Introduction

In 2018, William Nordhaus and Paul Romer were awarded the Nobel Prize in Economics for their work integrating climate change and innovation into long-run macroeconomic models (Nobel Prize Committee 2018). Despite this recognition, environmental economics has traditionally been considered a microeconomics field. As a consequence, there is less emphasis on macroeconomic issues in undergraduate environmental economics textbooks. One exception is the textbook by Harris et al. (2017) who provide a discussion of macroeconomic issues through the lens of both neoclassical economics and ecological economics. Ecological economists view the economic system as a subset of the broader ecological system. This approach implies that the economy should not grow larger than the biosphere or else the earth will face environmental destruction (Daly 2007). This conclusion differs from mainstream economic literature that indicates that technological change is simultaneously important for sustained growth and for limiting environmental impacts (Popp et al. 2010).

The reduced coverage of macroeconomic topics presents a significant content gap for three reasons. First, it limits the discussion of global environmental policy and climate change within undergraduate environmental economics courses. Theoretical work indicates that optimal environmental policy includes both carbon taxes and research subsidies, leading to cleaner production methods without sacrificing long-run growth (Acemoglu et al. 2012). Recent empirical results confirm that carbon taxes do not negatively impact GDP growth rates (Metcalf and Stock 2020b). In other words, the latest and most significant findings regarding economic growth and the environment are not yet part of standard undergraduate curriculum.

Second, less time is spent teaching growth topics to undergraduates in general. Acemoglu (2013) points out that growth is rarely taught in principles of economics courses, even though it is a topic many students have an interest in. Given the fact that curriculum for intermediate macroeconomic courses varies widely across textbooks and instructors, it is possible that students may not see the standard Solow model unless they take a course specific to economic development. Environmental economics courses are no exception to this deficiency.

Third, when topics related to growth and the environment are included in environmental economics textbooks, they are often presented descriptively without any discussion of theoretical models. For example, the textbook by Callan et al. (2013) provides a brief discussion of the environmental Kuznets curve (EKC), the hypothesized inverted U-shaped relationship between economic growth and environmental degradation. The authors review the empirical findings for this relationship yet there is no accompanying quantitative exercise regarding what may generate this relationship. This is commonplace among environmental economics textbooks.

Can we teach environmental macroeconomics solely using empirical examples? I argue that students deserve a complete picture of the topic that combines the elegance and intuition of economic theory with the real-world applications presented in most textbooks. Instructors often have multiple learning goals including content-driven objectives that help students gain the necessary foundation for future coursework and for understanding policy debates. Learning goals can also involve building analytical skills and challenging students with a new way of thinking (De Araujo et al. 2013). A simple theory model is well suited to expose students to topics not typically emphasized in environmental economics courses while also fostering critical thinking.

In this paper, I propose a simplified version of Brock and Taylor (2010)’s “green” Solow model that is accessible for undergraduates. This model provides theoretical support for the EKC hypothesis by including technological progress in abatement in the standard Solow model. I present the model in five systematic steps, using only basic algebra to reinforce the economic intuition. This requires a simplifying assumption and omissions, namely that the growth rate of emissions is equal to the growth rate of capital per effective worker. These exclusions are justified as they result in a version of the model that requires no calculus or differential equations to understand. I outline the model as a stand-alone content reference and explain how the material can be taught in multiple learning modalities.

There is often concern that the Solow model is too difficult, both intuitively and computationally, for undergraduates (Stein 2007). However, I argue that this restatement of the model is teachable to undergraduates since it builds upon micro-foundational insight and tools that are already consistently taught in principles and intermediate microeconomics. Since it extends what is widely regarded as a major workhorse model of modern macroeconomics, it is relatively simple to develop supplementary materials such as problem sets and exam questions. A further benefit is that the model provides theoretical support for the EKC, a topic that is already peripherally discussed in the environmental economics textbooks reviewed in the next section. Thus, it provides a complement to existing material.

The rest of this paper is organized as follows. I review the macroeconomics content that is included in a set of standard environmental economics textbooks and provide context regarding the inclusion of the “green” Solow model in undergraduate coursework. Next, I present the simplified model and outline recommendations for including the model in different course formats as an accompaniment to empirical applications. Finally, I offer concluding remarks.

