Treating Markets Mechanically – An Example

April 27, 2011

The aim of this post is to provide the transition from time-independence to time-dependence within a simple economic model for further reference.

For that purpose we consider a single consumer-worker. This agent obeys a time constraint on labour L and free time F

$L + F = 1.$

We introduce a utility function U as

$U = C^\alpha F^{1-\alpha}$

for a given $0< \alpha<1$. There is a budget constraint W given as price p times consumption C equals wage rate w  times labour L.

$p C = w L.$

The agent now maximizes $U$ such that $p C + w F = w$. Let us solve that. Lagrange equations yield

$-\frac{\partial U}{\partial C} = \lambda \frac{\partial W}{\partial C}$

and

$-\frac{\partial U}{\partial F} = \lambda \frac{\partial W}{\partial F}$

with constraint

$W = p C + w F -w = 0.$

Thus

$\frac{p}{w}=\frac{\alpha C^{\alpha-1} F^{1-\alpha}}{(1-\alpha) C^\alpha F^{-\alpha}}=\frac{\alpha F}{(1-\alpha) C}.$

Solving that for C and plugging it into the budget constraint yields

$\frac{\alpha}{(1-\alpha)}w F+w F -w =0$.

Solving this for F and again using the budget constraint shows that

$F=1-\alpha$

and

$C=\alpha\frac{w}{p}$

solves the maximization problem. So far there is no time evolution. To introduce such a dynamics we mimic mechanics and set $C=C(p,\dot{p})$. Demand is a function of price and its derivative. For economists the $\dot{p}$ comes from nowhere. Especially since it is not obvious at all how to define the derivative of a price evolution. For now it has to suffice that eventually we shall understand the derivative in a distributional sense and until then we treat it as a formal parameter.

The time-dependent utility function for the consumer-worker

$U = C^\alpha F^{1-\alpha}r^t$

for a discount rate $0. The agent now maximizes $\int_0^T U d t$ under the constraint $p C + w F = w$.

We make the following assumption due to S. Smale (for excess demand):

$\dot{p}=C \textnormal{ and }\dot{w}=L.$

Euler-Lagrange equations yield

$\frac{d}{d t}\frac{\partial U}{\partial C} = \lambda \frac{\partial W}{\partial C}$

and

$\frac{d}{d t}\frac{\partial U}{\partial L} = \lambda \frac{\partial W}{\partial L}$

with constraint

$W = p C - w L = 0.$

For the Lagrange multipliers we get

$\lambda =-p^{-1}\alpha r^t C^{\alpha-2}F^{-\alpha}((\alpha-1)(C\dot{F}-F\dot{C})-C F \log r)$

and

$\lambda =w^{-1}(\alpha-1) r^t C^{\alpha-1}F^{-1-\alpha}(\alpha(C\dot{F}-F\dot{C})-C F \log r).$

Equating, plugging in the constraint and dividing by $C F$ yields

$\frac{\dot{C}}{C}-\frac{\dot{F}}{F}= \frac{(\alpha w - p C)\log r}{\alpha(1-\alpha)w}$

First we discuss the case $r=1$. Then

$\frac{\dot{C}}{C}=\frac{\dot{F}}{F}$

and thus (consider $\frac{d}{d t}\ln C$) there is a positive, constant K such that C = K F and we get because of the budget constraint

$F = \frac{w}{p K + w}, C = \frac{w K }{p K + w}.$

The constant $K(p,w,\alpha)$ is unique and maximizes $\int_0^T C^\alpha F^{1-\alpha} ds = \int_0^T \frac{w K^\alpha}{p K + w} ds$. In equilibrium we have $p = p^*$ and $w = w^*$. Maximizing  K yields $\frac{d}{d K}\frac{w^* K^\alpha}{p^* K + w^*}= 0$ and thus $K=\frac{\alpha w^*}{(1-\alpha) p^*}$. Now

$F = 1-\alpha$

and

$C = \alpha \frac{w^*}{p^*}.$

The case $r<1$. In equilibrium $\dot{C}=\dot{F}=0$ we immediately obtain $C=\alpha \frac{w^*}{p^*}$. Plugging this into the budget constraint yields $F=1-\alpha$.

