Potential for economic and population growth without power growth

Suppose humans stopped increasing power consumption. How much would this limit economic or population growth?

Total energy consumption by humans per unit of time, i.e., aggregate power consumption (P), is identical to the product of: the human population (N), per capita output (income) per year (y), and the average amount of power used to produce a unit of output, which is equivalent to the reciprocal of average energy efficiency or productivity, () measured in economic output per unit of energy,

(1)

The units in are W = (people) x (USD/person) x (W/USD), where W stands for Watts (kg m² s⁻³), or Joules per second, and USD stands for real, purchasing power parity-adjusted United States Dollars. Taking the logarithm and time derivative, we can rewrite in terms of growth rates:

(2)

where the growth rate of quantity x is written . , the growth in per capita income, is the main measure of economic growth.

We estimate the geometric mean of power, population, and per capita income growth rates in this equation for the period 1965 to 2020 using global data on power from Our World in Data [19] and on population and per capita income from [20] in Fig 1. We calculate the implied growth rate of average productivity using . Power consumption is increasing at a rate of 2.07%/yr, slightly lower than Murphy’s [5] estimate of 2.3%/yr. Murphy’s slightly higher estimate results from adding data from the first half of the 1900s from [21] to our data. Either estimate is nearly enough to increase power consumption by a factor of 10 every century. Interestingly, the growth in power consumption results from 1.73% economic growth and 1.54% population growth per year, offset by 1.2% growth in average productivity per year.

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Fig 1. Global power consumption (P), population (N), per capita output (y), and average productivity of power () from 1965–2020.

Plot titles give geometric mean of growth rate in power (), population (), per capita output (), and average productivity of power (). Power data are from Our World in Data and population and output per capita data are from the World Bank.

https://doi.org/10.1371/journal.pone.0319217.g001

It is interesting to consider the possibility of population and economic growth in the limit of zero growth in power consumption. Any combination of and that yields will result in . Let us consider a scenario in which the recent historic growth rate in average productivity simply continues. If the population stays constant, per capita income could continue to grow at 1.2%/yr, roughly 69% of historical growth rates, without increasing energy consumption simply by exploiting increases in energy efficiency. Alternatively, if per capita income stopped increasing, the population could grow by 1.2%/yr, roughly 78% of its current growth rate, again only by exploiting increases in energy efficiency without increasing energy usage. Evidently, if historical increases in efficiency continue, they will be capable of driving significant economic and population growth even in the absence of growth in power consumption. Of course one cannot “choose” future growth in power usage, economic growth, and population growth. But this analysis suggests it is physically possible to have reasonable population and economic growth—even if power growth is contained–if historical progress in efficiency continues.

Delaying the limits to economic growth with slowing population growth

Those positing physical limits to growth often forecast dates by which human civilization will exhaust a key resource [4,5]. These forecasts tend to be based on forecasts of population growth from a period where death and birth rates diverged, so the population grew rapidly. However, the demographic transition has brought birth rates back down to death rates, or even below, in many countries [22].

The median forecasts of the United Nations, the Institute for Health Metrics and Evaluation, and the International Institute for Applied Systems Analysis all agree that global population growth will reach zero then become negative on a decadal timescale [11,12,22]. Although there is significant uncertainty in this type of forecast, particularly in the effect of high fertility sub-populations [23], these projections do motivate the consideration of a zero population growth scenario.

