventories shrink since buyers tend to outbid each other for the scarce product. And they lower prices when their inventories grow since firms underbid each other to sell to scarce buyers.

So that’s how firms set their prices. How do they decide on how much to produce?

Industrial capitalists, as a whole, own a portfolio of firms across all sectors of production. They maximise their profits by differentially increasing or reducing production in different sectors based on comparing their profit rates.

So profitable firms borrow more in order to increase their production with the expectation of earning even more profit. Similarly, loss-making firms reduce their borrowing and decrease their scale of production in order to reduce their losses.

So the level of production in different sectors is controlled by a scramble for profit.

The relation of the model to economic reality

All these relationships, which I’ve just described, fit together and form a complete, closed system of nonlinear differential equations.

In general, it’s really difficult to understand what kind of dynamics emerge from these kinds of relationships. And normally we can’t solve these equation systems algebraically so we need numerical methods. We then explore the behaviour of the model by choosing different starting parameters and then watching how the system evolves over time.

Before we take a look at an example, I need to say a few words about the relation between this model and empirical reality, in order to avoid misunderstandings.

A real economy is a complex system composed of lots of different mechanisms and open to an external environment. So the empirical data is shaped by lots of different things.

In economics we lack the causal powers to perform experiments. We can’t intervene and hold bits of an economy constant. So we must take a more indirect route, and imagine doing so, by adopting counterfactual assumptions that perform a theoretical ‘experiment’. And this is what the model does. For example, it assumes that the techniques of production are constant throughout.

The classical authors knew that many factors, not least ceaseless technical change, continually alters the conditions that define natural prices. But their theory of gravitation assumes that these factors are either absent or constant.

So it’s important to understand that this model will not directly correspond to what we observe in empirical reality. The purpose of the model is to capture a single underlying mechanism that affects, but doesn’t completely determine, what we empirically observe.

A dynamical system

OK, now we’ve got that out of the way, let’s see what happens!

Figure 1 on the last page of the handout, shows 4 graphs that plot the behaviour of an example economy.

This economy is really small. It only has 3 sectors, which produce corn, sugar and iron. Each sector needs certain inputs from other sectors. So they are all interconnected.

Figure 1(a) shows market prices over time. And here we can see that the price of corn and sugar initially rises, because they are in under-supply, but then falls because capitalists decide to increase their production. The price of iron falls throughout, since it’s in relative oversupply.

However, at around time t=10, the prices of all commodities stabilise. In fact, these stable prices are the natural prices.

So we’re beginning to see precisely what the classical authors conjectured: market prices vary with supply and demand, but they gravitate to stable, natural prices.

Figure 1(b) shows the scale of production in each sector. All sectors start out profit-making and so increase their output. We see some overshoot in the sugar sector, where there’s temporary oversupply.

But, here again, at around time t=10, output levels stabilise, at which point the supply of goods equals the overall demand for them.

Figure 1(c) shows the profits in each sector. As we note, all sectors start profitable. But profits begin to fall as supply adjusts to demand. You can notice that, for a brief period, the iron and sugar sectors are loss-making, due to temporary overproduction.

But the striking fact is that industrial profits (and losses) in all sectors eventually converge to zero. The scramble for profit has the unintended side-effect of reducing imbalances between supply and demand, and therefore the opportunity to earn scarcity rents. So in a perfectly competitive system, like this one, the pursuit of profit has the paradoxical consequence of causing profits to fall.

Figure 1(d) shows the wage rate. As the economy grows more of the available labour force is drawn into production, and the labour market tightens. Again, the wage eventually stabilises to its natural level.

We could look at other interesting properties, such as employment in different sectors, the level of outstanding loans and interest income, and the distribution of income between workers and capitalists. But I’ll stop here.

Once you start exploring the parameter-space of this model it rapidly becomes clear that capitalist competition is indeed stabilising. The scramble for profit, as the classical authors suggested, does function as an invisible hand.

We can even simulate external shocks to the model, such as forcing the oversupply of a particular commodity, or injecting a change in the techniques of production, so that a particular commodity takes more or less time to produce.

After the shock, we can watch the economy self-correct, and adapt to the new conditions, by gravitating toward a new steady-state where the employed labour force is again perfectly distributed to produce the final demand for consumption goods.