Interestingly enough, we get an equilibrium equal to the solution of the time-independent model. How justified is S. Smale’s assumption $C=\dot{p}$? Economists often use linear demand theory and set $C=T-p$. Both approaches seem to be incompatible and both have a draw back. When you scale prices (e.g. by introducing a new currency) demand should stay the same. This is not the case in both settings. One needs currency dependent constants that scale accordingly to fix that. One possibility to avoid that is $C=\frac{\dot{p}}{p}$. As usual, more options do not improve clarity and calculating the whole model in the general case, i.e. $C=C(p,\dot{p})$ is not totally conclusive either. For a solution of the Euler-Lagrange equations one obtains under moderate assumptions on the partial derivatives that

$\alpha (1-\alpha)w\frac{\partial C}{\partial \dot{p}}\left(\frac{\dot{C}}{C}-\frac{\dot{F}}{F}\right) = (\alpha w - p C)\left(\frac{\partial C}{\partial \dot{p}}\log r + \frac{d}{d t}\frac{\partial C}{\partial \dot{p}}-\frac{\partial C}{\partial p}\right).$

Linear demand has $\frac{\partial C}{\partial \dot{p}}=0$ and thus $\alpha w - p C=0$. The budget constraint implies $F=1-\alpha$ which is a constant. We thus can safely exclude linear demand from our considerations. The above equation cannot distinguish between Smale’s assumption and $C=\frac{\dot{p}}{p}$. However, hidden in the technical assumptions, there seems to be some advantage in Smale’s approach. It remains to clarify the price-scaling issue.

Some Not-Yet-Mathematical Problems in Science

April 21, 2010

When you ask a mountaineer why he is doing what he is doing a possible answer is: “Because it’s there“. Are mathematicians any different? Our motivation for doing mathematics also stems from the fact that there are still significant mathematical mountains. For example, the holy grail of mathematics, the Riemann Hypothesis is considered hard, interesting and a solution is worth a million dollar. Most likely new methods and ideas have to be developed and  they will most certainly contain surprises for the experts. Several fields in mathematics would take a gigantic leap forward.

However, the way we do mathematics or the way we think about mathematics will not be changed. Whatever the solution will be, we do not expect it to be as staggering as quantum theory was for physics or as breathtaking as evolution theory was for biology. This is certainly good news if one thinks of mathematics as a dogmatic science. On the other hand, there might be interesting insights if we only just bend our preconceptions on the axiomatic  approach or towards circularity a little. If successful, we might change our understanding of what mathematics actually is and how it should be done.

Are there widely accepted scientific problems for which we so far have no mathematical description or solution? Problems that are so hard, that maybe they will change the way we do mathematics? Problems that give us the impression that we only know what mathematics was in the past and that we shall never know what mathematics will look like in the future?

Without claim of completeness:

In biology, evolution captures the alteration of populations consisting of reproducing individuals and their environment over time by properties of the individuals. Well known concepts such as natural selection and “survival of the fittest” are justified by a plethora of observations and therefore the theory is a cornerstone of modern science. 150 years of research and there still is no comprehensive mathematical treatment of evolution. Why is this so hard? I have no answer to this, although two observations come to mind. 20th century mathematics is usually presented as a typed theory and thus promotes populations over individuals. From that viewpoint, evolution seems to consist of special cases only rather than to contain some abstract, mathematical content. Additionally, there are obvious circularities in the considered notions population/environment and survival/fitness. The biological content seems to be rather in these circles than in some mathematical bottom-up framework of axioms and theorems.

II. Market

In economics, markets are places where agents satisfy their demand for goods by trading them. For more than 100 years we have markets as  objects under scientific consideration. So far, we do not have a time evolution for prices and demands maybe even in terms of actions taken by agents and preferably extending existing results in microeconomics. What is the difficulty? Maybe the following observation sheds some light on the situation. It seems that correctly predicted prices lead to arbitrages which then change the behavior/demand of the agents. That circle seems to encode the basic essentials of economics. Theories with assumptions on either side, the prices or the demand, seem to assume away the problem and are then only loosely connected to observations. This reminds on games with incomplete information.  Making assumptions on opponents strategies greatly simplifies the search for ones own optimal strategy. However, justifying these assumptions is the hard part with not much mathematics involved so far.