Textbook Coverage

In preparing this paper, I surveyed macroeconomics content in seven popular environmental economics textbooks. I consider both the extent of coverage and whether or not analytic exercises are incorporated. I also report the relative position of the content within each specific text to gauge the perceived importance of these topics. In other words, I assume that a macroeconomics topic that is included toward the end of a textbook is less likely to be covered in class unless the instructor has a particular interest in the subject matter.

Tietenberg and Lewis (2018) briefly review the empirical evidence for the EKC relationship, as well as national income accounting measures that account for environmental impacts. These topics are included in the second to last chapter (20 of 21) and do not include any theory or problem-solving exercises. Hanley et al. (2013) provide similar coverage of these two topics but do so much earlier in their text, in chapter 6 of 13 total. The introductory level book by Field and Field (2017) reviews the data regarding environmental quality and economic growth, without actually mentioning the term “environmental Kuznets curve,” in the last chapter (21 of 21). Callan et al. (2013) also cover the EKC in the second to last chapter, 20 of 21, but include a brief discussion of national income accounting in a “text box” toward the end of the first chapter. They include one theory exercise in the end of chapter questions that utilizes a quadratic function to relate emissions to income per capita. Keohane and Olmstead (2016) provide a brief overview of the Limits to Growth Model, the EKC curve, national income accounting, and the implications of technology and intergenerational equity for sustainability. This information is included in chapter 11 (of 12) and contains discussion questions but no exercises. Kolstad (2011) provides the most rigorous macroeconomics content by introducing a theoretical model of productivity growth to examine how environmental regulation may contribute to productivity slowdowns. In addition, there is a corresponding end of chapter exercise for this model. He also covers the EKC and national income accounting similar to other texts. However, this content is included in the last chapter (20) of the book.

Overall, macroeconomics topics are included in mainstream environmental economics textbooks but they are not often covered with the same depth and rigor as microeconomics topics. When they are included, they are usually presented without any theory or analytic exercises and are located toward the end of the book. The main exception to this is Harris et al. (2017). In this text, the authors include macroeconomics content throughout the book, with the second chapter devoted to economic development and subsequent separate chapters on national income accounting, population growth, and economic impacts (labor market) of environmental protection. While the book does cover mainstream environmental economics, the authors place greater emphasis on ecological economics than other standard texts. In the environmental and resource economics education literature, González-Ramírez et al. (2021) find that sustainability and climate change are among the least developed pedagogy topics. Although Tsigaris and Wood (2016) develop a integrated assessment-Solow model to focus on climate change applications for undergraduates, there are few papers that adapt macroeconomic frameworks.

Background

At St. Lawrence University, a private liberal arts college, environmental economics is taught as an upper division elective with intermediate microeconomics as the only prerequisite course. The course is primarily taken by economics majors, for whom calculus is not currently required. The course averages 30 students per class section with one or two sections offered each fall. The course is normally taught in person but was conducted entirely online during the academic year 2020-2021 due to the COVID-19 pandemic. Students who choose a combined (interdisciplinary) major with the environmental studies department must take environmental economics as a required course.

I structure the course to include standard environmental economics topics including the theory of externalities, property rights, common property resources, public goods, valuation techniques, cost-benefit analysis, and an analysis of pollution control regulation. These topics tend to focus on efficiency. On the other hand, a macroeconomics framework is better suited to tackle issues related to both scale and efficiency. In recent years, students have expressed a greater interest in climate change, global warming, and how we reconcile economic growth with environmental quality overall. Thus, macroeconomics can help answer important questions including: How large should the economy be? Are there limits to growth? Can technology allow us to improve living standards while also improving environmental quality?

I also supplement class material with Daly (1991), who provides an ecological perspective on how large the macroeconomy should be, given natural capital constraints.Footnote 1 However, to illustrate a mainstream macroeconomic approach that emphasizes the role of technology in both growth and pollution control, I have students work through a simplified version of Brock and Taylor (2010)’s “green” Solow model. I present the model using elements from Mankiw (2012) and DeLong (2004) in order to adapt the analysis at an appropriate level for undergraduate students. My approach is very similar to Duncan (2015) who reformulates a model by Aguiar and Gopinath (2007) in order to teach business cycle macroeconomics to undergraduates. The following section provides a practical, stand-alone content reference for those teaching undergraduate electives in environmental economics or growth.

