Role of Reliable Energy in Capital Formation

14 Feb

Introduction

Capital formation, the accumulation of physical assets like machinery, infrastructure, and technology, is a fundamental driver of economic growth. It enhances productive capacity, increases output, and improves living standards. The dependability, reliability, and certainty of key inputs to production, particularly energy, are crucial factors influencing capital formation. Uncertainty in these inputs can deter investment, while reliability can stimulate it. This analysis explores the implications of input dependability on capital formation through mathematical models and proofs, referencing key literature in economics.

The role of energy in the production function

Firms invest in capital when they expect future returns to outweigh costs. Dependable inputs like energy reduce production uncertainty, making expected returns more predictable. Conversely, unreliable inputs increase risk, potentially deterring investment. A firm’s production function can be represented as: $Y = F(K, L, E) \label{eq:1}$ where $Y$ is output; $K$ is capital; $L$ is labor; and $E$ is energy input. Dependability in $E$ ensures that capital $K$ operates at optimal levels, maximizing $Y$ . Furthermore, economist Robert Solow showed that capital and labor accounted for 14% of economic growth in the industrial age.¹ Robert and Edward Ayres suggest that the increasing thermodynamic efficiency with which energy and raw materials are converted into work likely accounts for the remainder.² Therefore energy is a very important component of the production function.

Mathematical models of investment under uncertainty

Investment under uncertainty can be modeled using real options theory. ³. Firms hold the option to invest and will do so when the expected net present value (NPV) exceeds the option value of waiting. The traditional NPV rule states that a firm should invest if: $NPV = \sum_{t=0}^{\infty} \frac{R_t}{(1 + r)^t} - I > 0 \label{eq:2}$ where $R_t$ is the expected profit returned in period $t$ ; $r$ is the discount rate; and $I$ is the initial investment cost. The required return on investment increases with this risk as per the capital asset pricing model (CAPM): $r = r_f + \beta (r_m - r_f) \label{eq:13}$ where $r$ is the required return; $r_f$ the risk-free rate; $\beta$ is the beta coefficient, which is a measure of systematic risk; and $r_m$ is the expected market return. This implies that uncertainty in energy supply increases $\beta$ , raising the cost of capital $r$ , and reducing $NPV$ . Under uncertainty in energy input $E$ , the expected profits $R_t$ also become uncertain, and firms may require a higher threshold for investment. To account for this, we can define $E_t$ to be a stochastic process representing energy input uncertainty. Profits depend on $E_t$ : $q_t = P \cdot F(K, L, E_t) - C(E_t) - wL \label{eq:3}$ where $P$ is the output price; $C(E_t)$ is a cost function dependent on energy input; and $w$ is the wage rate. Using real options, the value of waiting to invest $(V_w)$ must be considered.⁴ ⁵ ⁶ The firm invests when: $NPV \geq V_w \label{eq:4}$

Assuming $E_t$ follows a geometric Brownian motion given by: $dE_t = \mu E_t dt + \sigma E_t dW_t \label{eq:5}$ where $\mu$ is percentage drift; $\sigma$ represents percentage volatility; and $dW_t$ is the Wiener process, the value of the investment opportunity $(F(E_t))$ satisfies the differential equation: $\frac{1}{2} \sigma^2 E_t^2 F''(E_t) + \mu E_t F'(E_t) - r F(E_t) + q_t(E_t) = 0 \label{eq:6}$

Solving this, we find the critical value $E^*$ at which the firm should invest.

Production function and input reliability

Consider a Cobb-Douglas production function: $Y = K^\alpha L^\beta E^\gamma \text{ with } \alpha + \beta + \gamma = 1 \label{eq:7}$ The marginal product of capital $(MPK)$ is: $MPK = \frac{\partial Y}{\partial K} = \alpha K^{\alpha - 1} L^\beta E^\gamma \label{eq:8}$ Dependable energy input $(E)$ increases $MPK$ , encouraging more capital investment.

Impact on capital accumulation dynamics

We can extend the Solow model to include energy input by starting with $\dot{K} = s Y - \delta K \label{eq:9}$ where $\dot{K}$ is the change in capital stock; $s$ is the savings rate; and $\delta$ is the depreciation rate, then substituting the production function to get $\dot{K} = s K^\alpha L^\beta E^\gamma - \delta K \label{eq:10}$ In the steady state, $\dot{K} = 0$ : $s K^\alpha L^\beta E^\gamma = \delta K \label{eq:11}$

Solving this for $K$ : $K = \frac{\delta}{(s L^\beta E^\gamma)^{\frac{1}{\alpha-2}}} \label{eq:12}$ This implies that dependable $E$ increases $E^\gamma$ , leading to a higher steady-state capital stock $K$ .