III. Cosmology

In physics, cosmology studies the dynamics and the evolution of the universe as a whole. By this definition an observer is part of the universe and thus ‘laws’ governing this universe put restrictions on what ‘observation’ actually is. On the other hand,  the observer develops models describing the universe and thus himself as part of this universe.  This circular observer/universe dependence is related to anthropic principles. The problem now consists in developing a correct argument to refute models essentially just by applying the observational fact of the existence of the observer. This is hard since the ‘logic’ an observer can use to reason about the universe is also part of the universe and might change if the universe is changed. As long as we do not know what ‘observation’ means it is not enough to exclude possible models just by the non-existence of a certain type of observer (e.g. carbon based life form), since there could be other observers experiencing their observations as e.g. we do ours and thus different models could still be ‘isomorphic as observed universes’ (whatever that means).

IV. Foundation of Physics

In physics, the outcome of conducted experiments and of developed theories  is compared according to established rules. If the comparison meets certain standards the theory is called physical and is said to describe reality. The outcome of an experiment is usually described in terms of fundamental units of measurement. Different social communities use different units. A widely accepted standard is SI. Within SI the unit of ‘mass’ is kilogram, which is defined by reference to a certain pile of platinum-iridium alloy stored at a certain place in France. This procedure puts severe limits to the accuracy of measurements (as of 2010 this is one part in 10^8). The ultimate goal therefore is to liberate the SI system’s dependency on this definition by developing a practical realization of the kilogram that can be reproduced in different laboratories by following a written specification. The units of measure in such a practical realization would have their magnitudes precisely defined and expressed in terms of fundamental physical constants. Why has nobody done that so far? The problem seems to be to describe the foundations of physics in physical terms. That circularity seems to be hard to overcome. Moreover, the above argumentation is not only valid for ‘mass’, but for any other physical quantity and one would end up with a huge network of interdependent ‘written specifications’.

V. Life, Intelligence and Conciousness

We know that there is life, that there is intelligence and that there is conciousness. Mathematics has little more to say about these. What is the problem? We might not have enough data, different intelligent life forms, to extract an abstract theory. However, we also might face a semantical problem. In 20th century mathematics we know of two ways to introduce notions. First, by description, like set or element. These notions are given by their relation to other described notions in a huge semantical circle.  No further justification for them is given. Second, by definition, like function or number. Defined notions come together with a substitution process and can, at least in principle, be eliminated from the theory. The notions in the title so far withstood any attempts to being defined. It seems that these notions ’emerge’ by some sort of ‘limit’ procedure from other notions. Of course, this is only speculation and anytime soon a reasonable definition of ‘life’, ‘intelligence’ and ‘conciousness’ might be found. If not, however, we  might take that as an indication that mathematics still has potential rather than limits.

A recent example of physical notions not being functions of each other

March 22, 2010

Let me quote John Baez from his recent issue “This Week’s Finds in Mathematical Physics (Week 294)”:

The point of these examples is that most linear resistors let us treat current as a function of voltage or voltage as a function of current, since R is neither zero nor infinite. But in the these two limiting cases – the short circuit and the open circuit – that’s not true. To fit these cases neatly in a unified framework, we shouldn’t think of the relation between current and voltage as defining a function. It’s just a relation!

That is another example of basic notions not being a function of each other.

Maybe that makes my last comment on the relation between ‘price of a good’ and ‘demand for a good’ not being a function of each other more accessible. Between these economic notions there is just a (commutation) relation. They are not functions of each other. I will certainly elaborate on this …

General Equilibrium in a Nutshell

February 17, 2010

My goal today is to tell the story of textbook general equilibrium theory and relate it to some of the things I have done so far in this blog.

Let me first quote from Stephen Smale, Mathematical Problems for the next Century, 1998:

The following problem is not one of pure mathematics, but lies on the interface of economics and mathematics. It has been solved only in quite limited situations.