How much would zero population growth delay the waste heat limit to power growth? Murphy [5] estimates the time to waste-heat constraint based on the surface temperature of Earth reaching the boiling point of water (373 K) due to waste heat from power usage. Additional heating by increased greenhouse gases can be ignored because power usage (and waste heat required to generate and use that power) is assumed to grow exponentially so that it eventually dwarfs other terms. Solar geoengineering would not be a solution because, even if all sunlight were blocked, the time to the temperature limit would only marginally lengthen due to the exponential growth in waste heat. Per Murphy [5], the surface temperature T (in Kelvin) of the Earth in t years can be approximated by:

(3)

where W m⁻² is the solar flux at the top of the atmosphere, is the planet’s albedo (reflectivity), W m⁻² is 4 times the power output per unit area of Earth’s surface at time ,  ×  W m⁻² K⁻⁴ is the Stefan-Boltzmann constant, and K is an adjustment that accounts for greenhouse gas warming. Murphy assumes that future growth in power will match historical growth in power, which is %/yr in the data he examines, leading to the projection that Earth will reach the waste heat limit to growth in years.

Murphy [5] acknowledges that is an extreme simplification of the climate system intended only for a qualitative estimate. It is worth considering a more sophisticated global climate model that includes three-dimensional variation, moisture, detailed radiative physics, and clouds. The physical process that would certainly limit human habitability is the runaway greenhouse [24], which is associated with the surface temperature reaching 373 K, but not exactly the same thing. Different global climate models suggest that must be increased by between 10% [25] and more than 21% [26] in order to cause a runaway greenhouse, most likely due to differences in cloud behavior. If we make the approximation that waste heat from power generation affects the climate equivalently to solar heating, following [5], we arrive at a time until the runaway greenhouse () via consideration of planetary energy balance as

(4)

where f is the fractional increase in necessary to cause a runaway greenhouse without excess heat from power generation. Using %/yr, we obtain a limit to growth due to the runaway greenhouse in years whether we assume f = 0 . 1 or , confirming that Murphy’s methodology [5] yields the correct order of magnitude.

Assuming that and that and remain at their recent historical values (1965–2020) yields %/yr. At this lower growth rate, it will take  ∼ 5 times longer to reach uninhabitable conditions, i.e., between years, providing an additional millennium of runway. Alternatively if the economic growth rate were reduced from %/yr to %/yr, a scenario we discussed in the last section, only a 27% reduction, the time to reach uninhabitable conditions due to power consumption would be delayed indefinitely in the model.

If population growth actually turns negative, the deadline can be completely lifted in a different way. In the “Empty Planet” solutions of Jones [27], human population can get stuck on a pathway of exponential collapse while GDP per capita increases and resource use is constant. If , then per capita income can grow indefinitely in the model with zero increase in power usage at the rate of . If current rates of productivity improvement continue, that means we can sustain existing rates of economic growth without increases in power consumption if the population growth rate is /yr. Of course it should be noted that global population collapse is not the optimal solution to the problem of unbounded increases in power consumption from a human perspective.

Long-term projections of climate, economic growth, and population are notoriously unreliable, so one should not take these deadlines as specific quantitative predictions. Instead, the appropriate take-away from this analysis is that plausible reductions in population growth can have multiplicative effects on the waste heat deadline, possibly even extending it to the distant future, especially when coupled with slight reductions in the economic growth rate.

Future productivity and limits to growth

A concern with relying on recent historical improvements in energy efficiency to sustain per capita income growth is that productivity growth might be slowing. The deceleration may be due to the exhaustion of low-hanging fruit or the bureaucratization of industry and regulatory approvals [28]. Has the literature found deceleration in productivity growth? How does it vary across sectors? How do we project that into future productivity growth? And how does this affect estimates of the time limits to growth?

To address these questions, we first elaborate on how power is transformed into output. Up to now we have considered a simple model in which economic output is related only to power consumption (). Economists, however, posit that output is an increasing function of number of workers or labor (L), M different capital inputs () such as power (), and total factor productivity (A): . Total factor productivity (TFP) is not just productivity of a single input such as power, but productivity of all inputs combined. TFP represents things like knowledge or technology, organization of firms, markets, and regulation, which all affect how well inputs can be translated into output. Our production function uses labor or employed population (L) rather than population (N), which was used in the identity in (1), because only employed persons contribute output.