The Simplified Model

The Production Function

The standard Solow model includes an aggregate production function where the supply of goods is a function of capital, K, and labor, L.

$$\begin{aligned} Y = F(K, BL) \end{aligned}$$
(1)

The parameter B represents labor augmenting technological progress. Thus, the term BL measures the effective number of workers. It reflects both the actual number of workers and the level of technology that each worker is equipped with (Mankiw 2012). In other words, technology makes each worker more productive. Instructors should explain three important assumptions regarding the standard Solow production function.

  1. 1.

    Constant returns to scale. If both labor and capital are doubled, output will double.

  2. 2.

    Diminishing returns for individual inputs. Suppose we hold the level of capital constant but keep adding workers. Each additional worker will increase output, but by less and less each time.

  3. 3.

    Exogenous technology. We assume that technology is determined outside of the model and we take its value as given.

The constant returns to scale assumption allows us to analyze all variables relative to the effective number of workers. We utilize lower case letters to denote variables that are divided by BL, namely output and capital per effective worker.

$$\begin{aligned}\frac{Y}{BL} &= F\left( \frac{K}{BL}, 1\right) \nonumber \\y &= f(k) \end{aligned}$$
(2)

It is important to note to students that land or natural resources are not included in the production function. Instead, we consider environmental impacts of pollution described below.

Pollution Abatement

In this economy, the production of each unit of output generates pollution. As a consequence, national income can be saved, consumed, or devoted to pollution reduction, known as abatement. If students have been taught material on marginal abatement cost curves (MACC), there is an opportunity to connect this material and explain how abatement costs are now represented as a constant fraction of output, \(\theta\). If this is the case then \((1-\theta )\) represents the fraction of output left over for consuming and saving. Incorporating this into the production function in per effective worker terms (dividing by BL) gives us

$$\begin{aligned} y = (1-\theta )f(k) \end{aligned}$$
(3)

Evolution of Capital and the Steady-State

It is useful to first show students the standard Solow equation as presented in Mankiw (2012), where capital accumulates via saving and investment. I explain to students that individuals in the economy can save by depositing money in financial institutions. These institutions lend the money to firms who purchase capital goods (investment). Thus, capital per effective worker evolves according to the following equation

$$\begin{aligned} \Delta k = sy-\delta k \end{aligned}$$
(4)

where \(\Delta k\) represents the change in capital per effective worker over time, s is a fixed (exogenous) saving rate, and \(\delta\) is the capital depreciation rate. Intuitively, the equation illustrates both the additions to capital (saving & investment) and deductions (depreciation). However, it is important to emphasize to students that there are other factors that may impact capital accumulation.

  1. 1.

    Population growth, n. Growth in the labor force leads to less capital per effective worker. In other words, the capital stock is spread more thinly over a larger population.

  2. 2.

    Technological progress, \(g_B\). Technological progress, the growth rate of technology, leads to less capital per effective worker because workers are now more efficient.

  3. 3.

    Pollution abatement, \(\theta\). The higher the fraction of output devoted to abatement, the less output is left over for saving and investment.

Incorporating these additional elements into the capital accumulation equation gives us

$$\begin{aligned}{} & {} \underbrace{\Delta k}_\text {change in capital per effective worker} = \underbrace{s(1-\theta )f(k)}_\text {gross savings function per effective worker}\nonumber -\underbrace{(\delta +n+g_B)k}_\text {capital diminishing function per effective worker} \end{aligned}$$
(5)

In the steady-state, \(\Delta k = 0\). In other words, there is a level of capital per effective worker, \(k^*\), such that the deductions from capital exactly offset investment (gross savings function). I explain to students that we call it the steady-state because capital per effective worker is no longer changing. The value \(k^*\) can be found by solving the following

$$\begin{aligned} s(1-\theta )f(k^*) = (\delta +n+g_B)k^* \end{aligned}$$
(6)

The standard Solow model indicates that technological progress causes the values of output and capital to rise together in the steady-state. We call this the balanced growth path. I use the following math rules from DeLong (2004) to help students understand the balanced growth path without calculus.

Rule 1: The growth rate of the product of two variables equals the sum of their growth rates.

Rule 2: The growth rate of the quotient of two variables equals the difference of their growth rates.