Extend the mathematical model of general equilibrium theory to include price adjustments.

What is the mathematical model of general equilibrium theory? Usually there are ${n}$ goods and prices ${p_i}$ attached to each good for ${1\leq i\leq n}$. Furthermore there is an excess demand function ${z:\mathbb{R}^n_+\rightarrow\mathbb{R}^n}$ (considered as the difference of demand and supply) from the set of prices to the set of goods. Excess demand satisfies three axioms:

1. Homogeneity of degree zero: ${z(\lambda p) = z(p)}$ for all ${0<\lambda\in\mathbb{R}}$ and all ${p\in \mathbb{R}^n_+}$.
2. Walras’ law: ${\sum_{i=1}^n p_i z_i(p)=0}$.
3. Positive demand for a free good: ${z_i(p)>0}$ if ${p_i=0}$.

For this vector field Hopf’s theorem ensures the existence of an equilibrium price vector ${p^*}$ such that ${z(p^*)=0}$. This is often refered to as “supply equals demand”. The problem now is to find a dynamical model explaining the time evolution of prices maybe even as actions of agents acting in the market. This model should preferable be compatible with the existing equilibrium theory.

How does this relate to what I have said so far in this blog? The essential question:

Is (excess) demand indeed a function of price?

To my knowledge there is hardly any economic evidence/experiment to settle this question. “What else could it be?” is certainly not a valid viewpoint. While there are reasons to believe this in the stationary setting, as soon as time hits the scene a functional relation needs justification. Agents may learn or exhibit other hard to explain behavior.

If we do not assume excess demand to be a function of price, how far do we get? Quite far, actually. The reason is axiom 1. Its content in everyday speech: demand does not depend on price-scaling. Since price-scalings form a group we are in a position to find a representation, for example, as linear operators on a vector space. Doing this we can derive a kind of uncertainty principle for demand and price. They cannot be measured simultaneously with arbitrary precision. This is certainly necessary if we wanted to define a functional relation between them. Moreover, as an invariant of the group action we get a two parameter family of operators. One of these parameters is readily identified as “endowment”, the other represents a willingness-to-pay/willingness-to-accept discrepancy. Goods are considered more valuable if we own them compared to if we want them. We get what economists call the endowment effect for free.

All this could be done without the other axioms. As soon as this problem is tackled I comment on these too. Not today however …

Utility and Time – Statement of the Problem

January 27, 2010

Ultimately our goal is to get some description of price evolution derived from first (economic) principles. In earlier posts (1, 2, 3) I have shown what can be deduced from ‘demand invariance under price-scaling’. As described there we still assume ${n}$ goods being traded in a market, hence there are prices ${p_i}$ and demands ${d_i}$ for ${1\leq i\leq n}$ attributed to these goods. That was the setting so far and now we are going to take the first steps into time.

We assume that good ${i}$ is consumed over time and describe consumption ${c_i(\cdot)}$ as a positive real function. Consumption of good ${i}$ from time ${a}$ to time ${b}$ is measured by ${\int_a^b c(s)ds}$. The participants in the market we call agents. An agent attributes to each consumption vector ${c}$ a utility ${u}$. Technically this is a positive, increasing and concave function. In all our examples ${u}$ and ${c}$ will be sufficiently differentiable. Utility is increasing since more consumption is considered better and it is concave since we assume ‘diminishing marginal utility‘. The latter does not always hold in economic situations. However, most introductory examples are concave and as a start this seems safe.
I assume ‘time impatience‘, that means, consumption now is better than consumption in the future. That assumption is not undisputed, but, as a model for the finite life span of the agents, this too seems safe for such an introductory text. Overall utility from time ${a}$ to time ${b}$ is measured by ${\int_a^b r^s u(c(s)) ds}$ for some discount rate ${0.

Agents have attached a wealth level ${w(\cdot)}$, that means for all time holds ${\sum_i^n p_i(t) c_i(t) = w_i(t)}$. They consume according to their prescribed wealth and they consume according to their demand (${c_i=d_i}$). The last assumption closes the gap between ${p}$ and ${c}$. We assume that demand is given as ${d_i=\dot{p_i}}$ and thus we obtain in summary ${c_i = \dot{p}_i}$.

