# The Intellectual problems with Adam Smith’s invisible hand and the question of omnipotency of free markets

’Equilibrium prevails if all plans and expectations of all economic subjects are fulfilled so that, at given data, no economic subject feels inclined to revise his plans or expectations.’-Erik Lindahl

Let us talk about Walrasian economic equilibrium and about the dynamic process that governs the evolution of market prices, possibly towards an equilibrium. As the quote above by Erik Lindahl (1891-1960), a Swedish economist, suggests, economic equilibrium is a rather general and abstract concept, which is a state where nobody is willing to move or alter her policy.

The neoclassical theory of economic equilibrium is arguably the intellectual cornerstone of modern economic theory. The modern theory of economic equilibrium, including the Welfare theorems, is the rigorous equivalent of the ’invisible hand’ of Adam Smith. Equilibrium concepts are common in any dynamical systems context, be it a Walrasian equilibrium in neoclassical economic theory, or a Nash equilibrium in game theory.  Analytically defined economic equilibrium is originally a concept that originated from the 19th century physical sciences.

Although the father of modern economics is indeed usually cited to be Adam Smith, the first semi-rigorous economic theory on equilibrium was put forward by Leon Walras in the late 19th century. The next major breakthrough was in 1954, when the existence of competitive equilibrium was proven by Arrow and Debreu. In spite of the fame of Debreu and Arrow, actually the mathematical theory of general equilibrium was put forward to a great extent by Abraham Wald in 1936. Moreover, Johnny von Neumann, 1945, contributed to the theory of equilibrium greatly as well.

The existence theorem for a competitive equilibrium is a substantial achievement as such, but nevertheless it does not tell much about the workings of a true market economy. We know very little about the adjustment process of prices as such. Especially little we do know about the dynamics of it. The canonical theory since 1940’s assumes that there is a ”trial and error” or ’tatonnement’ process, where a Walrasian auctioneer adjusts the prices so that the excess demand is driven to zero according to the following dynamics

$\frac{dp}{dt}=\lambda Z(p)$

where p is the price vector, lambda is a constant and Z is the excess demand function for the economy.

Even though the equation above seems rather innocent, Scarf in 1960, among others has shown that it is not generally globally stable. A globally stable equilibrium means (assuming it exists) in this context that if you start with some initial set of prices and excess demands, the dynamics will lead to an equilibrium always.

The lack of a stable equilibrium is a major problem, as it implies that in general, there is no market clearing. It is actually rather peculiar, that economic theory contains such an essential pathology, given how lightly people usually assume that demand and supply will balance each other. Therefore, we would need to advance on the dynamic adjustment process, because actually we know rather little about theoretical economies.

According to the Fields medalist and mathematician Stephen Smale, the problem of lack of knowledge with general equilibrium theory is really severe, and he has included it on his famous list of eighteen unsolved problems in mathematics.

In the 1970’s Stephen Smale published extensively on the problems around general equilibrium theory. In particular, he held the view that the main unsolved problem in mathematical economics was the lack of understanding of dynamics of general equilibrium.

’I feel equilibrium theory is far from satisfactory. For one thing the theory has not

successfully confronted the question, ”How is equilibrium reached?” Dynamic

considerations would seem necessary to resolve this problem.’ -Stephen Smale

There has been a consistent line of research in non-tatonnement through last decades, although thin, regarding Walrasian exchange and adjustment of prices. Even though the price adjustment process is of fundamental importance, the amount of research published on this particular issue is not however so large. It is worrying, as price adjustment towards equilibrium is a key issue, not least because the market clearing does not occur always.

So, my leftist friends, study more mathematics, and you can challenge Mr. Smith!

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# Is QE equivalent to “printing money” ?

Since the launch of quantitative easing (QE) by the major central banks, numerous economic commentators and especially journalists are equating QE and “money printing”. This is of course interesting, as money is generally very precious, yet not completely understood by the public and this is troublesome, given the paramount role of ‘money’ in society.

I will try to explain here why it is misleading to equate QE with money printing, and also I will try to explain how QE induces money creation.

For these purposes we need to build a very simple model of our monetary economy. Let us first suppose that we have a Central Bank (CB), a (commercial) bank and an institutional investor (II) be it a pension fund. Moreover, we implicitly assume that there are households, firms etc. who need capital markets for financing.

First of all, let us define a couple of important concepts (simplification)

-Central bank money =commercial banks’ deposits at the central bank + notes and coins in circulation. These are liabilities of the central bank.