Consider output per effective worker, \(y = \frac{Y}{BL}\). We can rewrite this expression as \(y \cdot BL = Y\). Applying Rule 1 gives us the growth rate of output as \(g_Y = g_y + g_B + n\). Since \(g_y =\Delta y = 0\) in the steady-state, the expression becomes

$$\begin{aligned} g_Y = g_B + n \end{aligned}$$
(7)

This illustrates the classic Solow model with technology and population growth result. The growth rate of output will be approximately equal to the growth rate of technology plus the growth rate of the labor force. We can also look at GDP per capita, or output per worker, \(\frac{Y}{L}\), to consider changes in the standard of living. Using Rule 2, the growth rate of income per capita is \(g_B + n - n\) or just \(g_B\). Intuitively, this indicates that technology is the key factor in overcoming diminishing returns, leading to sustained increases in living standards.

Sustainable Growth

In this model, we are not only concerned with sustained economic growth but also environmental quality. If the economy is growing, emissions are also increasing. However, technological advances related to abatement can help offset this environmental impact. For example, the invention of a new smoke stack filter may lead to reduced emissions for the same level of production. Let \(g_A\) represent technological progress in abatement. The growth rate of emissions, \(g_E\), along the balanced growth path can be written as

$$\begin{aligned} \underbrace{g_E}_\text {growth rate of emissions} = \underbrace{g_B+n}_{\text {scale effect} (g_{Y})} - \underbrace{g_A}_\text {technique effect} \end{aligned}$$
(8)

The first term in the equation, \(g_Y = g_B + n\), is called the scale effect. The more output is produced, the more emissions there will be. An increase in emissions is associated with larger GDP values. The second part of the equation \(g_A\) is called the technique effect. The development of cleaner production techniques will reduce pollution emissions. This is why it is subtracted from the first term, as it will reduce emissions growth.

We define sustainable growth as the balanced growth path generating both rising income per capita and an improving environment. Given this definition, the following conditions must hold in order for growth to be sustainable.

$$\begin{aligned} g_B > 0 \end{aligned}$$
(9)
$$\begin{aligned} g_A > g_B + n \end{aligned}$$
(10)
$$\begin{aligned} g_E < 0 \end{aligned}$$
(11)

Equation 9 indicates that there must be technological progress to generate sustained per capita income growth, while Eq. 10 illustrates that technological progress in abatement must exceed growth in aggregate output (\(g_y\)) in order for emissions to decline and output to increase. Together, these two conditions imply a negative emissions growth rate given by Equation (11).

The Environmental Kuznets Curve

To introduce the next section, it is useful to ask students the following question: Is economic growth good or bad for the environment? On one hand, a richer nation will utilize more energy and produce more waste. On the other hand, a richer nation may have the resources to invest in cleaner production techniques. If we consider environmental quality to be a normal good, the environmental Kuznets curve (EKC) hypothesis can provide some insight into answering this question. To convey this concept to students, I explain that the EKC is the hypothesized inverted U-shaped relationship between per capita emissions and per capita income. The basic idea is that a country’s environmental impacts will increase in the early stages of economic development. After a certain per capita income level, these environmental impacts will decrease as people demand increased environmental quality. In this section, we consider how the “green” Solow model provides theoretical support for this hypothesis.

If we divide Eq. 5 by k, we will have an expression for the growth rate of capital per effective worker, \(g_k\).

$$\begin{aligned} g_k = \frac{\Delta k}{k} = \frac{s(1-\theta )f(k)}{k} - (\delta +n+g_B) \end{aligned}$$
(12)

We assume that growth rate of aggregate emissions is equal to the growth rate of capital per effective worker. This is a simplification that avoids the differential equation for emissions that Brock and Taylor (2010) develop in their model.Footnote 2 The simplifying assumption implies the following relationship.

$$\begin{aligned} g_E = \frac{s(1-\theta )f(k)}{k} - (\delta +n+g_B) \end{aligned}$$
(13)

