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 …

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An Uncertainty Principle for Markets

December 9, 2009

Today our goal is to derive an exact formulation of an uncertainty principle in markets. To that purpose we have established in earlier posts a commutation relation between demand {d_i} and price {p_i} of a good {i} in a market. I state it again:

Prices {p_i} and demands {d_j} interact according to

\displaystyle  		 		[p_i,d_j]=i \mu_i p_i \delta_{i,j} \ \ \ \ \ (1)

for a fixed real {\mu_i\in\mathbb{R}}.

What I didn’t tell you so far is how measurement of market observables is supposed to work. Let me just close this gap. Measurement of an observable, e.g. the price of good {i}, in a market in state {\xi} (e.g. in this case selling a small quantity of good {i}) will result in a jump of the market into a new state {\zeta} being an eigenvector of the observable. The outcome of the measurement will be a real number {\zeta_i} (e.g. the price), the eigenvalue of the observable corresponding to {\zeta} with probability

\displaystyle \textnormal{prob }(\zeta_i)=\frac{\left\langle \xi|\zeta \right\rangle \left\langle \zeta|\xi\right\rangle}{\|\xi\|^2}.

For an observable {a} on {X} one can show that its mean value at state {\xi\in X} is given as

\displaystyle  	\overline{a_\xi}=\frac{\left\langle a \xi|\xi\right\rangle}{\|\xi\|^2}.

The dispersion of an observable {a} on {X} is given as

\displaystyle \overline{\left(\triangle a \right)^2_\xi}= \frac{\left\langle\left(a-\overline{a_\xi} \text{id}_X \right)^2 \xi|\xi\right\rangle}{\|\xi\|^2 }.

Now we are in the shape to state the uncertainty principle in markets. In essence it claims that prices and demands of a good cannot be measured with arbitrary precision. Moreover, an explicit lower bound on the maximal simultaneous precision is given. Its proof is essentially a straight forward application of Cauchy-Schwarz inequality.

Proposition. For a market in state {\xi} the dispersions of {p_i} and {d_i} satisfy

\displaystyle \overline{\left(\triangle p_i \right)^2_\xi} \, \overline{\left(\triangle d_i \right)^2_\xi} \geq \frac{\mu_i^2}{4} \| \sqrt{p_i} \xi\|^4.

In the asymmetric case {\mu_i\neq0}, the right-hand side is strictly larger than zero.

Proof. Since dispersion and mean do not depend on the norm of a state we can, without loss of generality, assume that {\|\xi\|=1} and obtain

\displaystyle \overline{\left(\triangle p_i \right)^2_\xi} \, \overline{\left(\triangle d_i \right)^2_\xi} = \left\langle\left(p_i - \overline{p_i} \text{id}_X \right)^2\xi|\xi\right\rangle \left\langle\left(d_i - \overline{d_i} \text{id}_X \right)^2\xi|\xi\right\rangle.

Now Cauchy-Schwarz inequality implies

\displaystyle \begin{array}{rcl} \overline{\left(\triangle p_i \right)^2_\xi}\, \overline{\left(\triangle d_i \right)^2_\xi} & \geq & \left\langle\left(p_i - \overline{p_i} \text{id}_X \right) \left(d_i - \overline{d_i} \text{id}_X \right) \xi|\xi\right\rangle \\ & & \qquad \times \left\langle\left(d_i - \overline{d_i} \text{id}_X \right) \left(p_i - \overline{p_i} \text{id}_X \right) \xi|\xi\right\rangle. \end{array}

Since {ab = \frac{1}{2}[a,b]_+ + \frac{1}{2i}i[a,b]} with {[a,b]_+=ab+ba} we obtain

\displaystyle \begin{array}{rcl} \overline{\left(\triangle p_i \right)^2_\xi} \, \overline{\left(\triangle d_i \right)^2_\xi} & \geq & \left\langle\frac{1}{2} [d_i - \overline{d_i} \text{id}_X, p_i - \overline{p_i} \text{id}_X]_+\xi|\xi\right\rangle^2 \\ & & \qquad +\left\langle\frac{1}{2i} [d_i - \overline{d_i} \text{id}_X, p_i - \overline{p_i} \text{id}_X]\xi|\xi\right\rangle^2 \end{array}

and since the first term is positive

\displaystyle \begin{array}{rcl} \overline{\left(\triangle p_i \right)^2_\xi} \, \overline{\left(\triangle d_i \right)^2_\xi} & \geq & \left\langle\frac{1}{2i} [d_i - \overline{d_i} \text{id}_X, p_i - \overline{p_i} \text{id}_X]\xi|\xi\right\rangle^2 \\ & \geq & \left\langle\frac{1}{2i} [d_i,p_i]\xi|\xi\right\rangle^2. \end{array}

Now (1) and the fact that positive observables have a square root yields the final inequality

\displaystyle \begin{array}{rcl} \overline{\left(\triangle p_i \right)^2_\xi} \, \overline{\left(\triangle d_i \right)^2_\xi} & \geq & \frac{\mu_i^2}{4} \|\sqrt{p_i} \xi\|^4. \end{array}

Since {\mu_i \|\sqrt{p_i} \xi\|^4} can only be zero if {\mu_i} is zero the proposition is proved.

