Some Implications of the Ayres Warr Model

Schinzy

Introduction

Neoclassical economists are complacent with respect to peak oil largely because of their model, or lack thereof, for economic production. The Ayres-Warr model of economic production (2006) gives cause for considerable concern with respect to the effect of peak oil on the economy. The object of this post is to familiarize people with the Ayres-Warr model of economic production and explore some of its consequences which are in stark contrast to neoclassic economic theory.

We begin with a short summary of economic production models.

The neoclassical economic production model is a variant of the following model proposed by Solow in 1956:

$$Y(t)= [K(t)]^\alpha [A(t)L(t)]^{1-\alpha}.$$ (2.1)

Where

• $Y(t)$ is the economic production.
• $K(t)$ is capital.
• $L(t)$ is labor.
• $A(t)$ is an exogenous multiplier associated with technology.
• $0 < \alpha < 1$.

Because the term $A(t)$ cannot be measured, this is not really a model. It is an equation defining $A(t)$. It is as if I modeled a force $F$ with the equation $F=mP$ where $m$ is the mass and $P$ is exogenous pixie dust having to do with movement. Since $P$ cannot be measured my equation defines pixie dust and is always correct. Such a relation may have abstract theoretical use, but cannot be used for computations. Given the vagueness of this model it is not surprising the economists have so much trouble predicting recessions and explaining economic growth.

According to Strahan [6], very little work was done to determine a useful model. In 1985 Kümmel et al [3] proposed something closer to a model:

$$Y(t)=[E(t)]^{\alpha(t)}[L(t)]^{\beta(t)}[K(t)]^{1-(\alpha(t)+\beta(t))}$$ (2.2)

Where $E(t)$ is energy production. The coefficients $\alpha (t)$ and $\beta(t)$ can be estimated for a certain periods using statistics. This is closer to a real model than (2.1), but the unexplained dependence of the exponents on time still makes the model very imprecise.

Ayres and Warr have done considerable work on a real production model [1]. In [3], they proposed the following model (among others we won't discuss):

$$Y(t)= [U(t)]^\alpha[L(t)]^\beta[K(t)]^{1-(\alpha+\beta)}$$ (2.3)

where $U(t)=e(t)E(t)$ is exergy production. There is probably a positive constant in front of the equation, but we will assume this constant to be one. The function $e(t)$ satisfies $0<e(t)<1$ and can be interpreted as efficiency. Ayres and Warr had previously estimated the coefficient $e(t)$ for the 20'th century as a function of technology. As an illustration, they estimate $e(1900)\approx .03$, $e(2000) \approx .2$. Equation (2.3) is in fact a real model that can be worked with as the exponents no longer depend on time (time dependence is contained in the term $e(t)$). It is said that all models are false, but some are useful. We will examen some consequences of the Ayres-Warr model in order to determine its usefulness.

Remark 2   The homogeneity (the fact that the exponents on the right hand side sum to one) of (2.3) makes the model consistent with the hypothesis that the units for all terms are the same. The units for $Y(t)$, $L(t)$, and $K(t)$ can be expressed in currency. The units of exergy are energy. Therefore the model is consistent with currency having units of energy.

To estimate the exponents, Ayres and Warr proceeded as follows:

1. They accumulated data for all the terms in (2.3) for the U.S. in the 20'th century.
2. They normalized all terms by dividing by the quantities in 1900, for example $y(t) \stackrel{{\rm {def}}}{=}Y(t)/Y(1900)$.
3. They took the log of equation (2.3) (normalized) to obtain the equation $\log y(t) = \alpha \log u(t) + \beta \log l(t) + (1-(\alpha+\beta)) k(t)$.
4. They performed statistical linear regression to estimate the coefficients $\alpha$ and $\beta$.

Statistical tests affirm that the data is consistent with the model. Numerically, they obtained: $\alpha = .7$, $\beta = .02$, and thus $1 -(\alpha + \beta)= .28$. The exponent for $\beta$ is so small that the term $L(t)$ could be left out of the model, but this will not effect the analysis presented below. Note that the largest exponent corresponds to exergy. Labor is calculated using population data, so leaving labor out of the model would indicate that economic output is largly independent of population size. This can be explained if population is one possible expression of economic (and hence exergy) production, but not a necessay one.

