Irreducible complexities: from Gödel and Turing to the paradigm of Imperfect Knowledge Economics

Abstract

The paper discusses synthetically an epistemological question in the field of economics: how to translate a real problem into formal terms without a substantial loss of significance for its solution in policy making. The discussion will challenge the plausibility of the basic assumption of ultra-rational subjects who act independently of each other and will propose a research programme in Imperfect Knowledge Economics (in an Appendix the latter is compared with the mainstream model founded on Rational Expectation Hypothesis). References to some results of the debate in mathematics, but also in computer science, are produced in support.

Appendix: Ike models vs Reh models

Starting from Frydman and Goldberg (2008) we can directly compare Ike and Reh models on the basis of a basic supply and demand analysis. The equilibrium market price at a point in time is:

$$\begin{aligned} P_t =a_t +b_t X_t +c_t {\hat{P}}_{t\left| {t+1} \right. }, \end{aligned}$$

where:

• $X_{t}$ is a set of causal variables:¹³

  $$\begin{aligned} X_t =\mu ^{X}+X_{t-1} +\varepsilon _t^X \end{aligned}$$

• ${\hat{P}}_{t\left| {t+1} \right. }$ is an aggregate of market forecasts settled starting from a vector of causal variables that characterizes the union of information sets used by market participants:

  $$\begin{aligned} {\hat{P}}_{t\left| {t+1} \right. } =\alpha _t +\beta _t Z_t, \end{aligned}$$

  where $Z_t =\mu ^{Z}+Z_{t-1} +\varepsilon _t^{Z}$

Applying the Reh (the superscript “RE” denotes a Reh representation), ${\hat{P}}_{t\left| {t+1} \right. }^{RE} =E(P_{t+1} \left| {X_t )} \right.$ and the market price at $t +1$ is:

$$\begin{aligned} P_{t+1}^{\textit{RE}} =\frac{a(1-c)+b\mu ^{X}}{(1-c)^{2}}+\frac{b}{(1-c)}X_t +\varepsilon _{t+1}, \end{aligned}$$

where $\varepsilon _{t+1} =\frac{b}{(1-c)}\varepsilon _{t+1}^X$. The average representative forecast of the price formed at $t$ is:

$$\begin{aligned} {\hat{P}}_{t\left| {t+1} \right. }^{\textit{RE}} =\frac{a(1-c)+b\mu ^{X}}{(1-c)^{2}}+\frac{b}{(1-c)}X_t \end{aligned}$$

The implication is that $Z_t =X_t$.

If we assume a diversity of the individual forecasting strategies (e.g. type 1 and type 2 individuals), we have that:

$$\begin{aligned} {\hat{P}}_{t\left| {t+1} \right. }^{\textit{RE}} =\omega {\hat{P}}_{t\left| {t+1} \right. }^1 +(1-\omega ){\hat{P}}_{t\left| {t+1} \right. }^2 =\omega (\alpha ^{1}+\beta ^{1}X_t )+(1-\omega )(\alpha ^{2}+\beta ^{2}X_t ), \end{aligned}$$

where the weight $0<\omega >1$ represents the importance of type 1 individuals in influencing the equilibrium price. Then, unless $\omega =0$, the model presumes persistent forecasting errors.

The Ike (the superscript “IK” denotes Ike) representation of the aggregate of individual’s forecasts ${\hat{P}}_{t\left| {t+1} \right. }^{\textit{IK}}$ is the aggregation of individual’s forecast that can change over time according to movements in the causal variables $(Z)$ or revisions of forecasting strategies ($\beta$), so that:¹⁴

$$\begin{aligned} {\hat{P}}_{t\left| {t+1} \right. }^{\textit{IK}} \longrightarrow P_{t\left| {t+1} \right. }^i =\beta _t^i Z_t^i, \end{aligned}$$

where $i$, in the context of asset, or currency (the last will be our reference point) market, can be seen, alternatively, as a bull or a bear strategy (we will adopt this hypothesis shortly).

Since the revisions and diversity of forecasting strategies characterize real world markets, we can write market price equation as:

$$\begin{aligned} P_t =P_t^{\textit{RE}} +c\left( {\hat{P}}_{t\left| {t+1} \right. }^{\textit{IK}} -{\hat{P}}_{t\left| {t+1} \right. }^{\textit{RE}}\right) , \end{aligned}$$

where $P_t^{\textit{RE}}$, in the context of currency markets, can be seen as the PPP exchange rate.

