DIRTY INPUTS IN POWER PRODUCTION AND ITS CLEAN UP: A SIMPLE MODEL

1. Introduction

Modern industrial growth has had major implications for consumption of energy. Initially, this fuel consumption (mainly coal and wood) was done in situ; for instance, the steam engines used coal, and other industries used coal at the location where production was undertaken. Over time as centers for industrial production proliferated, it was found that centralized production of electricity worked better. Power was then sent to various points of demand along a grid.

Over time oil became an important source of energy. It powered vehicles, planes, etc. Oil and natural gas continue to be used as sources of energy, side by side with coal. These fossil fuels, in course of burning, release impurities into the atmosphere — e.g., sulphur dioxide, oxides of nitrogen, carbon monoxide and dioxide. These joint products are sources of pollution both at the local and global level. The health effects of these pollutants depend on other local conditions — e.g., chalky soil is better for absorbing sulphur dioxide; a choppy North Sea is better in dealing with an oil-spill than a placid Mediterranean.

Local pollution often falls as society becomes richer — as illustrated by the so-called Environmental Kuznets Curve (EKC), an inverted U-shaped relationship between per capita income and the level of pollution. Carbon dioxide, the primary global pollutant, shows no such inverted U-shaped relationship with an economy’s rising income level.

As countries industrialized, there was an increase in the demand for electricity (as well as transport). To the extent this was met from coal-fired power stations, a lot of local pollution was generated. As postulated by the EKC, as societies became richer they undertook abatement activity (e.g., use of scrubbers to reduce sulphur emissions).

For developing countries, this posed a challenge and thermal power plants became major polluters. For example, in 2007, thermal power accounted for 33% of all industrial dust discharge and 56% of all industrial sulfur dioxide emissions in China (Wen and Liu, 2012). In 2010–2011 in India, 111 coal-fired power released 580 ktons of particulates with diameter less than 2.5$\mu$m (PM${}_{2.5}$), 2,100 ktons of sulfur dioxides, 2,000 ktons of nitrogen oxides, 1,100 ktons of carbon monoxide and 665 million tons of carbon dioxide (Guttikunda and Jawahar, 2014). There is a lot of data to show the effects pollution has on health of citizens but this was met with procrastination, with the low level of incomes being cited the cause for inaction. But over time, coal-fired power stations, such as those in China and India, have been removed from densely populated urban areas.

Global warming provided a reason for scrapping the coal-based electricity plants. Coal is the dirtiest fossil fuel from the perspective of global warming. The recent Glasgow conference made a strong pitch for discontinuing the use of coal; in the event, a compromise solution was arrived at whereby the coal-based plants would be phased out over time. If the use of coal is going to be terminated, what are the clean fuels available? There are a number of candidates — hydroelectric and nuclear power (these have been in use for decades); and, more recently, solar and wind-based powered generation.

In this paper, we want to focus on middle income developing economies. These countries, ranging from the really larger ones of China and India to distinctly smaller ones such as Indonesia and Bangladesh, have witnessed a high growth of produced power. China and India have sought to register rates of economic growth close to double-digit levels and often succeeded. Given the nearly one-to-one relationship between GDP and power production, such growth has been based on enhanced capacity for power generation (with some hiccups). Much of this enhancement has been in coal-based capacity. But at the same time, these countries have paid a heavy price: the emissions from coal or other fossil fuel-based power generation have contributed significantly to both increase in morbidity and mortality (for India see Guttikunda et al., 2015; and for China see Chen et al., 2015 and Miao et al., 2019). From now on, these countries need to switch their reliance to cleaner renewable resources in the drive to generate more power for sustaining (a) increase in per capita welfare levels and (b) economic growth.

This switch is also consistent with their pledges for the Paris agreement and the developments in climate diplomacy after Paris: India enhancing the share of non-fossil sources in power production to 40% by 2030 (Climate Action Tracker, 2021) and China reaching an emissions peak in regard to carbon dioxide in 2030 (Climate Action Tracker, 2021). The recent treaty at COP 26 in Glasgow builds on the pledges made under the Treaty of Paris (Ramesh, 2021; UNFCCC, 2021). This pact attacks coal-based emissions directly instead of limiting its focus to just non-fossil fuel-based emissions. It takes into account the fact that the world is 1.1^∘C warmer now than it was in the pre-industrialization era of 1850–1900 and has tried therefore to limit further increase in temperature to only 1.5^∘C. This would take time for a country such as India, where coal-based plants are still a major source of electric power, and would have to be phased out through a switch to plants based on other fuels.

Both India and China have already demonstrated that they can generate significant amounts of power through renewable resources — sun, wind and hydel sources. This is exemplified by Chinese behavior in regard to enhancing production of power in 2012–2020 and Indian behavior for the same period in 2015–2020 during which addition to solar power production was 70 times that of the capacity in the initial year.