There is a subtlety in using statistics. Statistics can tell you if your model is consistent with the data, statistics will not say anything about causality. For example if you do a linear regression of spending as a function of wealth, you might find a very good fit. But a rich man will probably not become poor if he decreases his spending, and a poor man will probably not become rich by increasing his spending. The neoclassical response to these results would be to rewrite (2.3)

$$U(t)=\left ( Y(t)[L(t)]^{-\beta}[K(t)]^{\alpha+\beta-1} \right )^{\frac{1}{\alpha}}$$ (2.4)

assuming that exergy is the dependant variable and that exogenous technology makes the economy grow. Then one could compute future exergy production based on the future size of the economy. This seems ludicrous given the dependence of the economy on exergy production, but this is precisely how agencies like the IEA predict future demand for oil. In fact many people make this error which is the primary reason for this post.

In [2], Ayres and Warr produced the feedback diagram in Figure 1 to illustrate how they believe cause and effect work in their model.

Figure 1: Ayres Warr 2006

We suggest another feedback mechanism below.

Analysis

We now analyze some consequences of (2.3) on prices.

Let $p(t)$ be the price per unit of oil, $S(t)$ be the quantity (supply) of oil produced, and $c(t)$ be the proportion of GNP spent on oil. Then we have $0<c(t)<1$ and

$$p(t)S(t)=c(t)Y(t).$$ (3.5)

We make the following simplifying assumptions:

A1

All Energy comes from oil produced in quantity $S(t)$.

A2

$S(t)=C_1E(t)$, where $C_1>0$.

A3

$[L(t)]^\beta[K(t)]^{1-(\alpha+\beta)}=C_2$.

Combining (3.5) and (2.3) and solving for $p(t)$ using our assumptions, we obtain:

$$p(t)= C_3 c(t)[e(t)]^{.7}[E(t)]^{-.3}$$ (3.6)

where $C_3$ is a positive constant.

Consequences

The first remark is that all economists agree that energy production is important part of economic activity and drops in supply can cause recessions. This dependence does not appear in neoclassical growth models, but does appear in the Ayres-Warr production model, a clear advantage for the Ayres-Warr model.

From equation (3.6), assuming $c(t)$ to be constant, we obtain:

• If $E(t)$ decreases by 10% then $p(t)$ increases by 3%.
• If $E(t)$ decreases by 50% then $p(t)$ increases by 23%.

This contrasts to neoclassical theory where much larger price increases are expected: a 10% decrease in energy supply would cause a 10% increase in price, a halving of supply should double the price. The explanation for the smaller price change comes from the fact that according to the Ayres Warr model, a decrease in energy supplies produces a decrease in economic production (see (2.3)) and thus a decrease in demand. For example if $E(t)$ decreases by 50%, economic production decreases by 38%.

Next we observe that if $e(t)$ increases by 10% then both $p(t)$ and $Y(t)$ increase by 7%. This is an excellent result for the Ayres-Warr model because this initially surprising fact has been noted many times. In the 19'th century, as steam engines became more efficient, they proliferated and the market for coal increased. The same can be said of computer technology, as processing power became cheaper (more efficient) the market for computers increased rather than decreased. This can be explained as follows, if a liter of oil can do a great deal of work, it becomes more valuable, so that efficiency is in fact the best investment for producing economic growth. In standard neoclassical economic theory, scarcity creates value, in the Ayres-Warr model, the ability to do work creates value. Efficiency is bounded above by one, so there is a theoretical limit to how much efficiency can drive economic growth. This observation implies that efficiency is not a good tool with which to control global emissions.

Feedback mechanism

We conjecture the following feedback mechanism. Recent posts on the work of James Hamilton [4] regarding recessions linked to increases in the proportion of GNP devoted to oil supports the notion that the function $c(t)$ is intimately involved in the feedback mechanism. Our analysis is based on the following two hypotheses:

H1

When exergy is produced in increased quantities, the value of the work produced exceeds the marginal cost of increased exergy.

