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<r\leq1. 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.


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<r<1}.

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<T\leq\infty}, discount rate {0<r<1}, 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.


Scientific Laws

September 2, 2009

As I have told you earlier, my guest is very sceptical about our scientific achievements. What follows are the notes I took, when he gave me a short summary of what he considers ‘our strategy’.

In modern understanding of science, the fundamental laws seem to be consequences of various symmetries of quantities like time, space or similar objects. To make this idea more precise scientists often use mathematical arguments, thereby choosing some set {X} as state space encoding all necessary information on the considered system. The system then is thought to evolve in time on a differentiable {n}-dimensional path {x_i(t)\in X} for all {t\in\mathbb{R}} and {1\leq i \leq n\in\mathbb{N}}. Quite frequently there is a so-called Lagrange function {L} on the domain { X^n \times X^n \times \mathbb{R} } and a constraint function {W} on the same domain. The path {x(\cdot)} is required to minimizes or maximizes the integral

\displaystyle  \int_0^T L\left(x(s),\dot{x}(s),s\right)ds

under the constraint

\displaystyle  W\left(x(s),\dot{x}(s),s\right)=0.

(Under some technical assumptions) a path does exactly that, if it satisfies the Euler-Lagrange equations

\displaystyle \frac{d}{dt}\frac{\partial L}{\partial \dot{x}_i}-\frac{\partial L}{\partial x_i}=\lambda \frac{\partial W}{\partial \dot{x}_i}

for some function {\lambda} depending on {X^n \times X^n \times \mathbb{R}}.

Define {y_i:=\frac{\partial L}{\partial \dot{x_i}}} and observe that (under suitable assumptions) this transformation is invertible, i.e. the {\dot{x}_i} can be expressed as functions of {x_i, y_i} and {t}. Next, define the Hamilton operator

\displaystyle  H(x,y,t) = \sum_{i=1}^n \dot{x}_i(x,y,t) y_i - L(x,\dot{x}(x,y,t),t)

as the Legendre transform of {L}. The Legendre transformation is (under some mild technical assumptions) invertible.

Now, (under less mild assumptions, namely holonomic constraints) two things happen. The canonical equations

\displaystyle \frac{d x_i}{d t} = - \frac{\partial H}{\partial y_i} \left(=[x_i, H]\right), \frac{d y_i}{d t} = \frac{\partial H}{\partial x_i}\left(=[y_i, H]\right),\frac{d H}{dt} = -\frac{\partial L}{\partial t}

are equivalent to the Euler Lagrange equations. Here {[\cdot,\cdot]} denotes the commutator bracket {[a,b]:= ab-ba}. Furthermore, if {L} does not explicitly depend on time, then {H} is a constant. That is the aforementioned symmetry. {H}, the energy, is invariant under time translations.

Given all that, the solution of the minimisation or maximisation problem can then be given (either in the Heisenberg picture) as

\displaystyle  x(t) = e^{t H} x(0) e^{-t H}, y(t) = e^{t H} y(0) e^{-t H}

or (in the in this case equivalent Schrödinger picture,) as an equation on the state space

\displaystyle  u(t)= e^{t H}u(0).

This description is equivalent (under mild technical assumptions) to the following initial value problem:

\displaystyle  \dot{u}(t)=H u(t), u(0) = u_0\in X.

where the operator {H} is the ‘law’. More technically, the law is the generator of a strongly continuous (semi-)group of (in this case linear and unitary) operators acting on (the Hilbert space) {X}. As an example of this process he mentioned the Schrödinger equation governing quantum mechanical processes.

His conclusion was that the frequently appearing ‘technical assumptions’ in the above derivation make it highly unlikely for laws to exist even for systems with, what he calls, no emergent properties. ‘If that was true’, I thought ‘then … bye bye theory of everything!’ He explained further, that under no reasonable circumstances it is possible to extrapolate these laws to the emergent situation. I am not sure, whether I understand completely what he means by that, but his summary on how we find scientific laws is in my opinion way too simple. It can’t be true and I told him.

With just a couple of ink strokes he derived the commutation relations for exchange markets from microeconomic theory. That left me speechless, since I always thought, that there cannot be ‘market laws’. Markets are on principle unpredictable! They are, or?