-Public Money = Deposits of households and firms and government at the commercial banks. These are liabilities of the commercial banks. This is nowadays what we call casually ‘money’, given that the use of cash is diminishing.

We assume that initially the commercial bank (all banks in the economy) in the economy holds 100 euros worth of bonds issued by the government and firms. The pension fund holds lets say 300 euros worth of same bonds. So the total outstanding amount of bonds in the economy is 100+300=400.

The initial balance sheet of the commercial bank is assumed to be consisting of some assets, say loans worth of 900 euros and the loans and bonds held are financed with 950 euros of deposits and 50 euros of equity. The balance sheet of the commercial bank looks like this:

Assets                                 Liabilities

Loans 900                          Equity 50

Bonds 100                          Deposits 950

The central bank then proceeds with QE. It buys the whole portfolio of bonds from the commercial bank crediting the bank’s account at the central bank with 100 euros. The balance sheet of the bank looks like this:

Assets                                Liabilities

Loans 900                         Equity 50

CB deposits 100              Deposits 950

Now the commercial bank is not happy, as the account at the central bank is costly (-0,4% p.a. in the Euro area), so it wants to compensate this negative carry by buying a lot of government bonds from the pension fund. This is the so called “portfolio (rebalancing) effect”. Do note that the sovereign bonds are “risk free” in the sense that they do not require any regulatory capital. The pension fund holds a checkings account at the commercial bank.

The bank now credits the pension fund’s account with 200 euros and gets in return 200 euros worth of sovereign bonds. Now the bank’s balance sheet looks like this

Assets                                  Liabilities

CB deposits 100                Equity 50

Bonds 200                         Deposits 1150

Loans 900

The same economic effect for the bank can be achieved by extending new credit to households and firms. It improves the net interest income and therefore the bank has an incentive to expand its balance sheet, if there is sufficient capital available. This effect is especially strong now, because of negative interest rates.

So we notice that balance sheet of the commercial bank did not increase directly due to QE and hence that the amount of public money did not increase directly due to QE. However, when the banking system is induced to buy new bonds or to issue new loans, due to rebalancing effects the amount of public money is increased as a second-order phenomenon.

In other words, QE is not “money printing” as such, because the first-order effect just increases the amount of central bank deposit money, which is not used by the public.

On the other hand, if the central bank wants to buy all/large amounts of bonds held by the bank and the pension fund, the bank needs to buy these bonds first from the pension fund in order to sell them to the CB, which creates new public money (the bank credits the pension fund’s account). On the other hand, this money is at the hands of the pension fund and as such does not create directly any additional purchasing power as the fund needs to have certain amount of assets to cover its pension-related liabilities.

Is the debt burden lower for central goverments due to QE ?

No, not really. Even though most central banks are owned by the central government, the debt is still there and if the CB decides to taper, i.e. to start divesting and let its balance sheet shrink, the maturing government debt must be refinanced by selling new bonds to private investors. Therefore QE is not about cancelling government debt. Some of coupon interest ends up cancelled because the CB profits are distributed to the central government.

Problems with QE

Even though QE might work to fight deflation, it is not particularly efficient at that. Moreover, QE creates huge amounts of central bank money, which usually causes a euphoria in the asset markets as banks want to buy bonds and stocks with their excess reserves. Over-valued asset markets might cause problems in terms of financial stability. Moreover, as QE inflates asset prices, the distribution of wealth tends to favour the rich as the rich own more assets than the poor. This is not trivial in terms of social justice. Finally, at least in the Eurozone, QE has increased the level of TARGET2 imbalances and the general level of credit risk for the central banks. This might be trouble in terms of budgetary sovereignty of national parliaments. Moreover, the solvency of various entities might be zombie-solvency, because debt sustainability might be very different with a “normal” yield curve. This is especially so as QE includes corporate bonds as well.

Therefore fiscal policies (lowering taxes/increasing public investment) would be in general more efficient and socially justifiable solutions fighting deflation. Of course in the Eurozone this is very difficult.

Conclusion:

QE in its purest form (no buying of bonds by the banks) is not equal to money printing as it only increases the amount of central bank money deposits. However, in practice because of of the portfolio rebalancing and because of the scale of QE , banks do create new money to fund their bond-buying from institutional investors.

# The Maximum Entropy principle in modelling and estimating probabilities of default for banks

I am now finally proceeding with my PhD dissertation in systems analysis and operations research. What I found originally interesting, was estimating probabilities of default for a group of banks using logistic regression, see my presentation at University of Cambridge Judge Business School, Lindgren [2016].

