# How increasing returns leads to non-neutrality of money

In 1980 an economist called Yew-Kwang Ng published a paper called Macroeconomics with Non-perfect Competition to little acclaim. The paper was cited a few times, and someone wrote a response that entirely missed the point of his argument. The main argument of his paper was largely ignored. Yew-Kwang Ng has written essentially the same paper many times since 1980, and every time it has been ignored.

I think that this paper is probably one of the most important papers in macroeconomics written in the last 50 years. In this post I will summarize the main ideas of the paper, and talk about how and why the conclusions he reaches have continued to be ignored by the mainstream. I believe the ideas within this paper should revolutionize the understanding of inflation within economics, and potentially provide support to the MMT view of inflation.

Yew-Kwang Ng begins by illustrating why money is neutral in a very simple neoclassical model of the economy. In this economy there is no capital so production is only a function of labour. The function is chosen such that costs and marginal costs are increasing in output. Workers are paid the marginal product of their labour. The real wage is given by an inverse labour supply curve that determines how much consumers are willing to work at each level of real wages. This model encapsulates the main points of standard neoclassical models in the simplest possible form.

The production function, inverse labour supply function, and the fact that workers are paid their marginal product uniquely determine the level of real output. There is no way for monetary factors to affect real variables in the model. Yew-Kwang Ng adds a very simple monetarist financial side to his model, where there is a certain amount of money in the economy. Because the real output in the economy is fixed the monetary side only determines the price level. Although Yew-Kwang adds a monetarist money supply function to his model different specifications of the financial side of the model could be added and the result would be the same. Whatever the details of the financial side of the model they can only determine the price level.

The next section of the paper modifies the above framework to include imperfect competition. The analysis proceeds in a manner similar to what we saw in this post when we looked at Dixit-Stiglitz competition: demand functions are assumed to be such that they limit firm size. Yew-Kwang is considering a much more general framework and defines his cost and labour cost functions in much more general terms. After some mathematical analysis he concludes that if a certain condition holds, money can be non-neutral, and increases in the money supply can actually increase output and employment without increasing prices.

Ng’s condition involves the inverse labour supply function $\psi(N)=w$ which gives the real wage $w$ as a function of the employment level $N$, and the production function $F(N)=Q$ which gives total production $Q$ as a function of the employment level. The condition is

$\frac{d \psi}{dn} \frac{N}{\psi}-\frac{d^2 F}{d^2N}\frac{N}{\frac{dF}{dN}}<=0$

When I first encountered this condition I was not sure how to intuitively interpret it. But knowing that non-increasing costs often lead to interesting results I decided to check under what conditions marginal costs are non-increasing in the mathematical framework Ng uses.

Since there is only one input the costs to produce an amount $Q=F(N)$ are equal to the labour required to produce that amount of product (which will be the inverse of the production function above), times the cost of that labour (the wage). Costs are therefore given by

$C(Q) =N(Q)w=F^{-1}(q)\psi (N)$,

$F^-1(q) =l(q)$ is the amount of workers it takes to make $Q$ units of output, otherwise known as the inverse production function (the -1 here denotes the inverse function).

Taking the derivative to find marginal cost and using the formula for the derivative of the inverse function we obtain

$\frac{dc(q)}{dq}=\psi \frac{dN}{dq}=\frac{\psi}{\frac{dF(N)}{dN}}$

or that marginal cost is equal to wage over the marginal product of labour. Note that in this stage we are assuming that each firm is small so it cannot affect wages. We are interested in knowing under which conditions marginal costs will be decreasing, ie the conditions under which the derivative of the above expression will be negative. At this state we are interested in how the marginal costs of a firm respond to changes in total output so we do not treat the wage as constant.

$\frac{d^2c(q)}{d^2q}=\frac{d\psi}{dN} \frac{dN}{dq} \frac{dN}{dq}-\frac{\psi}{\frac{dF}{dN}^2} \frac{d^2F}{d^2N} \frac{dN}{dq}$

Cancelling out some terms and rearranging

$=\frac{d\psi}{dN}\frac{1}{\psi}-\frac{d^2F}{d^2N}\frac{1}{\frac{dF}{dN}}$

Since we are interested in when this is less than one we can multiply through by $N$ since the number of people employed is always positive. Thus, costs are non-increasing when

$=\frac{d\psi}{dN}\frac{N}{\psi}-\frac{d^2F}{d^2N}\frac{N}{\frac{dF}{dN}}<=0$

This condition is exactly equal to the condition Ng derives in his paper under which money is non-neutral! In other words, Ng shows that when marginal costs are constant or decreasing money is non-neutral. The connections with what we discussed earlier are immediately obvious. A firm with constant or decreasing costs is not limited by supply: it wants to produce as much as possible at the current price. It turns out that if firms are in that situation money can have an effect on the real economy.