Common economic growth models often further assume that, in the long run, outputs have Cobb-Douglas form [29], i.e., are log-linear in inputs

(5)

where and . Here, Y is measured in dollars, each capital input i is in its unit , and total factor productivity (TFP), A, is in USD/( ). The reason for this functional form assumption is that the share of output spent on capital and labor have been remarkably stable over the last century [18] despite growth in TFP, and profit maximization or cost minimization with Cobb-Douglas production using stable but time-varying A is consistent with that observation. While there are criticisms of the Cobb-Douglas function [30,31], we use it mainly to illustrate a challenge with mapping productivity changes to time limits on growth; the same qualitative behavior can be achieved with more general production functions, albeit with more mathematical complexity that distracts from our point here. The are often called capital and labor shares because profit maximization or cost minimization with Cobb-Douglas technology implies that spending on each input is a fixed share of output y corresponding to the exponent on that input. We assume Θ = 1, which implies constant returns to scale in inputs, because that is also consistent with long-term data [32]. If we take logs and then time derivatives of (5), we obtain an equation that relates the growth of output to growth in TFP and inputs:

(6)

Here, is growth in output per employee. For simplicity, we have aggregated the stock of all capital inputs other than power and called it non-power capital: . We define as non-power capital share. We call power and power factor share .

We can estimate historical factor shares () and TFP growth () consistent with our assumption of a Cobb-Douglas production function using panel data (with annual data on a fixed number of countries) on inputs and outputs from multiple countries across prior years. Specifically, we estimate (6) using ordinary least squares linear regression and data on power consumption from the U.S. Energy Information Administration [19] and on output, employed populations, capital stock, and labor share from 66 countries consistently in the Penn World Tables (PWT) 10.01 [33] from 1965–2019. From 1965–1991, this includes the USSR. From 1965–1991, we substitute Russia plus the former Commonwealth of Independent States for the USSR, for a total of 79 states. These countries account for roughly 94.4% of global output, 80.2% of total population and 96.3% power consumption in 2019. Although TFP growth is not observed directly in our data, it can be approximated by the estimated constant in our regression. We estimate that power’s share of output () is 1.99%, the remaining capital share () is 53.81%, and the labor share () is 44.2% (see S1 Table). Our estimate of power share is close to a recent measurement of 4% by Thunder Energy [34]. Our labor share estimate is somewhat low (thus our capital share estimate somewhat high) relative to the literature ( ≈ 2 ∕ 3) [29], but the labor share has been declining and the capital share rising since the 1980s such that [35] estimates labor share  ≤ 0 . 6 in the 4 largest economies. Importantly, our estimates imply that historic TFP growth was 1% during this period, similar, but slightly smaller than our estimate of 1.2% growth in average productivity of power globally.

We are interested in future—not past—TFP growth. This can be forecasted from historical data by modeling the drivers of innovation. Many economic growth models [27,36] posit that innovation is related to the amount of working people and existing ideas:

(7)

where Ẋ = ∂X ∕ ∂t, t is time, and α > 0. This equation uses working population rather than total population because the non-working population is thought not to contribute to production and thus to improvements in total factor productivity. In what follows, we allow TFP growth to vary over time, but we continue to hold working population, income and non-power capital stock growth constant. (7) permits the returns to working population to be diminishing if λ < 1. Moreover, TFP growth is more difficult as countries produce more ideas if β > 0, a term economists call dynamic diminishing returns. Although (7) is simple, it is flexible enough to account for a range of different models of innovation [36] and to estimate trends in productivity growth [28].

We can use estimates of historical factor shares () and determinants of future productivity growth (λ , β) to calculate the time it would take for power levels to reach the waste heat limit for a range of future TFP growth rates. First, we solve the TFP growth Eq (7) for A(t). Specifically, we multiply (7) by and integrate with respect to time: , where is TFP at . We choose 2019 to be the initial time (t = 0) because that is the last date at which we have an estimate of TFP (, S3 Table) from the Penn World Tables (PWT) version 10.01 [33,37]. Substituting in , where is L at , yields future TFP as a function of current values of working population and TFP:

(8)

Combining (8) with estimates of (λ , β), we use (7) to calculate future productivity growth .

