An Imaginary Realistic Market

• Financial crisis : 1929 the Great Crash, 2008 the not so great Crash…

• Economic theory: General theory (Keynes, 1936), General equilibrium (Arrow-Debreu-McKenzie, 1950s), No arbitrage (Black, Scholes et al, still ongoing).

• “A financial crisis is any of a broad variety of situations in which some financial assets suddenly lose a large part of their nominal value”. - from Wikipedia

• Nominal values are quantified by prices in the market. False beliefs of the prices or the market.

Price is determined by the supply and the demand.

• Distortions in the supply, or the demand, or both.

Supply or demand compose of individuals' choices.

• Defects in Axiom of Choice and Zorn's lemma. (Good luck!)

Selective sloppy explainations of

1. a generic exchange economy,

2. a complex price,

3. an invisible hand,

4. a seemingly equilibrium market,

5. an extension and a singularity,

6. a market structural distribution.

Quite likely we'll run out of time at point 2.

What I meant is

1. a polynomial economy \( \mathbb{V} \), where supply equals demand \( \mathbf{f}(\mathbf{x})-\mathbf{x}^{d}=0 \);

2. a price vector \( \mathbf{h}\in\mathbb{C}^{K} \);

3. an exponential map \( \exp(-\mathbf{h}^{\top} \mathbf{X}^{d}) \);

4. a process driven by exponential families \( \frac{\exp(-\mathbf{h}^{\top} \mathbf{X}^{d})}{\mathbb{E}\left[\exp(-\mathbf{h}^{\top} \mathbf{X}^{d})\right]} \);

5. a coupling process of two \( \mathbb{V} \) and \( \tilde{\mathbb{V}} \), and a singularity in this coupling process;

6. a distribution of the market structure induced by the beliefs.

• Cobb– Douglas function in two sectors \( x_{1}^{a_{1}}x_{2}^{a_{2}} \).

• A general production in \( K \) sectors: \( x_{1}=f_{1}(x_{1},\dots,x_{K}) \), \( \dots \), \( x_{K}=f_{K}(x_{1},\dots x_{K}) \) where \( f_{i} \) is a polynomial function of \( (x_{1},\dots,x_{K}) \): \[ \mathbf{x}=\mathbf{f}(\mathbf{x}),\quad\mathbf{f}\in\mathbb{R}[\mathbf{x}]. \]

• The aggregate demand for each sector \( l \): \[ x_{l}^{d}=\mathbb{E}[X^{d}(\omega_{l})],\mbox{ where }X^{d}(\omega_{l})\sim\mathbb{P}(\omega_{l}) \] \( \mathbb{P}(\omega_{l})=\mathbb{P}(\omega_{l}^{1})\cdots\mathbb{P}(\omega_{l}^{n_{l}}) \) for the endowments of \( n_{l} \) participants. \[ X^{d}(\omega_{l})=\sum_{i=1}^{n_{l}}X^{d}(\omega_{l}^{i})\sim\mathbb{P}(\omega_{l}^{1})\cdots\mathbb{P}(\omega_{l}^{n_{l}})\;\mbox{(infinite divisibility)}. \]

• Economy: the set of \( (\mathbf{f},\mathbf{x}^{d}) \) satisfying \( \mathbf{f}(\mathbf{x})=\mathbf{x}^{d} \) is \( \mathbb{V}(\mathbf{f},\mathbf{x}^{d}) \).

• The market clear in a linear form \[ \mathbf{h}(\mathbf{x})^{\top}\left(\mathbf{f}(\mathbf{x})-\mathbf{x}^{d}\right)=0. \]

• Unfortunately, the solution set of \( \mathbf{h}(\mathbf{x}) \) is incomplete. That means market may not know how to price an exchange \( \mathbf{f}(\mathbf{x})=\mathbf{x}^{d} \).

• Supply must be real. Demand must be real. But when the supply meets the demand,\[ \mathbf{f}(x_{1},x_{2})=\left[\begin{array}{c} \frac{1}{2}x_{1}+x_{2}\\ x_{1}^{2}+10 \end{array}\right]=\left[\begin{array}{c} 9\\ 6 \end{array}\right]=\mathbf{x}^{d}. \]

• The solution is \( x_{1}=\sqrt{-4}=2\mathrm{i} \).

• Let the complex-valued price \[ h_{1}(x_{1},x_{2})=(x_{1}^{2}-4)(x_{1}+\sqrt{-4}). \] Then \( h_{1}(x_{1},x_{2})(x_{1}-2i)=x_{1}^{4}-16 \) has a realistic solution.