Yew-Kwang Ng spends most of his time discussing the case where costs are constant. The reason for this is that without constant costs the economy does not reach an equilibrium so typical static equilibrium analysis does not apply well. What occurs is that an expansion in demand causes prices to decrease, which causes a further expansion in demand and so on. While such a scenario is certainly interesting, it relies on the idea that firms lower prices in response to demand, and do so immediately.

Is such an assumption realistic? It depends on our model of competition, which, as I discussed earlier, is less straightforward with decreasing costs than when costs are increasing. It certainly seems reasonable to expect that competitive pressures allow a firm to make increased profits in response to an increase in demand, and not be forced to immediately reduce costs. If competitive pressures constraining markups work only over the longer term, or if prices are relatively inflexible then firms might keep prices constant and instead make greater profits. In such a case a lot of the conclusions of the constant cost case will apply to the case of decreasing costs.

The case of decreasing costs could potentially provide the explanation for certain historical events, for example, during the antebellum period in the US there was, according to some economic historians, a deflationary expansion. That time was a period of greatly increasing market size due to improved transportation, which would likely mean that increasing returns were significant.

The idea that price stability halts expansionary or contractionary cycles also is in line with Keynes view on the role of price stability in macroeconomics. While in modern macro models price inflexibility is what causes the economy to not reach equilibrium, Keynes stated that with price inflexibility the economy would be even more unstable, and depressions would likely be even worse. The contractionary/expansionary nature of the economy in the case where costs are decreasing seems to support his point.

Of course the economy cannot expand without limit. Eventually unproduced inputs of production will become scarce. This can be modelled by assuming that the prices of inputs increase rapidly beyond a certain point. We can thus divide the economy into two regimes: a Keynesian regime, where increases in efficiency due to increasing returns are large enough to make up for the increasing costs of inputs, and a neoclassical regime, where money is neutral, and the economy is at capacity. This description of the economy matches what MMT proponents say: the economy is often far from capacity, where inflation is not a concern, and price controls are likely to be effective. But if demand increases enough the economy will reach a point of full capacity, where further increases in demand will cause inflation.

By precisely deriving the mathematical conditions that characterize the capacity of the economy the above understanding also could enable, at least in theory, the measurement of the capacity of the economy. This would enable better policy solutions, since supply side policies could be implemented once capacity is reached and Keynesian stimulus could be enacted before that.

There might of course still be some inflation before the economy reaches capacity if firms’ markups are increasing. However inflation in that regime is different from inflation once the economy reaches capacity, and has very different solutions. Price controls, and policies to increase competition would likely be effective at reducing inflation before the economy reaches capacity. After the economy reaches capacity such policies would likely lead to shortages.

Despite the interesting implications of the paper above, basically all macro papers written since then have assumed increasing costs. Typically this is done by assuming constant returns to scale and preferences which make labour and capital cost more as their quantity increases. Why have economists ignored the case of decreasing costs despite the interesting implications? I think a large part of the reason is the extent that the microeconomics which economists spend so much time studying, and which forms the basis of their macroeconomic theories, is based on the assumption of convexity. Economists barely see examples of increasing returns in their education, and learn a set of tools and techniques that do not apply in their presence. Additionally, economists develop intuitions based on years of ignoring increasing returns to scale, and these intuitions tend to guide their research. Finally I believe the tendency towards increased mathematical rigour in economics journals makes it difficult to publish ideas that are a little more difficult to model in a formal mathematical way. New research approaches that necessarily have less of a history of formal modelling are thus likely to have difficulty gaining traction, especially if those ideas are complicated enough that rigorous modelling is naturally more difficult.

There are a few macroeconomists who have incorporated increasing returns into their macroeconomic theory. Roger Farmer has written many papers on indeterminacy under increasing returns to scale. Another interesting paper was written by three economists and found that an increase in government spending would boost long run output. While I disagree with some of the modelling choices made by each of these papers these papers show the types of interesting and Keynesian results that can be found by incorporating increasing returns into macroeconomic models.

These results would seem particularly relevant to the current macroeconomic climate, where central banks have had difficulty increasing inflation. Even mainstream economists such as Narayana Kocherlakota have admitted that the MMT view of inflation seems to better fit the available macroeconomic data. If economists want to explain the current lack of inflation they would do well to incorporate nonincreasing costs into their models. Increasing returns is also likely to be particularly relevant as the technology and software industries make up a larger and larger portion of the economy. Those industries likely have extremely high levels of returns to scale.

I started this blog largely because I have had difficulty finding any economists interested in exploring the macroeconomic implications of increasing returns. Post-Keynesians are generally uninterested in microfounding their models with this level of rigour, and when I mention these ideas to mainstream economists I tend to get blank stares, or some version of the competition argument I discussed in a previous post. If any macroeconomists are interested in these ideas I would love the opportunities to collaborate with someone in exploring them further.