Second, we solve for power, P(t), given TFP levels, A(t), using (5) and assuming constant growth rates for income, employed population, and capital stock net of power.

(9)

Third, we use Murphy’s [5] waste heat limit from (3).

In order to use (9), we must specify λ and β values, which are highly uncertain. Bloom et al. [28] estimate β by holding the innovative returns to growth in working population (λ) constant and comparing inputs such as labor and R&D expenditures into innovation and outputs of innovation across sectors and data sets, including outputs such as patents, new drugs, and agricultural land productivity. Their study is among the most comprehensive estimates of deceleration in productivity growth across sectors and time. Given observed R&D input/output ratios, their median estimate across sectors suggests large, self-limiting reductions in research productivity as TFP rises (β = 2 . 5) when working population growth provides constant innovative returns (λ = 1). Bloom et al. [28] cannot identify both λ and β with data labor inputs and research outputs, so they constrain λ and estimate β. Their baseline value of λ is 1, and they obtain estimates of β that range from 0 . 9 to 7 . 2. We follow their lead and constrain the innovative return to working population at λ = 1, but consider forecasts under different values of β that represent notable or representative economic sectors.

Given the large range in plausible values of β, Fig 2 explores how different paths for future TFP growth affect the date of the waste heat limit. It shows 6 paths demonstrating that even small changes in β have an immense impact on the time to the waste heat limit:

1. A path that continues historical rates of TFP growth: (red dashed);
2. A  ( λ , β )  pair that produces a TFP growth path that approximates historical growth rates: (blue);
3. The median of the sector-specific (λ , β) estimated by Bloom et al. [28]: λ=1, β=2.5 (green), which corresponds roughly to the path of cotton production according to [28].
4. A path with equal values of λ = 1 and β = 1 (purple), which is roughly the path of publicly listed U.S. firms according to [28].
5. Two paths, with λ = 1 but β just below and just above 1 to illustrate how sensitive future power consumption is to this parameter: β = 1 . 1 in orange and β = 0 . 9 in cyan.

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Fig 2. Projected TFP growth (), TFP levels (A(t)), growth in power consumption (), and power levels (P(t)) over 1000 years under different parameter values for innovative returns to working population levels (λ) and dynamic diminishing returns to innovation (β).

The legend provides parameter values for each projection. The black line indicates the waste heat limit. Forecasts start at 2019 levels of working population, power consumption and TFP (see S3 Table). Forecasts assume that production is governed by a Cobb-Douglas function (5) with factor shares estimated using data from 1965–2019 (see S2 Table), that TFP growth is governed by (7), and that output, working population and factor growth rates follow historical values from 1965–2018 (see S2 Table).

https://doi.org/10.1371/journal.pone.0319217.g002

Forecasts start at 2019 levels of working population, power consumption and TFP (see S3 Table). Forecasts assume that production is governed by a Cobb-Douglas function (5) with factor shares estimated using data from 1965–2019 (see S2 Table); that TFP levels are governed by Eq (8), TFP growth by Eq (7), power by Eq (9), and power growth is derived from the time derivative of Eq (9); and that output, working population and growth rates of factors other than power follow historical values from 1965–2018 (see S2 Table).

Path 1 (red dashed) shows what happens when we use a Cobb-Douglas function and project out using historical estimates of TFP growth. Historical rates of TFP growth portend the waste heat limit in  ∼ 300 years, similar to what Murphy predicts. This forecast implies that a Cobb-Douglas model and a focus on TFP growth can approximate forecasts based on historical growth in average product. The fact that we can approximate historic path 1 with path 2 (, in blue) demonstrates that we can augment the Cobb-Douglas function with the TFP growth model in (7) and match our forecasts to Murphy’s [5] forecast.

