Elementary Microeconomics of the Talmudic Rule

TL;DR

This paper analyzes the Talmudic 1/N investment rule using microeconomic theory, deriving a utility function, examining supply and demand effects, and exploring liquidity provision benefits.

Contribution

It introduces a Cobb-Douglas utility function for Talmudic investors and analyzes their market behavior and liquidity provision strategies.

Findings

Derived a Cobb-Douglas utility function for Talmudic agents.

Compared individual supply and demand to market depth.

Showed benefits of liquidity provision for Talmudic investors.

Abstract

This paper takes a look at the Talmudic rule aka the 1/N rule aka the uniform investment strategy from the viewpoint of elementary microeconomics. Specifically, we derive the cardinal utility function for a Talmud-obeying agent which happens to have the Cobb-Douglas form. Further, we investigate individual supply and demand due to rebalancing and compare them to market depth of an exchange. Finally, we discuss how operating as a liquidity provider can benefit the Talmud-obeying agent with every exchange transaction in terms of the identified utility function.

Equations

$$\frac{\Delta m}{\Delta q}$$

$$\frac{dm}{dq}$$

$$\frac{dm}{m}+\frac{dq}{q}$$

$$\ln m+\ln q$$

$$mq$$

$$p_s=\frac{p_0{q}_0^2}{(q_0-q_s{)}^2},p_d=\frac{p_0{q}_0^2}{(q_0+q_d{)}^2}.$$

$${\int }_0^{\infty }\frac{p_0{q}_0^2}{(q_0+q_d{)}^2}d{q}_d=p_0{q}_0.$$

$$\frac{d{p}_s}{d{q}_s}$$

$$u_0$$

$$\frac{u_1}{u_0}=\frac{u(p_1{q}_1,q_1)}{u(p_0{q}_0,q_0)}=\frac{\sqrt{m_1{q}_1}}{\sqrt{m_0{q}_0}}=\frac{p_0+p_1}{2\sqrt{p_0{p}_1}}>1.$$

$$\%\Delta u=2\cdot \frac{|u_0-u_1|}{u_0+u_1},\%\Delta p=2\cdot \frac{|p_0-p_1|}{p_0+p_1}.$$

$$\frac{u_1}{u_0}$$

$$\%\Delta u$$

$$p$$

$$\frac{\Delta m}{\Delta q}$$

$$\frac{dm}{dq}$$

$$\alpha \cdot \frac{dm}{m}+\beta \cdot \frac{dq}{q}$$

$$\alpha \ln m+\beta \ln q$$

$$m^{\alpha }{q}^{\beta }$$

$$u(m,q)$$

$$f(t)$$

$$f(1)$$

$$u(m,q)$$

$$p_s$$

$$q_s$$

$$\frac{d{p}_s}{d{q}_s}$$

$$\frac{u_1}{u_0}=\frac{\alpha p_0+\beta p_1}{p_0^{\alpha }{p}_1^{\beta }}\Rightarrow \%\Delta u=\frac{\alpha \beta }{2}\cdot (\%\Delta p{)}^2+O\left((\%\Delta p{)}^3\right)\text{at \%Δp=0}.$$

Full text

1 Deriving the cardinal utility function

The Talmudic rule111 “A person should always divide his money into three; he should bury one-third in the ground, and invest one-third in business, and keep one-third in his possession” (Bava Metzia 42a). requires one to store equal amount of goods and money with respect to the market price of the goods at any given time. Let us assume that an agent has initially bought a quantity $q_{0}$ of goods at the price $p_{0}$ and is left with $m_{0}$ units of money. Then we can derive the cardinal utility function $u(m,q)$ of that agent measured in the units of money. First, the initial configuration requires the budget $2m_{0}=2p_{0}q_{0}$, thus we have $u(m_{0},q_{0})=2p_{0}q_{0}$. Second, having $km$ of money and $kq$ of goods should be exactly $k$ times more preferable than having $m$ of money and $q$ of goods. Finally, notice that the Talmud-obeying agent should always be ready to exchange $|\Delta q|$ of goods for $|\Delta m|$ of money (and vice versa) as long as the exchange rate is equal to the ratio between the resulting amounts of money and goods in their possession:

[TABLE]

The latter means that $u(m,q)$ remains constant along hyperbolas $mq=C$. Thus we have $u(m,q)=f(mq)$ for some function $f(t)$. As noticed earlier, $u(km,kq)=ku(m,q)$, hence $f(t)=\sqrt{t}f(1)$ and $u(m,q)=f(1)\sqrt{mq}$. Taking into account $u(p_{0}q_{0},q_{0})=2p_{0}q_{0}$, we conclude with $f(1)=2\sqrt{p_{0}}$ and $u(m,q)=2\sqrt{p_{0}mq}$. Note that the latter utility function has the Cobb–Douglas form $Am^{\alpha}q^{1-\alpha}$ with efficiency $A=2\sqrt{p_{0}}$ and elasticity $\alpha=1/2$. The resulting indifference map is illustrated in Figure 2. See also Appendix A.