• The imaginary price \( \mathbf{y} \) is invisible as it has the form \( \mathrm{i}\mathbf{y} \).

• But a function on imaginary value may induce a real force.

• Two sentiments for the price: positive (阳, bullish) and negative (阴, bearish).\[ \min_{\mbox{阴}}\max_{\mbox{阳}}\mbox{阴}(\mathbf{y})\mbox{阳}(\mathbf{y}) \] The solution of this minimax game is the exponential map (invisible hand, Prop. 1).

• For the complex-valued pricing demand \( \mathbf{h}{}^{\top}\mathbf{x}^{d}\in\mathbb{C} \): \[ e^{\mathbf{h}{}^{\top}\mathbf{x}^{d}}=e^{\mathbf{p}^{\top}\mathbf{x}^{d}}e^{\mathrm{i}\mathbf{y}{}^{\top}\mathbf{x}^{d}}=e^{\mathbf{p}^{\top}\mathbf{x}^{d}}\left(\cos\mathbf{y}{}^{\top}\mathbf{x}^{d}+\mathrm{i}\sin\mathbf{y}{}^{\top}\mathbf{x}^{d}\right). \]

• One is pushed by the invisible hand onto a manifold.

• One would have a different vision of one's market power and one's new vision would influence the market.

• If lucky enough, the market may be driven into equilibrium by this kind of influences.

• The aggregate demand \( \mathbf{x}^{d}=\mathbb{E}\left[\mathbf{X}^{d}\right] \) is from many individual demands in \( K \) sectors.

• Market power of stochastic demands: \( e^{-\mathbf{h}{}^{\top}\mathbf{X}^{d}} \).

• It is a harmonic entity, namely satisfying law of conservation (Th. 2).

• The visible part of \( e^{-\mathbf{h}{}^{\top}\mathbf{X}^{d}} \) is: \( e^{-\mathbf{p}{}^{\top}\mathbf{X}^{d}}\cos-\mathbf{y}{}^{\top}\mathbf{X}^{d} \).

• The recognizable/measurable part of \( e^{-\mathbf{h}{}^{\top}\mathbf{X}^{d}} \) is: \( e^{-\mathbf{p}{}^{\top}\mathbf{X}^{d}} \).

• A probabilistic belief in the exponential family: \[ \Pi=\frac{e^{-\mathbf{p}{}^{\top}\mathbf{X}^{d}}}{\mathbb{E}\left[e^{-\mathbf{p}{}^{\top}\mathbf{X}^{d}}\right]}=\frac{e^{-\mathbf{p}{}^{\top}\mathbf{X}^{d}}}{e^{-\mathbf{p}{}^{\top}\mathbf{f}}}. \]

• When \( \mathbf{y}=0 \), \( \Pi \) can induce a stochastic law that solve the equilibrium condition (Th. 4): \[ \mathbb{E}\left[\log\Pi\right]=-\mathbf{p}{}^{\top}\left(\mathbf{x}^{d}-\mathbf{f}\right)=0. \]

• When new sectors appear, the existing economy can absorb them.

• Coupling \( \mathbb{V}(\mathbf{f},\mathbf{x}^{d}) \) and \( \mathbb{V}(\tilde{\mathbf{f}},\tilde{\mathbf{x}}^{d}) \). A new economy \( \mathbb{V}((\mathbf{f},\tilde{\mathbf{f}}),(\mathbf{x}^{d},\tilde{\mathbf{x}}^{d})) \).

• A new belief of \( (\mathbf{X}^{d},\tilde{\mathbf{X}}^{d}) \) will guide the market movements (Th. 5).

Singularity

• Mixing two probabilistic beliefs may create singular points.

• No definitive new belief.

• Arbitrary guides for the market movements.

• Often occurs in the process of innovation or bubbles.

• A further type of beliefs on the market structure.

• Given a specific market growth pattern (\( \alpha \)-growth), the distribution is a Dirichlet distribution\[ \mbox{Dir}(\bar{\alpha}_{1},\dots,\bar{\alpha}_{K}) \] where \( \bar{\alpha}_{i} \) is the growth parameter of the \( i \)-sector's demand.

• The stochastic version of \( K \) sectors' structure is close to a stochastic game with \( K \) players.

• A utopian view of the limit of the market extension (\( K\rightarrow\infty \) and all \( \bar{\alpha}_{i}=\frac{1}{K} \)) is given by a prime number distribution.

• Conjecture: Some sectors are primes. New industries rely on these primes. New primes would appear but they are harder to be discovered.