Turing’s Decidability

Abstract

Entrepreneurship is “the” source of information within the market process. However, it is not explicit, written or available to everyone. Although Finance professionals continuously seek to formalize entrepreneurship, there are necessary conditions to establish formalization. We examine them and show that entrepreneurship is not “decidable”. We discover that an entrepreneur is anyone who creates decidable problems or discovers previously undecidable ones and makes them decidable. This discovery comes in handy when one has to choose in which part of the capital structure (of a business) to invest.

Appendices

Appendix I: Monetary Policy and Capital Structure

Interestingly, the so-called quantitative easing (QE) policies of the twenty-first century shed light on the impact of monetary policy on capital structure decisions. QE, in the United States, consists in monetizing US federal government deficits:

In Fig. 3.5, the amount of US dollars in circulation increases when US sovereign debt is purchased by the Fed. The US dollars created are subsequently deposited in the banking system and the wonders of fractional reserve expand them multiple times. In so doing, the “loan” interest rate drops. But since 2009, given the magnitude of QE and the impact of the Basel III regulatory framework, the loan interest rate dropped so significantly that the profitability of the banking system plunged. Forced by law to increase capital and liquidity, and with lower interest rate margins, corporate banks embarked on a path of steady diminishing returns, protected by a blanket of lower expected loan losses. This situation led banks to look beyond common places, to lend to high-yield companies. In so doing, they reversed the natural relationship between decidability and capital structure, discussed in this lesson. Banks, partly funded themselves with demand deposits and partly with debt, could only remain profitable if they earned additional investment banking fees. Therefore, roles reversed and banks became de facto the new providers of venture capital for those sectors that could afford capital markets services (i.e. origination, advisory, hedging). They are usually undecidable ventures and the flavour of the day at that time¹⁰ happened to be the US and Canadian oil and gas shale, exploration and production companies. Within the exploration and production (E&P) sub-space, the reversal in the decidability/capital structure suffered the most. The vehicle was secured lending in the form of borrowing base credit lines (Figs. 3.6, 3.7):

Fig. 3.5

Balance sheets of the Fed and US government under QE

Fig. 3.6

Oil exploration and production without quantitative easing, under normal conditions

Fig. 3.7

Oil exploration and production with quantitative easing and Basel III: The recipe for a lending crisis caused by policy makers

Indeed, the crash in the price of oil and gas was caused by factors beyond monetary policy. But, had the profitability of the banking sector not been decimated by regulations, banks would have not been pushed to the borrowing base lending sector in the volumes and at the prices they did. The contagion that took place during 2015 would have thus been much more limited.

Appendix II: Decidability and Economies of Scale

In any regular course of microeconomics, economies of scale may be formalized using the following function:

$$ Y=A\ {L}^{\alpha }\ {K}^{\beta } $$

Where:

• Y = total production (the real value of all goods produced)

• L = labour input (the total number of person-hours worked)

• K = capital input (the real value of all machinery, equip.)

• A = total factor productivity

• α and β are the output elasticities of capital and labour, respectively. These values are constants determined by available technology.

If

$$ \alpha \kern0.5em +\kern0.5em \beta =\kern0.5em 1, $$

The production function has constant returns to scale, meaning that doubling the usage of capital K and labour L will also double output Y.

For example, when capital and labour double:

$$ {\displaystyle \begin{array}{l}A\ {\left(2\kern0.5em \times \kern0.5em L\right)}^{\alpha }{\left(2\kern0.5em \times \kern0.5em K\right)}^{\beta }=\\ {}A\ {2}^a\times \kern0.5em {L}^{\alpha}\kern0.5em \times \kern0.5em {2}^{\beta}\times {K}^{\beta }=\\ {}A\ {2}^{\left(\alpha +\beta \right)}\kern0.5em \times \kern0.5em {L}^{\alpha }\ {K}^{\beta}\end{array}} $$

With a + β = 1 and reorganizing 2 A L^α K^β = 2 Y, which shows that Y (i.e. production) doubles too.

It is therefore easy to see that if:

$$ \alpha \kern0.5em +\kern0.5em \beta <1,\kern0.5em \mathrm{returns}\kern0.5em \mathrm{to}\kern0.5em \mathrm{scale}\ \mathrm{are}\ \mathrm{decreasing}, $$

And if:

$$ \alpha \kern0.5em +\kern0.5em \beta >1,\kern0.5em \mathrm{returns}\kern0.5em \mathrm{to}\kern0.5em \mathrm{scale}\ \mathrm{are}\ \mathrm{increasing}. $$

The above equation is known as the Cobb-Douglas production function, after Paul Douglas and Charles Cobb.¹¹ It assumes that there is already a decidable production process that has been identified, which by way of inputting fixed factors of production in a given proportion delivers the produce with a given return. This is by all means a mechanistic view, devoid of any economic sense. But this perspective has been taught and used for so long that we have come to believe in it unquestionably. It doesn’t have any economic sense because, to begin with, it is static. It does not explain how a product came to be, nor how will it remain competitive in the market. In other words, it completely misses the role of entrepreneurship and it assumes no flexibility in terms of resource reallocation. The fact that we cannot formalize such flexibility is not an excuse to ignore it.

With the development of information technology and the internet, a concept similar to “economies of scale” appeared, although this time it did have economic sense. I am referring to the concept of a “platform”. In a way, an online platform is a social institution. Its value derives from the spontaneous social cooperation that it brings to life and is therefore, by its very nature, a concrete form of undecidability. The fact that online platforms are valued by investors is proof that we humans acknowledge the concept of decidability. It is the cooperation that a platform enables what makes it a scalable resource, in a true economic, not mechanistic sense. As an investor, this is the real economies of scale that I look for in my investments allocated to equity.