Path 3 (green) depicts strongly diminishing rates of productivity improvements, implied by median (λ , β) estimates across sectors from Bloom et al. [28]. The waste heat limit occurs in years in this scenario. Path 4 (purple) illustrates a balanced path in which employed population growth pushes innovation roughly at the rate that past innovation increases the difficulty of future innovation. This balanced path delays the waste heat limit beyond a millennium. Finally, paths 5 and 6 illustrate how small changes in parameters around a balanced path can have massive effects on forecasts of the waste heat limit. Slightly greater dynamic diminishing returns to innovation (β = 1 . 1) triggers the waste heat limit in under 200 years, while slightly smaller dynamic diminishing returns to innovation (β = 0 . 9) means historic economic and employed population growth rates never drive power usage to the waste heat limit.

The projected date of the waste heat limit is highly sensitive not just to values of (λ , β), but also to factor shares and economic growth. Fig 3 explores this sensitivity by perturbing these model parameters from their historical values. Panels A and B show that as the factor share of capital or power rises at the expense of labor share, the date of the waste heat limit is delayed. This is because employee income growth is increasing in capital and power factor shares (6), so those shares reduce pressure on power growth to meet the requirement of constant growth in employee income. The black vertical lines and dashed lines in each figure indicate our estimates of the mean and confidence intervals for relevant factor shares. In Panel A, the confidence intervals span a wide range of implied waste heat limit dates. In panel B, the range is quite small. Panels C and D show that increasing the innovative return to working population growth (λ) increases τ and diminishing innovative returns with greater innovations (β) reduces the time to the waste heat limit. The vertical lines in panels C and D show the estimates of λ = 1 and that track historical power growth rates. An important aspect of Fig 3 is that the system contains a singularity such that small changes in the parameters can lead to an infinite time to the heat limit, as demonstrated by the path with (λ , β)=(1,0.9) in Fig 2. We interpret these infinite time horizons as indications in our simple model that the waste heat limit is not an important constraint on at least a millennial timescale.

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Fig 3. Projected time until the waste heat limit under different parameter values for capital () and power () factor shares, innovative returns to working population levels (λ), dynamic diminishing returns to innovation (β), growth in working population () and growth in total output ().

Legends provide parameter values for each projection. Solid and dashed black lines in Panels A-D present historical mean and 95% confidence intervals for the parameter specified on the x-axis. The solid black lines in Panels E-F present the historical mean growth in working population and output, respectively. Forecasts start at 2019 levels of working population, power consumption and TFP (see S3 Table). Forecasts assume that production is governed by a Cobb-Douglas function (5) with factor shares estimated using data from 1965–2019 (see S1 Table), that TFP growth is governed by (7), and that output, working population and factor growth rates follow historical values from 1965–2018 (see S2 Table). Each plot changes just the parameter indicated on the x-axis, holding other parameters constant.

https://doi.org/10.1371/journal.pone.0319217.g003

The effect of the growth rate of the working population on the time to the waste heat limit is particularly interesting. Fig 3E shows that, consistent with the intuition of Malthusians [2,4,5], there is a solution in which significant (employed) population reduction (negative ) forestalls the waste heat limit. However, consistent with economic intuition [6], there is also a solution in which faster (employed) population growth can forestall the waste heat limit just as effectively. If innovation from more people can outpace the rate at which innovation reduces research productivity, then the waste heat limit can be substantially delayed in the model. But the model in (7) suggests there are two ways for working population-based innovation to keep pace: either increase the innovative returns to working population (λ, as illustrated in Fig 3C) or increase population growth (Fig 3E). Although total (and thus employed) population growth rates are slowing, the recent historical rate of working population growth (black line) is so close to the elevated level required to evade the waste heat limit that it may be feasible to pursue the optimistic scenario with both population and economic growth.