Fig. 1 illustrates how this relationship generates the EKC diagram when growth is sustainable (\(g_E<0\)). The top panel shows the relationship between \(g_E\) and capital per effective worker, k, given by Eq. 13. It includes a negatively sloped line shown by the saving locus, \(\frac{s(1-\theta )f(k)}{k}\), and a horizontal line with the height \((\delta +n+g_B)\). The lower panel illustrates how the level of emissions changes as capital per effective worker increases. Consider the point labeled T. To the left of the point T, \(\frac{s(1-\theta )f(k)}{k} > (\delta +n+g_B)\). This implies that \(g_E > 0\). This means emissions levels must be rising on the lower panel graph. To the right of T, \(\frac{s(1-\theta )f(k)}{k} < (\delta +n+g_B)\). This indicates that \(g_E < 0\). This means that to the right of T, emissions levels must be falling. In other words, the growth rate of emissions must be zero at point T, representing a turning point in the EKC in the bottom panel. Note that the simplifying assumption of \(g_E = g_k\) now implies that k(T) is the steady-state level of capital per effective worker. In Brock and Taylor (2010), emissions peak as the economy approaches its balanced growth path when \(-g_E\) is small.

Fig. 1
figure 1

EKC illustration modified from Brock and Taylor (2010)

This analysis illustrates how the dynamics of the “green” Solow model traces out an EKC. The simplifying assumption allows instructors to emphasize clarity over nuance and focus on the comparative statics of policy changes (Mankiw 2016). For example, if environmental regulations become permanently more stringent, a larger fraction of output must be devoted to abatement. This increase in \(\theta\) will lower \((1-\theta )\), shifting the savings locus to the left. Figure 2a illustrates how this causes the peak in emissions (turning point) to occur sooner.Footnote 3 In other words, emissions start to decline at a lower level of income per capita because abatement uses up resources that would have otherwise been allocated toward investment that drives growth.

It is also interesting for students to examine the impact of a government policy that raises the saving rate. Figure 2b shows that an increase in s shifts the savings locus to the right. This causes the peak in emissions to occur later and raises the level of capital per effective worker in the steady-state. Higher savings leads to rapid capital accumulation and faster output growth, meaning that emissions do not decline until diminishing returns to capital set in.

Importantly, this model illustrates to students that the key factors of the standard Solow model, diminishing returns and technology, may contribute to a well-known empirical finding in environmental economics, the EKC. Brock and Taylor (2010) describe how with rapid initial growth, the impact of abatement technology is overwhelmed. This causes an increase in emissions. As countries develop and approach their balanced growth path, slower growth allows technological progress in abatement to take over. As a consequence, we see falling emissions levels as income per capita rises.

A number of omissions allow for a concise presentation of the model. Specifically, I do not include the equation and properties for pollution emissions. This requires defining four additional variables and two functions, with several substitution steps in between.Footnote 4 This provides a version of the model that is less overwhelming in terms of equations and symbols. I should also note that I do not discuss initial conditions or an analysis of the turning points on the EKC, which would require an understanding of differential equations. My goal is simply to expose students to the fundamental connection between neoclassical growth theory and environmental impacts.

Fig. 2
figure 2

Comparative statics

Discussion

In this section, I discuss how to adapt the model to different class formats and difficulty levels as well as how to pair the model with empirical results and activities.

Class Formats and Levels

I taught the model in environmental economics courses offered over five semesters. For in-person lectures, the model can be taught over two 90 min class periods or three 60 min class periods. The model can also be taught in a distance learning environment to larger class sizes. Due to the COVID-19 pandemic, I delivered this material asynchronously in online sections of environmental economics for two semesters. I covered the model in four brief videos that utilize a combination of slides and written lecture notes using an iPad. Complete lecture slides and view access to the videos are available upon request. The videos cover the following topics.

  • Production function & pollution abatement

  • Capital accumulation & steady-state values

  • Sustained growth & sustainable growth

  • EKC & comparative statics

These “mini” lectures convey all of the material from the previous section in a compact format that maximizes engagement because the videos are all under ten minutes in length and can be paused or re-watched. This allows students time to digest more complicated aspects of the model while giving them greater control over their study time. This may be particularly important in larger class sizes where one-on-one engagement is challenging. There is a high fixed cost to produce these videos, but they can be implemented in online, hybrid, or flipped classroom environments. There is evidence that flipped classroom formats may increase student performance in both principles and upper-level economics courses (Balaban et al. 2016; Craft and Linask 2020; Wozny et al. 2018) but future research is needed to determine their effectiveness for growth topics (Mikek 2022).