Admittedly, that was a bit dry, but it does the job and that is sometimes all that is necessary in mathematics. Now the pace increases and we are heading with giant leaps towards the time evolution equations for markets …


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 …


Prices and Demands (Part II)

October 19, 2009

In part one we have seen that in microeconomics if we treat demand and price as observables on a Hilbert space, then not both of them can be bounded linear operators. Especially, since all linear operators on a finite dimensional Hilbert space are bounded, the state space for our market cannot be finite dimensional. All this is a consequence of a single assertion, namely demand invariance under price-scaling and somehow resembles the situation in quantum theory.

So far our considerations concerning prices and demands were quite abstract. To do some real computations we need a representation of these concepts. As promised last week, I now provide such a representation for the observables price {p_i}, demand {d_i} and excess demand {z_i} on an appropriate Hilbert space.

The results so far, lead to the following approach: The Hilbert space is given as {X = L^2(\mathbb{R}^n)}. This, in a way, is the simplest non-finite dimensional Hilbert space and therefore an unsurprising first choice. Thus, the state of the market with {n\in\mathbb{N}} goods is described by a function {\xi\in X}. Let the coordinates of {\xi} be denoted as {(x_1,\ldots,x_n)\in \mathbb{R}^n}, then the demand {d_i: D(d_i)\rightarrow X} is given as a differential operator

\displaystyle  	d_i \xi = -i \mu_i \frac{d}{d x_i} \xi

with domain

\displaystyle  	D(d_i) = \{ \xi\in X: \xi \textnormal{ absolutely continuous and } \xi' \in X \}.

The excess demand operator {z_i=d_i-\omega_i \textnormal{id}_X} has the same domain as {d_i}.
Define the function {e_i:\mathbb{R}^n\rightarrow \mathbb{R}} as {e_i(x)=e^{x_i}}. Then, the price operator {p_i:D(p_i)\rightarrow X} is given as a multiplication operator

\displaystyle  	p_i \xi = e_i \cdot \xi

with domain

\displaystyle  	D(p_i) = \{ \xi\in X: e_i \cdot \xi \in X \}.

All operators {p_i, d_i, z_i} are self-adjoint, {p_i} is positive, the commutator satisfies

\displaystyle \left[p_i,z_i\right]\xi = \left[p_i,d_i\right]\xi = -i \mu_i \left(e_i \cdot \frac{d}{d x_i}\xi - e_i \cdot \xi - e_i \cdot \frac{d}{d x_i} \xi\right)= i \mu_i p_i \xi

and thus the market axioms are fulfilled.

That still might look a little abstract if one is not used to Functional Analysis. The corresponding representation in quantum mechanics however lead to major new insights into the field.

Next time I shall give a formal comparison of microeconomics and quantum mechanics. As you can imagine by now, they are similar on some abstract level. However, there are also some striking discrepancies like the different commutation relations and thus the step towards the desired Schrödinger type equation for markets is not straight forward.

Stay tuned …


Prices and Demands (Part I)

October 5, 2009

You might remember, that we are looking for general laws describing the behavior of markets. To that purpose, at first glance, it is not self-evident to describe such markets, i.e. prices {p} of and demands {d} for goods, by observables acting on some Hilbert space {X}. However, as we have seen in Commutation Relations in Markets and in Scientific Laws there might be good reasons for doing this and if we do it, the observables {p} and {d} have to satisfy the following axioms:

  • (MA1) The price {p_i} of good {i} is a positive observable on {X} for all goods {1\leq i\leq n}.
  • (MA2) The demand {d_i} of good {i} is an observable on {X} for all goods {1\leq i\leq n}.
  • (MA3) The endowment {\omega_i} of good {i} is a real number {\omega_i \in\mathbb{R}} for all goods {1\leq i\leq n}.
  • (MA4) Prices {p_i} and demands {d_j} interact according to

    \displaystyle  			\left[p_i, d_j\right]=i \mu_i p_i \delta_{i,j}

    for a fixed real {\mu_i\in\mathbb{R}}.
    Up to now, we have also learned that the demand {d} essentially is the generator of the price-scaling group and that mathematicians often represent groups as linear operators acting on vector spaces to get an idea of what is going on. Let us just do that and find some ‘matrices’ to represent price-scaling. Maybe we then get a better understanding of such market descriptions.