Of course, one should not view these waste heat limit dates as specific quantitative predictions. They are based on low-order models that are valuable primarily for the qualitative intuition they provide. Our goal is to show, first, that an alternative solution exists in which survival is coupled with perhaps a millennium or more of economic and population growth, and second, that much more research is required before one can make credible predictions about the waste heat limit and the end of economic growth.

Growth due to substitution to other resources

One might object that inputs into production of power are but one set of limited resources. Even if limits on power turn out not to constrain growth, perhaps other resources could. The converse of this warning is that, even if we run out of specific resources on Earth, we may be able to continue to grow per capita income by substitution to other resources.

In this section, we demonstrate how flexible production technology has historically been, and then discuss how production might adapt to permit economic growth even as certain resources become depleted. We show that historical growth in average productivity combined with stability in resource prices demonstrates a capacity for human civilization to substantially improve how efficiently it uses specific inputs or change how it combines inputs to produce outputs. These adaptations suggest that exhaustion of specific resources, which will raise input prices and perhaps even block improvements in input efficiency, need not stop per capita income growth.

For this section, we adopt a more flexible production function that embeds the Cobb-Douglas function used in the prior section and allows us to see what drives substitution of resource inputs [29]. The constant elasticity of substitution (CES) function is:

(10)

where i indexes a total of R resource inputs , are effective factor shares, are factor-specific productivities (in contrast to total factor productivity or average product of factors), ρ = ( σ − 1 ) ∕ σ, and σ is the elasticity of substitution between resources.

A production process can be thought of as converting a series of inputs (e.g., copper) into a service or function (like conductivity) that helps build an output (such as a cable). Think of as the effective amount of an input required to produce service i, i.e., the amount of conductivity extracted from a unit of copper. This could change if one figures out a better way to use copper to create conductivity or finds an alternative to copper that is more conductive and/or more abundant. By contrast, is the relative importance of an intermediate service in the production of the output. This could change as cables go from copper to fiber optic for certain purposes such as transporting information. There may be changes in input productivity () or in the conversion of intermediate services as reflected in input shares () over time. Going forward we drop the total factor productivity term (A) because it can be absorbed into the input-specific productivities ().

If σ ≥ 1 (and so ρ ∈ ( 0 , 1 ] ), factors can substitute for each other, with greater σ implying more substitutability. As σ → ∞, any amount of factor i can be replaced with factor j at some constant rate. If σ < 1, factors are less easy to substitute: the more one wants to substitute factor i for factor j, the more units of i are required for each unit of j.

If firms in the economy choose inputs to minimize costs, due to either market pressures or effective internal management, then they will equate the relative marginal product (marginal rate of technical substitution) and the relative prices of resources (marginal rate of substitution): for resource inputs i and j, where indicates marginal product of input i, and is the resource price of input i. The relative marginal product of inputs i and j with CES production is , where is the average product of resource i. If we plug the equation for relative marginal product into the cost minimization condition, take logs, and then take a time derivative, we find that

(11)

i.e., input shares are chosen so that the growth in relative resource price is equal to growth in relative effective factor shares and the weighted average of growth in relative factor productivity and in average factor productivity .

Recent historical data on resource prices and average factor productivities suggests that the global economy has experienced either a substantial change in effective input mix or improvement in specific input productivities. Input prices have been largely constant for decades, if not centuries [38], implying . Moreover, growth in average productivity has varied substantially across resources (Fig 4). Suppose we examine a resource i that has experienced slower average productivity growth relative to input j, i.e., . Then the ratio is also declining and (11) implies

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Fig 4. Use of 18 different commodities, relative to use of power, over time.

Legends indicate specific commodities. Ratio is units of specific commodity divided by units of power, with the ratio normalized to be 1 in 1965. Data are from the US Geological Survey.

https://doi.org/10.1371/journal.pone.0319217.g004

(12)

Slower growth in relative average productivity of factor i implies greater use of that input () because the ratio of average productivity of inputs i and j is equal to the ratio of inputs j and i (). Greater relative use of input i can be explained by a mix of substitution towards that input () and improvement in the efficiency with which an input produces that factor (). The mix is governed by the elasticity of substitution ( ± ), because r = { 1 , 3 , 8 }  and s = 5. Assuming inputs are substitutes, lower substitution elasticity implies that changes in factor mix are driven more by change in factor shares than by relative input-specific productivities.