This material can also be adjusted based students’ requisite knowledge and level of quantitative preparation. If students are less familiar with macroeconomics, instructors can teach the basics of the standard Solow model first. If this is done the overview of the production function can be omitted and a discussion of pollution can immediately commence. Undergraduates tend to be more familiar with static models and transitioning to a dynamic framework can be conceptually challenging. It can be useful to review how growth rates are computed to bring everyone up to speed. For example, instructors can explain that \(\Delta k\) shows us how capital per effective worker changes from one time period to the next, whereas the growth rate of capital per effective worker, \(\frac{\Delta k}{k}\), refers to its proportional rate of change (Romer 2012). This is entirely different from \(k^*\), which is the level of capital per effective worker in the steady-state.

Instructors can also choose to emphasize the comparative statics and policy implications of Fig. 2 or can focus on the algebra with a Cobb–Douglas production function in more advanced courses. I include this material in the Appendix. It may be useful for students to be able solve for the steady-state level of capital and output per effective worker because it highlights the fact that changes in fixed exogenous factors, such as \(\delta\), s, and \(\theta\) can impact the levels, but not growth rates of these variables in the long run. I find that reviewing basic exponent properties (i.e., product, quotient, power, and negative exponent rules) before introducing a functional form goes a long way.

Empirics

There is evidence that using actual data can be a successful pedagogical tool for teaching growth theory to undergraduates as it allows students to connect technical material with the real world (Elmslie and Tebaldi 2010). I find it critical to discuss well-known empirical results related to economic growth and the environment to complement the model. Brock and Taylor (2010) derive an estimating equation from their theoretical model. Thus, the “green” Solow regression results can be discussed in class. However, Grossman and Krueger (1995) are more accessible to undergraduates who have taken only one or two semesters of statistics. The paper uses a reduced form model to study the relationship between economic growth and environmental quality, the same EKC relationship that the “green” Solow model generates. I ask students to read Grossman and Krueger (1995) and write a one-page summary in their own words. In class, we discuss how the authors measure environmental quality and the dollar value of per capita income needed to reach the turning point of the EKC (the authors estimate less than $8,000), which does not require a deep discussion of econometric details. More recent work by Metcalf and Stock (2020a) can also be incorporated to motivate the model. Based on European data, they present several accessible graphs that illustrate that a carbon tax would not harm the US economy. This is an example of a policy that supports sustainable growth, a concept that the “green” Solow model introduces.

Since the Solow model emphasizes the saving rate, I also discuss adjusted net savings (ANS) as a savings measure that accounts for environmental degradation. This measure is included in Harris et al. (2017) as part of the chapter on national income and environmental accounting. As an applied exercise, I ask students to gather gross savings and resource depletion data from the World Bank World Development Indicators for five countries (the United States, China, India, Chile, and Saudi Arabia). The selection of countries can vary based on what the instructor prefers to emphasize. I ask them to compute the ANS measure using Excel and answer the following questions: Which country has the highest ANS rate and why? Does a high ANS value necessarily mean that resource depletion and pollution impacts are relatively low? Explain using examples from the data. Which countries have the highest rates of resource depletion? What does this say about how much these countries are actually investing in their future?

These empirical results and exercises help generate in-depth class discussions about how to reconcile external costs with economic growth. To conclude the topic, I assign the Planet Money podcast episode titled Will economic growth destroy the planet? (Blumberg and Smith 2011). The episode features Herman Daly, who argues that growth is unsustainable, and Robert Mendelsohn, who emphasizes carbon pricing. I ask students to listen to the episode and identify aspects of the EKC that are discussed by both economists to evaluate how their viewpoints differ.

Conclusion

In this paper, I present a simplified version of Brock and Taylor (2010)’s “green” Solow model that is suitable for an undergraduate environmental economics course or an economic growth course. I simplify the model by assuming the growth rate of aggregate emissions is equal to the growth rate of capital per effective worker. As a consequence, students do not need calculus or differential equations to work through the model. The results illustrate to students the importance of technology, for both growth and pollution abatement, while using a standard model to generate the EKC, a theory that is already discussed in many environmental economics textbooks. Thus, it fills a significant gap by introducing students to environmental macroeconomics with a model that can be taught across multiple modes of instruction. Instructors can choose to focus on the comparative statics of the model or on the algebraic manipulations for more advanced students. Integrating the model with a discussion of empirical results is helpful for making theory more tangible.