    Unfortunately, there is a famous result of H. Wielandt stating that if linear operators {A,B} satisfy a commutation relation {[A,B]=\textnormal{id}_X}, then not both can be bounded simultaneously. Since there are no unbounded linear operators on finite dimensional vector spaces, the Hilbert space then must be infinite dimensional.

    Just as a side note, you might notice that (scalar multiples of) momentum and position in quantum dynamics satisfy the above commutation relation and this is the reason why we cannot represent these observables as matrices. On our hunt for market laws that might be a little setback. Certainly not for mathematicians or physicists, but probably for economists. As far as i know, infinite dimensional Hilbert spaces are up to now not part of their syllabus.

    Today we are not doing in quantum mechanics and thus there is still hope. If we divide market axiom 4 by {i \mu} we observe that our commutation relation is {[A,B]=A} which is certainly different from the quantum situation.
    Let us do some mathematics.

    Assume {A,B} to be bounded linear operators (acting on some Banach space) with {A^n\neq 0} for all {n\in \mathbb{N}}. Assume furthermore {[A,B]= A} and as induction hypothesis {[A^n,B]= nA^n}. Then {[A^{n+1},B]= A[A^n,B]+[A,B]A^n=nA^{n+1}+A^{n+1}=(n+1)A^{n+1}}. The norm estimate {n \|A^n\| = \|[A^n,B]\|\leq 2 \|A^n\| \|B\|} yields a contradiction since {A^n\neq 0} for all {n\in\mathbb{N}}. Therefore, either at least one of the operators {A} and {B} is unbounded and/or the commutation relation {[A,B]=A} does not hold.

    In our situation the commutation relation holds as stated in market axiom 4. Since, by market axiom 1, the observable {p_i} is positive we obtain {p_i^n\neq 0} for {n\in\mathbb{N}} and hence, at least one of the operators {p_i} and {d_i} is unbounded.

    Oh, oh … not nice. But, it could have been worse. Next week I choose a Hilbert space and give you a representation of price and demand as unbounded operators on this Hilbert space. Things will look much more down-to-earth then. Promised …


  • Commutation Relations in Markets

    September 8, 2009

    To derive commutation relations in microeconomics we first have to reach sure ground. What is a minimal set of assumptions we need to derive something interesting, but still comprehensive enough to describe something meaningful?

    In a market, it is definitely safe to assume that we have {n} goods for some number {1\leq n \in \mathbb{N}}. This goods are being traded and therefore we need to talk about prices and demand. Call {p_i} the price of good {i} and {d_i} the demand for good {i} and let {1\leq i \leq n}. What else do we need?

    Sure, we need a lot more, but not now! As we have seen in Scientifc Laws all we need now is a symmetry between price and demand. The key to this symmetry is found in any basic text book like e.g. Microeconomic Theory by A. Mas-Colell, M.D. Whinston and is called invariance of demand under price-scaling. What is meant by that? Let me just give you an example. When continental europe introduced the Euro currency, many nations swapped their national currency for the new Euro. In Germany, 1 Euro was worth 1.95583 Deutsche Mark. All prices, wages, debts aso. where scaled by {\frac{1}{1.95583}}. The day after, no increase of demand for fridges, cars, credits aso. was observed. That was no surprise for economists. Where should a change of demand come from? A redefinition of the currency is not enough to generate demand. That is generally believed and a pillar in the following argumentation.

    Just for the sake of completeness let me emphasize that price scaling, as introduced above form a group. Whenever we scale by a factor {\alpha\in\mathbb{R}_{>0}} and then scale by a factor {\beta\in\mathbb{R}_{>0}} we obtain a scaling by the factor {\alpha\beta}. Scaling with 1 is the neutral element and for each scale factor {\alpha\in\mathbb{R}_{>0}} we can go back by scaling with {\frac{1}{\alpha}}.

    As mathematicians, we often represent abstract groups (like the above price-scaling) as linear operators acting on some vector space. To that purpose, we choose the state of the market to be given by a non-zero vector {\xi} in a Hilbert space {X} with inner product denoted by {\left\langle \cdot | \cdot \right\rangle}. Of course, in the moment you can think of {X} as a finite dimensional Hilbert space {\mathbb{R}^n} or {\mathbb{C}^n}. On the other hand, it is always good to be suspicious and fixing the dimension to be finite might be premature. Observables are self-adjoint operators on this Hilbert space and satisfy the following axioms:

  • (MA1) The price {p_i} of good {i} is a positive observable on {X} for all goods {1\leq i\leq n}.
  • (MA2) The demand {d_i} of good {i} is an observable on {X} for all goods {1\leq i\leq n}.
    A positive observable {a} on {X} is an observable with {\langle a\xi | \xi \rangle>0} for all {0\neq\xi} in the domain {D(a)} of {a}.