Suppose that, in the long-run, production technology is best described by a Cobb-Douglass function (r = 5) but with changing shape (i.e., changing factor shares). This assumption is one way to reconcile historical evidence on changing factor shares [30,39]. It is also consistent with a Pareto distribution of ideas that can lead to productivity improvements [32]. With unit elasticity of substitution, past changes in average productivity over time suggest one-for-one changes in factor shares as (12) collapses to . In words, the long-term decrease in relative use of factors such as land and lead and increase in relative use of aluminum and lithium suggest substantial substitution away from the former factors towards the latter.

Indeed, historical changes in input proportions are consistent with factor shares changing quickly. Using factor shares for power as a benchmark, Table 1 suggests that the factor share for land fell on average 1.83% per year and for aluminum rose on average 2.25% per year in the last 60 years. This implies that the factor share of power rose 3x relative to land over this period, while the factor share of aluminum rose nearly 4x relative to power. If we ignore pre-1975 dips in lithium production, use of lithium grew roughly 8x relative to power.

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Table 1. Growth rate in quantity used, average productivity, and factor share ratios (relative to power) for 18 commodities over time.

https://doi.org/10.1371/journal.pone.0313772.t001

Past is not necessarily prologue, as suggested by the non-monotonicity in resource ratios in Fig 4. However, historical rates of change suggest a degree of plasticity in technology that inspires confidence in humankind’s ability to adapt to resource limits without sacrificing economic growth, although this process can involve shocks and disruptions.

The value of a dollar

Dollars are a unit of exchange that, among other things, simplifies the expression of prices. Suppose there are only 2 goods, and the exchange rate is 4 units of good 1 for 3 units of good 2. We can say the relative price of good 2 is 4/3 of good 1: 1 unit of good 2 costs 4/3 units of good 1. If there is also a good 3 that can be exchanged for 2 units of good 1, then the price of good 2 can also be expressed as 1/2 of good 3. However, we can define a medium of exchange like a dollar to simplify prices, which can now all be expressed in dollars rather than in each of the other goods. Suppose we define a dollar to be equal to the value of 1 unit of good 1. Then the price of good 2 is simply $(4/3)N = { 1 , 2 , 3 , 4 } $1.33 and the price of good 3 is $(2/1)*(4/3) > $2.67. Because the dollar is no longer pegged to a commodity like gold, the so-called quantity theory of money says that the value of a dollar will be a function of the supply and demand for a dollar as a medium of exchange rather than gold. Although there are other theories of the value of money, they do not affect relative prices, so the analysis in this paper does not depend materially on the theory used.

There are two complications when valuing a dollar. First, the relative dollar prices of specific goods and services depend on the relative demand and supply for those different items. If the quantity of dollars or the quantity of goods and services changes over time, the dollar price of a given item will change. Prices may go up (inflation) or they may go down (deflation). Real prices, defined as prices expressed in the value of a dollar in a given year, address this problem by eliminating temporal changes in the value of a dollar. Second, different locations may use different units of exchange, e.g., the US uses a dollar and England uses a pound. This creates two difficulties: we need to know the amount of dollars one may trade for one pound and relative prices may differ across countries because transportation costs make it difficult for markets to equalize relative prices worldwide. A purchasing power parity (PPP) exchange rate addresses these problems. It is defined as the exchange rate that adjusts for both the nominal currency exchange rate and differences in relative prices for a fixed bundle of commonly consumed goods. Thus, a PPP-adjusted dollar value of output in a country like England in a year is the value of England’s output in pounds that year times the PPP exchange rate for dollars to pounds.