H2

Oil is addictive in the sense that once consumers and business get used to using it, it takes very large price increases to cause people to buy less. Thus when energy prices rise, $c(t)$ increases.

H1 essentially says that we do not increase exergy production unless we can make a profit. Thus the sales price exeeds the cost of production. If consumers are willing to buy exergy, it is to do work which is valued at least at the price of the exergy.

H2 is observed empirically and contrasts with other sectors of the economy in which high prices will immediately decrease demand and for which a price rise could actually decrease the proportion of the economy devoted to that sector, for example the luxury sector. H2 is related to H1 in that it occurs because the value of work provided by exergy is much larger than the marginal cost of producing exergy, therefore most consumers would pay higher than market prices for oil if they had no choice.

When oil is produced in increasing quantities, exergy increases. Therefore by H1, more goods and services are produced and the economy becomes larger.

When oil is produced in decreasing quantities, the price initially goes up rapidly. By H2, $c(t)$ increases, but no new goods or services are associated with the increase in $c(t)$. Therefore spending must be cut elsewhere to balance budgets. Thus earnings in other parts of the economy come under pressure creating a double vice for consumers: at the same time their energy costs are rising, earnings come under pressure and their jobs become less secure causing them to become cautious and decrease spending even more. Moreover, some industries and/or consumers are unable to pay the higher price, and therefore less goods and services are available in the economy meaning the economy has decreased.

We note that there are two ways to increase exergy supplies, one way is to increase energy supplies $E(t)$, the other is to increase efficiency $e(t)$. Increasing energy supplies decreases prices, thus increasing economic output. Increasing efficiency increases both prices and economic output. It is excellent for both energy producers and the economy. We speculate that increasing efficiency stimulates the economy horizontally, that is, it expands the middle class. Just as progress in computer technology created a feedback loop in which a larger number of consumers purchased computers stimulated research into better computing technology, the 7 fold increase in efficiency in the use of energy during the 20'th century created a feed back loop in which increased efficiency brought more people into the middle class increasing the usefulness (and hence price) of energy stimulating production. Thus when Dick Cheney said that energy efficiency was virtuous but was not part of a solution to energy problems, he was not only wrong about the solution part, he was disserving the energy producers he was trying to serve. Note that $e(t) < 1$, so another 7 fold increase in $e(t)$ is not possible.

Finally we note that similar increases in $c(t)$ might have different effects on the economy at different times. Gail's post about the amount of debt in the economy suggests that the current economy is fragile, and that small changes in $c(t)$ might cause large numbers of people to fall out of the middle class.

Numerical experiment

We perform some simple numerical experiments to illustrate how changes in energy supplies affect prices and economic output in the Ayres-Warr model.

We first discretize time, then we model supply $S_t$ recursively as follows.

$$S(t)=S_t= S_{t-1}(1 - \frac{p_{t-1}}{p_{t-2}} a_t + \frac{p_{t-2}}{p_{t-1}} b_t).$$ (6.7)

Where $a$ is the aggregate rate of decline in oil fields with declining production and $b$ is the aggregate rate of increase in production for oil fields. We assume that if prices are increasing, producers will decrease the decline rate and increase the production rate. Inversely, if prices are declining, producers will increase the decline rate and decrease the rate of increase. This model was not produced using data, it was the first model we happened to think of with the preceding characteristics.

By mathematical induction we obtain

$$S_t = S_0\Pi_{i=0}^{t-1} (1- \frac{p_{1+i}}{p_i}a_{i+1} + \frac{p_i}{p_{i+1}} b_{i+1}).$$ (6.8)

Combining with (3.6) yields

$$p_t = C \Pi_{i=0}^{t-1} (1- \frac{p_{1+i}}{p_i}a_{i+1} + \frac{p_i}{p_{i+1}} b_{i+1})^{-.3}.$$ (6.9)

In each experiment we assume $S_0 = p_0 = Y_0 = 1$, $c(t)$ to be constant, and use 10 time intervals (the model starts at $t=2$). In each graph $Y_t$ is black, $p_t$ is red, and $S_t$ is blue.