Now we are in the shape to state the problem: agents in a market maximize utility

$\displaystyle \int_0^T r^s u(c(s)) ds$

according to the constraint

$\displaystyle \sum_{i=1}^n p_i(t) c_i(t) - w(t) = 0$

for given time ${0, discount rate ${0, wealth function ${w}$ and utility function ${u}$.

Voilà, we end up with a constrained Euler-Lagrange equation. But beware! There are a couple of traps jamming all intuition we might have from mechanics or similar theories with conserved energy (understood as the Legendre transform of the Lagrangian). I certainly elaborate on this in one of the next entries.

Mechanics and Markets

November 25, 2009

When we talk about markets we often use terms like equilibrium or even market force. We choose this terminology for a reason. The analogy to the well established theories of mechanics and quantum mechanics is intended and the pictures we have in mind are a pendulum or even a simple spring. Their restoring forces seem to model the market forces and therefore we frequently observe argumentations very similar to:

if prices increase, then demand decreases and vice versa finally, because of some process still to be described, the market settles down in an equilibrium (called Walrasian price equilibrium).

As a start, that sounds convincing. There just remains one big question. Is that a good picture? Or, even more to the point:

Are there any justifications for the existence of market forces?

Rather than answering this question (regular readers know my standpoint anyway) I would like to justify why this question is actually reasonable and should be asked and answered. In physics this question is answered to the positive, in economics the situation is a little blurry to say the least. I continue by comparing mechanics with economics in catchwords. Thereby pointing out similarities, but also discrepancies and, in a way, recalling ‘the story so far’.

Basic notions

Let me start with two of the fundamental notions in mechanics, namely position and momentum. In earlier posts we have identified their counterparts in economics as price and demand.

Symmetries

In mechanics the intuition is that momentum is invariant under translation of position. In economics we need demand invariance under price-scaling.

Commutation relations

These symmetries lead to commutation relations of the form ${[A,B]=\text{id}}$ in quantum mechanics and ${[A,B]=A}$ in economics (cf. here). This difference is essential and has a huge impact, albeit not immediately.

Bounded representations

Both commutation relations imply that the symmetry groups do not have representations on a finite-dimensional vector space (cf. here).

Unbounded representations

While there are no bounded representations, we get unbounded representations on the Hilbert space ${L^2(\mathbb{R}^n)}$ of square integrable functions. Momentum and demand operators are differential operators, whereas position and price are (different) multiplication operators (cf. here).

Uncertainty principle

The uncertainty principle of quantum mechanics is well-known. So far I didn’t write about that here in the blog, but in economics the commutation relations imply inequalities which can also be interpreted as some sort of uncertainty principle. I shall come back to this later.

Time evolution

As described in scientific laws to get the time evolution in quantum mechanics one chooses an action, one uses Legendre transform to obtain the energy, one derives the canonical equations and essentially plugs in the above representation to obtain Schrödingers equation governing the time evolution of a quantum system. That surely sounds more complicated than it actually is.

Why can’t we just do that for markets and obtain market equations governing their time evolution? Now, there are a couple of technical difficulties. The most prominent probably is that the Legendre transform of a market action is not invariant under time translation. Hence, in markets there is no conservation of energy. This fact alone makes the usage of a term like market force a little obscure. What is meant by force if there is no energy or at least no energy conservation?

That essentially is the programme for the rest of the year. I shall spell out the maths behind the uncertainty principle for markets and then delve into the technical details of obtaining a time evolution for markets.

Stay tuned …

Finally Top Ten!

October 31, 2009

Last week I was teaching 8 hours mathematics a day, but that could not prevent me from checking my e-mail. What I found was a link to the following top ten list:

Top Ten List

It contains a joint work with Christian Schwarz on 10th place which is not overly remarkable. The paper is about a fairly general proof of the non-existence of market equilibria.

The twist is that the list also contains a ‘handful’ of papers proving the existence of said market equilibria.