    By a famous result of E. Noether, symmetries and invariants are closely tied together. What are the market invariants of the asymmetric market under the price-scaling symmetry? To see this, let {\left(U_i(\alpha)\right)_{0 < \alpha\in \mathbb{R}}} be a strongly continuous family of unitary operators on {X} such that

    \displaystyle  	U_i^{-1}(\alpha)p_i U_i(\alpha)=\alpha p_i.

    The family {U_i(\cdot)} satisfies the following properties for all {\alpha>0} and {\beta>0}:

    • {U_i(1)= \textnormal{id}_X}
    • {U_i(\alpha)U_i(\beta)=U_i(\alpha\beta)=U_i(\beta)U_i(\alpha)}
    • {U_i^{-1}(\alpha) = U_i\left(\frac{1}{\alpha}\right)}

    Define {T_i(t):=U_i(e^t)} and observe

    • {T_i(0)= \textnormal{id}_X}
    • {T_i(t)T_i(s)=T_i(t+s)=T_i(s)T_i(t)}
    • {T_i^{-1}(t) = T_i(-t)}

    This yields {T_i} to be a strongly continuous group of unitary operators acting on {X}. Thus, the theorem of Stone ensures the existence of a skew-adjoint generator {A_i}. Set {\alpha = e^t} and with {U(\alpha)=T(\ln \alpha)} it follows that

    \displaystyle \begin{array}{rcl} p_i & = & \frac{d}{d\alpha}\left(U_i^{-1}(\alpha)p_i U_i(\alpha)\right) \\ & = & \frac{d}{d\alpha}\left(T_i(-\ln \alpha)p_i T_i(\ln \alpha)\right) \\ & = & - \frac{1}{\alpha} T_i(-\ln \alpha) A_i p_i T_i(\ln \alpha) + \frac{1}{\alpha} T_i(-\ln \alpha)p_i A_i T_i(\ln \alpha). \end{array}

    Evaluation at {\alpha=1} yields

    \displaystyle  	 	\left[p_i, A_i\right] = p_i. \ \ \ \ \ (1)

    Since a generator commutes with the strongly continuous group it generates it is easily seen that {\beta_i A_i + \gamma_i\textnormal{id}_X} also commutes with {U_i(\alpha)} for any {\beta_i,\gamma_i\in\mathbb{C}}. Hence {\beta_i A_i + \gamma_i\textnormal{id}_X} represents a market invariant under price-scaling.

    Now we derive an economic interpretation of {A_i}. We know already that {\beta_i A_i + \gamma_i\textnormal{id}_X} represents a market invariant under price-scaling for any {\beta_i,\gamma_i\in\mathbb{C}}. Since {A_i} is skew-adjoint and {\beta_i A_i + \gamma_i\textnormal{id}_X} needs to be an observable, we get that {\beta_i = i \mu_i } and {\gamma_i = \omega_i} for some {\mu_i, \omega_i \in\mathbb{R}}. Furthermore, since scaling of one price does not influence scaling of the others (i.e., {\left[p_i, U_j(\alpha)\right]=0} for {i\neq j}) we can use (1) and obtain

    \displaystyle  	\left[p_i, i \mu_i A_j - \omega_i \textnormal{id}_X\right] = i \mu_i p_i \delta_{i,j}.

    The operator {i \mu_i A_i + \omega_i \textnormal{id}_X} is an observable and is invariant under price-scaling. Economic intuition therefore leads us to identify this operator with the demand respectively excess demand for good {i} if {\mu_i\neq 0}. The real parameter {\omega_i} is identified as endowment. The other real parameter {\mu_i} represents a new feature. Intuitively it measures the difference of first selling and then buying a good versus first buying and then selling that good.
    The observations in the last paragraph yield the final axioms.

  • (MA3) The endowment {\omega_i} of good {i} is a real number {\omega_i \in\mathbb{R}} for all goods {1\leq i\leq n}.
  • (MA4) Prices {p_i} and demands {d_j} interact according to

    \displaystyle  			\left[p_i, d_j\right]=i \mu_i p_i \delta_{i,j}

    for a fixed real {\mu_i\in\mathbb{R}}.
    There are still a lot of things to say, e.g. on how measurements are done, on the dimension of the Hilbert space {X}, on representations of demand {d_i} and price {p_i} as operators and on a comparison to the commutation relations of quantum mechanics. Stay tuned …