For the first experiment, Figure 2, we take $a=0$ and $b=.1$, so that production is increasing.

Figure 2: $a=0$ $b=.1$

Price moves down 20% as quantity increases by 120% and economic production by 60%. This is similar to what one would expect from the neoclassical model as well. The difference being that the economic growth would come from exogenous technology rather than increased exergy production.

In Figure 3, we use $a=.1$ and $b=0$, production is falling sharply.

Figure 3: $a=.1$ $b=0$

After 8 time units, the price moves up 30% as quantity drops by 55% and production drops by almost 50%. This contrasts starkly with what one would expect from the neoclassical model. If production dropped by 50% and economic production where constant, one would expect a 100% price rise rather than a 30% price rise. If economic growth occurred, the price rise would be higher.

In Figure 4, we model the peak oil scenario with $b_t$ decreasing: $b_t = .1 - t \times .01$ and $a_t$ increasing: $a_t = .01 \times t$.

Figure 4: Ayres Warr peak

The graph looks asymmetric because the data is asymmetric. The post peak decline rate is slightly faster than the rate of increase with, $S_2=1$, $S_5=1.106$ max, $S_8=.997$, and $S_{10} = .83$.

In Figure 5 we make a neoclassical simulation with 2% growth per time unit starting with $t=3$. Increased economic growth leads to higher prices but decline rates are sensibly the same.

Figure 5: neoclassical peak

Comparison with empirical data

Between 1998 and July 2008, oil price was multiplied by 15, oil production was up 10% and world GDP was up by about 50%. This is much closer to the neoclassical prediction than to the Ayres-Warr model. However Since July 2008 oil price has declined by 2/3 bringing the total price rise to 500%. This is possibly consistent with the Ayres-Warr model as we made some simplifying assumptions (oil is only 35% of energy production). Furthermore the current economic contraction is not over and we do not yet know it's full extent. In any case, the Ayres Warr model should not be seen as an exact model, it should be seen as a time averaged model (missing perhaps a stochastic term) because the feed back mechanisms take some time to work themselves through.

It is possible that the Ayres Warr growth model is a good tool for determining financial bubbles and troughs, periods in which the markets over or underestimate the value of assets due to an over or under estimation of future economic output.

We make one last remark. Equation (3.6) can be seen as a paradigm for economic growth. It indicates two ways for producers in any industry to raise prices. One way is to increase efficiency, bringing more people into the market which gets harder over time, the other is to raise $c(t)$, or the proportion of GNP devoted to the given industry. For example the financial sector has gone from $c_f(1980) \approx .05$ to $c_f(2007) \approx .08$ (something Jérome à Paris once called "capturing wealth'' as opposed to "creating wealth''). It could be that a major industry which is too successful at raising $c(t)$ without a corresponding rise in goods, services, or efficiency can cause recessions. This is not really news, it is why we have antitrust laws.

Bibliography

1

Robert Ayres and Benjamin Warr.
The Economic Growth Engine: How Energy and Work Drive Material Prosperity.
Edward Elgar Publishing, 2009.

2

Robert Ayres and Benjamin Warr.
The Economic Growth Engine: How Energy and Work Drive Material Prosperity.
Structural Change and Economic Dynamics, 2006.

3

Robert Ayres and Benjamin Warr.
Economic Growth, Technological Progress and Energy Use in the US over the Last Century: Identifying Common Trends and Structural Change in Macroeconomic Time Series.
INSEAD, 2006.

4

J. Hamilton Causes and consequences of the oil shock of 2007-08.

Brookings Papers on Economic Activity, 2009.

5

R. Kümmel, W. Strassl, Grossner, Eichhorn.
Technical progress and energy dependant production functions.
Journal of Economics, 1985.

6

David Strahan.
The Last Oil Shock.
John Murray, 2007.

Achnowledgements:

The author would like to thank Adrien Blanchet, Manfred Pfluegal, and David Epstein for stimulating discussions.