And perhaps we shouldn’t be too surprised by this result because it could not be otherwise, for if capitalism lacked any mechanism to distribute labour then, it would, as Marx suggested, “perish” in a matter of “weeks”.

On the one hand it’s a small miracle that monetary exchange can coordinate millions of independent economic activities in this way. Yet at the same time, it’s a necessary condition for its continued existence as a social institution.

And before we wander off into the panglossian realms of market fundamentalism we should note that converging to a state where supply equals demand tells us nothing about whether the distribution of goods meets actual social needs, or whether the wage system is just, or even an efficient way to organise an economy, or whether allocating a good portion of the working day to producing luxuries for an idle class of exploiters is a sensible thing to do.

And we should also note that the scramble for profit has no inherent tendency to stabilise at full employment. In fact, this model can stabilise to any level of employment, which is a typical Keynesian result. So, without intervention, the total labour of society can be enormously misallocated by markets.

The law of value

But the classical authors were basically right. A capitalist system has a homeostatic kernel that coordinates economic activities toward useful ends.

I want to draw to a close by returning to the law of value, and the questions raised by Marx’s letter to Kugelmann.

Marx defines the labour value of a commodity as the new labour added to the labour value of the means of production, which gets transferred to the output. We can translate this definition into a formal equation, and then solve to calculate the labour values in this model. Once we’ve done that, we can then compare labour values and market prices over time.

Figure 2, at the bottom of the last page of the handout, plots market prices divided by their labour values during gravitation (for the same 3-sector economy).

In Figure 2(a), on the left hand side, I’ve set the interest rate to zero (which eliminates capitalist profit in equilibrium). In this case, the market prices of all commodities divided by the respective labour values converge to the equilibrium wage rate. This means that natural prices are proportional to labour values. So they directly express labour time.

This situation reproduces Adam Smith’s famous thought experiment of the exchange of beaver and deer between hunters in pre-civilized times, and before the appearance of capitalist profits. Smith argued that, in these conditions, a simple or pure labour theory of value must hold.

Figure 2(b), on the right hand side, plots the general case, with a positive interest rate. Here, market prices converge towards labour values, and get quite close to them, but systematically deviate from them, even at the natural price equilibrium.

This situation reproduces David Ricardo’s 93% labour theory of value. Ricardo claimed that labour values account for most of the structure of natural prices, with a bit left over that’s unrelated to labour time, and which is the capitalists “reward for waiting”.

Now Marx fully accepted that natural prices diverge from labour values. But he argued this doesn’t invalidate the law of value because aggregates of prices and labour values still have a proportionate relationship. For example, Marx claimed that total monetary profit is proportional to the total surplus labour, and so on. Individual commodity prices are transformed labour values, but in the aggregate everything works out OK. And this transformation theory was immediately controversial — and remains so to this day.

But now we can understand why Marx held a labour theory of value, and yet claimed that prices are never identical with labour values.

The first reason is that, in empirical reality, economies never reach their natural price equilibrium. We only observe market prices, which are determined by supply and demand, and other accidental causes.

The second reason is that, even if an economy did reach its natural price equilibrium, we would still fail to observe a simple relationship between prices and labour values. Because capitalist profit distorts prices away from labour values.

Nonetheless, the law of value is always operating behind the scenes continually pushing market prices towards their natural prices, and allocating, and reallocating, labour to different sectors of production. And this equilibrium state, which attracts market prices, has a determinate relationship to labour values.

So a bird may appear to defy the law of gravity but is nonetheless always subject to it. In the same way, prices may appear to defy the law of value, but are also always subject to it.

To quote Marx:

in the midst of all the accidental and ever fluctuating exchange relations between the products, the labour time socially necessary for their production forcibly asserts itself like an over-riding law of Nature. The law of gravity thus asserts itself when a house falls about our ears. The determination of the magnitude of value by labour time is therefore a secret, hidden under the apparent fluctuations in the relative values of commodities.

So Marx’s invisible hand is the law of value: it explains how a market economy coordinates the division of labour, and therefore why the prices of commodities bear a lawful relationship to the labour time required to produce them.

So that’s it. There’s many threads I haven’t followed up. For example, what mechanisms destabilise capitalism? How can a labour theory of value handle non-reproducibles, such as land, or unique works of art? Can we find empirical evidence for the law of value? And so on.


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