Now we can explain a total change in an individual’s forecast as:

$$\begin{aligned} {\hat{P}}_{t\left| {t+1} \right. }^i -{\hat{P}}_{t-1\left| t \right. }^i =D\!{\hat{P}}_{t\left| {t+1} \right. }^i +\varepsilon _t^{i}, \end{aligned}$$

where the trend change $D\!{\hat{P}}_{t\left| {t+1} \right. }^i =\Delta \beta _t^i Z_t^i +\beta _{t-1}^i \mu ^{Z^{i}}(\Delta \beta _t^{i}$ shows the revisions of forecasting strategy, while $\beta _{t-1}^i \mu ^{Z^{i}}$ is the baseline drift).

Ike recognizes conservatism constraints in revision of forecasting strategies, so that:

$$\begin{aligned} \left| {\Delta \beta _t^i Z_t^i } \right| <\left| {\beta _{t-1}^i \mu ^{Z^{i}}} \right| \quad \mathrm {and} \quad \left| {\Delta \beta _t^i \mu ^{Z^{i}}} \right| <\left| {\beta _{t-1}^i \mu ^{Z^{i}}} \right| \end{aligned}$$

How Frydman–Goldberg note, “economists sometimes recognize that changes in causal variables may lead to revisions of individual’s forecasting strategies. But, when they do, they rely on pre-existing rules, like Baye’s formula. By contrast, our Ike formulation recognizes that individuals do not endlessly obey pre-existing rules in deciding on when and how to alter their forecasting strategies. Indeed, the decision to revise one’s forecasting strategy depends on many factors, including prior forecasting success, economic and political developments, emotions, or ... the size of the departure of the exchange rate from PPP” (2008, p. 50).

Within the specified framework we can suppose markets populated by bulls and bears taking, respectively, long $(L)$ or short $(S)$ positions, with expected returns given by:

$$\begin{aligned} {\hat{R}}_{t\left| {t+1} \right. }^L&= {\hat{P}}_{t\left| {t+1} \right. }^L -P_{t} >0 \\ {\hat{R}}_{t\left| {t+1} \right. }^S&= P_t -{\hat{P}}_{t\left| {t+1} \right. }^S >0, \end{aligned}$$

where ${\hat{P}}_{t\left| {t+1} \right. }^L =\beta _t^L Z_t^L$ and ${\hat{P}}_{t\left| {t+1} \right. }^S =\beta _t^S Z_t^S$ represent aggregates of the exchange rate forecasts of two types of individuals.

An individual’s expected loss from speculation can be written respectively as:

$$\begin{aligned} {\hat{l}}_{t\left| {t+1} \right. }^{i,L}&= E_t^{i} \left[ R_{t+1}^L <0\left| {Z_t^i } \right. \right] <0 \\ {\hat{l}}_{t\left| {t+1} \right. }^{i,S}&= E_t^{i} \left[ R_{t+1}^S <0\left| {Z_t^i } \right. \right] <0, \end{aligned}$$

where $E_t^i \Big [R_{t+1}^L <0\left| {Z_t^i } \right. \Big ]$ and $E_t^i \Big [R_{t+1}^S >0\left| {Z_t^i } \right. \Big ]$ denote the expected value of the realizations on $R_{t+1}^L$ and $R_{t+1}^S$, given by $R_{t+1}^L =P_{t+1} -P_t$ and $R_{t+1}^S =P_t -P_{t+1}$

Individual bull and bear radically revises her forecasting strategies if:

$$\begin{aligned}&\frac{D{\hat{l}}_{t\left| {t+1} \right. }^{i,L} }{Dg{\hat{a}}p_t^{i,L} }<0 \\&\frac{D{\hat{l}}_{t\left| {t+1} \right. }^{i,S} }{Dg{\hat{a}}p_t^{i,S} }>0, \end{aligned}$$

where $g{\hat{a}}p_t^i ={\hat{P}}_{t\left| {t+1} \right. }^i -{\hat{P}}_t^{\textit{i,PPT}}$ while ${\hat{P}}_t^{\textit{i,PPP}}$ denotes an individual’s assessment at $t$ of the PPP exchange rate.