Conclusion

Mathematical models demonstrate that input uncertainty raises the cost of capital, reduces expected profits, and deters investment. Studies have also shown that energy supply uncertainty negatively affects investment. For example, Nicholas Bloom found that uncertainty reduces investment and employment⁷, whereas John Elder & Apostolos Serletis demonstrated that energy price volatility decreases capital formation.⁸

A current real world example of this concept is the energy crisis in Germany. With an ambitious plan to move to renewable energy sources, Germany decommissioned its coal and nuclear power plants. This decision was made on the basis that Germany could use cheap Russian natural gas as a transition fuel because of infrastructure put in place when the country was divided post World War II. However, a slow bureaucratic process, COVID-19, and then the sanctions on Russian gas because of the war in Ukraine has led to Germany lagging behind on its green energy transition targets and forced into a situation where uncertainty caused energy prices to rise by 35%, severely impacting their industry and economy.⁹

This shows that dependability, reliability, and certainty of key production inputs, especially energy, significantly influence capital formation. Governments can ensure input dependability by invest in reliable energy infrastructure and diversifying energy sources minimizes the impact of disruptions. Capital formation can be encouraged with tax credits and subsidies designed to offset higher costs due to input uncertainty. Stable regulatory environments can reduce overall investment risk. Ensuring reliable energy supplies is vital for sustained economic growth and capital accumulation.

References

Abramovitz, Moses. Thinking about growth: And other essays on economic growth and welfare. Cambridge University Press, 1989.
Ayres, Robert U. and Edward H. Ayres. Crossing the energy divide: moving from fossil fuel dependence to a clean-energy future. Pearson Prentice Hall, 2009.
Bloom, Nicholas. The impact of uncertainty shocks. econometrica 77.3 (2009): 623-685.
Dixit, Avinash K. and Robert S. Pindyck. Investment under uncertainty. Princeton university press, 1994.
Elder, John, and Apostolos Serletis. Oil price uncertainty. Journal of money, credit and banking 42.6 (2010): 1137-1159.
Foo Sing, Tien, and Kanak Patel. Empirical evaluation of the value of waiting to invest. Journal of Property Investment & Finance 19.6 (2001): 535-553.
Ingersoll Jr, Jonathan E., and Stephen A. Ross. Waiting to invest: Investment and uncertainty. Journal of Business (1992): 1-29.
Lontay, Oliver. Germany’s Energy Crisis: Europe’s Leading Economy is Falling Behind Harvard International Review. 2024.
https://hir.harvard.edu/germanys-energy-crisis-europes-leading-economy-is-falling-behind/
McDonald, Robert, and Daniel Siegel. The value of waiting to invest. The quarterly journal of economics 101.4 (1986): 707-727.

Bibliography

Bernanke, Ben S. Irreversibility, uncertainty, and cyclical investment. The quarterly journal of economics 98.1 (1983): 85-106.
Hartman, Richard. The effects of price and cost uncertainty on investment. Journal of economic theory 5.2 (1972): 258-266.
Pindyck, Robert S. Irreversible investment, capacity choice, and the value of the firm. (1986).
Solow, Robert M. A contribution to the theory of economic growth. The quarterly journal of economics 70.1 (1956): 65-94.

Moses Abramovitz, Thinking about growth: And other essays on economic growth and welfare. (Cambridge University Press, 1989).
Robert U. Ayres and Edward H. Ayres. Crossing the energy divide: moving from fossil fuel dependence to a clean-energy future. (Pearson Prentice Hall, 2009).
Avinash K. Dixit and Robert S. Pindyck. Investment under uncertainty. (Princeton university press, 1994).
Tien Foo Sing and Kanak Patel. Empirical evaluation of the value of waiting to invest. Journal of Property Investment & Finance 19.6 (2001): 535-553.
Jonathan E. Ingersoll Jr and Stephen A. Ross. Waiting to invest: Investment and uncertainty. Journal of Business (1992): 1-29.
Robert McDonald and Daniel Siegel. The value of waiting to invest. The quarterly journal of economics 101.4 (1986): 707-727.
Nicholas Bloom. The impact of uncertainty shocks. (econometrica 77.3 (2009)) pages 623-685.
John Elder and Apostolos Serletis. Oil price uncertainty. (Journal of money, credit and banking 42.6 (2010)) pages 1137-1159.
Oliver Lontay. Germany’s Energy Crisis: Europe’s Leading Economy is Falling Behind Harvard International Review. 2024.

Andrew Scobie

Enoda Ltd Founder, Chief Technology & Product Officer