2 Supply and demand due to rebalancing

Note that the indifference curve $mq=m_{0}q_{0}$ is defined in terms of amounts $m$ and $q$ that are in possession of our Talmud-obeying agent, so it cannot be used directly to find the supply and demand curves. Instead, we should consider supplied $q_{s}$ and demanded $q_{d}$ quantities due to rebalancing required by the Talmudic rule when the price changes.

Specifically, when the price goes up, our agent is to sell some $q_{s}=q_{0}-q$ of their $q_{0}$ units of goods, resulting in $q$ units left in their possession. Note that since $q>0$, we have $q_{s}<q_{0}$. Conversely, when the price goes down, the agent buys $q_{d}=q-q_{0}$ units of goods in addition to their $q_{0}$ units and is left with the total of $q$ units. As the Talmudic rule implies $m=pq$ at any given price $p$, we can rewrite the indifference curve as $pq^{2}=p_{0}q_{0}^{2}$ and use it in order to obtain the following supply and demand curves:

[TABLE]

These supply and demand curves are illustrated in Figure 2. Notice that due to limited resources $q_{0}$ and $m_{0}$ in possession of our Talmud-obeying agent, the supply curve has an asymptote $q=q_{0}$, whereas the total area below the demand curve is equal to $m_{0}$:

[TABLE]

Figure 2 might look familiar. Indeed, it resembles market depth that is provided by exchanges, when the current price is equal to $p_{0}$. Notice that near $p=p_{0}$, our supply and demand curves behave linearly with the same absolute slope. The similar behavior can be noticed about market depth at relatively stable conditions near the current price. The latter observation about market depth has a rather curious consequence. Namely, we can estimate the budget $u_{0}=2p_{0}q_{0}$ that is needed for a Talmud-obeying agent in order to provide as much liquidity near the current price as the whole exchange does:

[TABLE]

3 Operating as a market maker

In the previous sections, we applied the Talmudic rule to infinitesimal changes in price. However, situation is slightly different for any finite non-zero change in price. As discussed in [1, Section 2], the Talmudic rule then implies that the geometric mean of the quantity of goods and the amount of money in possession always increases:

[TABLE]

Let us denote the relative differences in utility and price, respectively, as follows:

[TABLE]

Then, we can use (1) to calculate the relative growth in utility $\%\Delta u$ after the transaction that corresponds to a relative change in price $\%\Delta p$, assuming $p_{1}>p_{0}$:

[TABLE]

For example, $\%\Delta p=2\%$ results in $\%\Delta u\approx 0.5\text{\textpertenthousand}$. With such a transaction once a day on average and assuming negligible transaction fees, the annual interest rate in terms of our Talmudic utility is about $1.84\%$. As one would expect, the market maker has to face a trade-off between the frequency of transactions and the utility growth per transaction. In turn, [1] suggests that to find the optimal values of $\%\Delta p$ may not be a trivial problem.

Appendix A The weighted Talmudic rule

This section considers an arbitrarily weighted version of the Talmudic rule. Specifically, let $\alpha,\beta\in(0,1)$ and $\alpha+\beta=1$. Then the weighted Talmudic rule would require one to store portion $\alpha$ of value in money and $\beta$ in goods with respect to the market price.

Let $u(m,q)$ be the corresponding cardinal utility function. If $u_{0}=u(m_{0},q_{0})$, then we have $m_{0}=\alpha u_{0}$ and $p_{0}q_{0}=\beta u_{0}$. Note $u(km,kq)=ku(m,q)$. The rest is straightforward:

[TABLE]

Further, we use the indifference curve $m^{\alpha}q^{\beta}=m_{0}^{\alpha}q_{0}^{\beta}$ with $q_{s}=q_{0}-q$ and $q_{d}=q-q_{0}$ in order to obtain the supply and demand curves due to weighted rebalancing:

[TABLE]

For a finite change in price from $p_{0}$ to $p_{1}$, we have the ratio between $u_{1}$ and $u_{0}$ equal to the weighted arithmetic mean of the prices divided by their weighted geometric mean:

[TABLE]

Let us notice that the case of $\alpha=\beta=1/2$ is special in the following two aspects. First, it is the only proportion in which $u_{1}/u_{0}$ ratio is invariant to the direction of change in price, the latter being crucial for unpredictable markets. Second, $\alpha=\beta=1/2$ maximizes $\%\Delta u$ for small changes in prices which can be very useful for liquidity providers.