When we consider a statistical model for probability of default (PD) of a business entity or of a bank, we need to argue why we assume a specific statistical model for the data generating process. After we have identified a statistical model, estimation and inference is usually rather straightforward, although it might be computationally burdensome. In this article, I explain why logistic regression specification is a very natural one in terms of maximum entropy based statistical inference. The additional benefit is that we can use the machinery of statistical mechanics, as we will interpret the model through the Gibbs measure. This framework allows us to find expressions for various potentially useful concepts like enthalpy and free energy, usually based on the information codified in the partition function Z. Logistic regression is a very simple model for neural networks and this could be ultimately very useful paradigm in finance as well. Markets could be seen as a huge, adapting non-linear neural processing totality.

The principle of maximum entropy

I will follow in the steps of Jaynes [1957], who argued that the a priori distribution should be the one that maximizes entropy given some constraints. Entropy is a concept that originated from 19th century thermal physics  and statistical mechanics as a measure of disorder, but in a larger perspective it can be considered as an expectation related to surprisal, in terms of information theory. We usually consider information be related to the logarithm of probability because of its certain algebraic properties. For a thorough discussion, see for example the famous work by Claude Shannon.

In a discrete probability space we define entropy as

$S(p_i)=-\sum_{i=1}^{n}p_i \log{p_i}$

Where we define ‘surprisal’ to be $\log{\frac{1}{p_i}}$. Note that if an event is certain, surprisal is zero, and if probability is close to zero, surprisal grows very fast towards infinity. The intuition for entropy is therefore the average surprisal, when sampling. The idea now is to find a priori distribution, if we now nothing about it except some expectation based on the distribution. If we are prudent, we should assume the distribution is the one that maximizes entropy.

Consider now an expectation, call it energy

$\langle E \rangle = \sum_{i=1}^{n}E_ip_i$

If we now maximize entropy given a fixed constraint of average energy, we have the following Lagrangian

$L(p_i)=-\sum_{i=1}^{n}p_i \log{p_i}-\beta \left(\sum_{i=1}^{n}E_ip_i-\langle E \rangle \right)-a\left(\sum_{i=1}^{n}p_i-1\right)$

The last constraint is there to ensure that the probability measure is normalized to unity.

The maximization problem is straightforward and the entropy maximizing distribution is the Boltzmann distribution, or the Gibbs distribution

$p_i=\frac{e^{-\beta E_i}}{Z(\beta)}$

where $Z(\beta)$ is the partition function that ensures the distribution is normalised to 1.

Logistic model

We now consider a binary choice model for the problem of default. At any instant, the entity is in default or not. We assign these probabilites to be $p_i$ and $1-p_i$ respectively. Now let us assign energies for such two states of the world. We have energies $E_1$ and $E_2$. The partition function is therefore

$Z(\beta)=e^{-\beta E_1}+e^{-\beta E_2}$

If we now substitute this in to the Gibbs distribution, we will have

$p_1=\frac{e^{-\beta E_1}}{e^{-\beta E_1}+e^{-\beta E_2}}$

This can be simplified to be

$p_1=\frac{1}{1+e^{-\beta(E_2-E_1)}}$

This is the logistic curve, whose argument is the difference in energy. There is the Lagrangian coefficient beta, which in physics is the inverse temperature, here it can be used to balance the units to give a unitless probability.

Let us now identify the energies. Given that the probability of default is dependent on the difference, we should somehow relate these two concepts to risk and capital. So we could for example choose that $E_2$ represents total risk and $E_1$ represents capital-like variable. When risk is large compared to capital, probability of default is close to unity.

So in other words we might choose

$E_2=\vec{w}\cdot \vec{x}$ and $E_1=\theta$

Where risk is a weighted sum of incoming sources of risk and theta is a measure of capital. Given these specifications, we have the model

$p_1=\frac{1}{1+e^{-\beta(\vec{w}\cdot \vec{x}-\theta)}}$

We can use the logit transform to form a linear regression model

$\log{\frac{p_1}{1-p_1}}=\beta(\vec{w}\cdot \vec{x}-\theta) +\epsilon$

We can assume that the additive noise term is IID, normal, standardised, if we assume a multiplicative IID lognormal noise in the original specification. This is feasible. The multicollinarity issues of the risk vector can be ignored, because I mainly care about forecasting systemic risk. In the era of machine learning, black box modelling is OK!

What next?

This is the framework I feel intuitively is the logical foundation for my empirical studies of systemic risk. I need to consider if I could somehow make use of the statistical mechanics framework further.

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