In short, economic imperatives, the clauses of international agreements and demonstrated capabilities all point in the same direction — that in the future these countries will not only not enhance their capacity for power generation from dirty inputs significantly but will bring about almost all capacity increase by investing in clean or renewable inputs such as solar, wind or hydel energy.

In this paper, we model an economy faced with this compulsion. It is noteworthy that the level at which the capacity for producing dirty power is fixed is based on the historical magnitudes of demand for power and how the national government has responded to such magnitudes by fixing targets for power production. The strength of the response to demand is extremely important: as of 2019, according to United Nations data, the Indian population at 1.37 billion was a mere 4% less than China at 1.43 billion (Economic Times, 2021); their per capita incomes at purchasing power parity in 2019, according to the World Bank, were $6,997 and $16,804 with the former number around 41.6% of the latter (Statistics Times Website, 2021); yet China, the center of global manufacturing, produces more than five times the magnitude of power produced by India (Statista Website, 2021a,b). We use our model to determine the impact of (i) short run targets for production of dirty power and (ii) the demand for power on emissions and emission intensity of power production in the long run, when clean inputs for power production are brought into use.

The assumption regarding cost advantage when we compare dirty power to clean power is important for the outcomes of our model. Recent estimates indicate that per unit cost for solar power generation is lower than that for most coal-based plants. But this might be misleading. First, expansion of renewables-based power capacity often involves certain logistical problems: given that generation of power based on renewables such as solar and wind energy is only limited to certain hours of the day, an increase in the proportion of all power production accounted for by power based on such renewable sources requires costly alterations in the grid. Second, solar plants require large land area and therefore have to be built in remote locations which have to be then connected to the grid through lengthy and costly transmission cables. In India, government provision of large tracts of land to companies for solar plants involves costly dialogue and disputes with villagers.

Thus, the mentioned cost advantage may be lost or overwhelmed as the government has to make up the losses involved in disputes or compensation by ensuring that the consumer price represents a higher mark-up over producer price for solar power than for thermal power.¹ Note that companies also get hurt by the mentioned disputes as the operation of the newly built solar power plants gets stuck during the period of dispute; the equity of associated private firms not yielding any return for a significant period of time represents a cost to these companies. In other words, the average variable operational cost for firms producing solar power is probably much higher than that for firms producing thermal power, even though the inequality is reversed if we consider variable costs associated only with the process of production. This is reflected in the assumptions of our model presented below with its results.

2. Model

Consider the production of ‘electric power’ in a stylized model where the required resource input and therefore technology is ‘clean’ or ‘dirty’. The model built here actually applies more generally to production of any resource-based service based on the same possibilities in regard to technology/resource inputs.

The resource input/technology therefore is of type i, where i refers to ‘clean’(c) or ‘dirty’(d). An example of a clean input is solar energy and of a dirty input is coal. Let $z_i$ refer to the amount of input type i employed in the production process of that type. We assume that output of the resource-based service is therefore given by

where $L_i$ is the amount of labor employed, and $A_i$ is a coefficient. $A_i{L}_i^{{\beta }_i}$ denotes the magnitude of an intermediate input which is combined with employment of amount $z_i$ (units suitably chosen) of the relevant resource input in the ratio 1:1 to produce an equal amount of electric power. To make our understanding more complete, we define a unit of resource input of type i to be the amount which is used to manufacture 1 unit of electric power; thus, $z_i$ units of this input will be used to manufacture $z_i=y_i$ units of electric power but that manufacture would also require ‘operation and maintenance’ of fixed capital using labor given by $A_i{L}_i^{{\beta }_i}$, equal in magnitude to the level of output produced.

As mentioned, a ‘dirty input’ could be coal and we can consider the amount of coal used to produce 1 kilowatt hour (KWH) of power as the unit for measuring the input of coal in the production process. The same would hold for other dirty inputs. Let the cost to the firm for a unit of the dirty input be $t_d$. In regard to clean inputs such as solar power or wind, the firm will not have to directly pay for the input but will have to invest in equipment whose quantity would vary with the amount of energy trapped by the firm’s plant per unit time. Suppose a certain fixed surface area of solar panel lasts for a time period which is X hours long and produces 1 KWH per unit time (1 hour). The use of this surface area per unit time is then defined as a unit of clean input. If the firm pays $p_c$ to buy this surface area of solar panel, then obviously the unit cost of the clean input is $p_c(\frac{1+r}{X})=t_c$, where r is a suitable rate of return which captures the opportunity cost of investment in solar panels.

Our model is stylized and illustrative and though there are many types of resource inputs (coal, natural gas, solar energy, wind energy etc.) in the real world which can be used as alternatives to produce electric power, we consider a world in which there are only two types, which as mentioned we refer to as ‘clean’ and ‘dirty’. We shall consider $t_c>t_d$ in constructing our stylized world, again borrowing from the real world in which use of renewables such as solar energy is associated with various costs not associated with dirty manufacture of power: partial reconstruction of the grid, laying of long transmission cables, disputes with villagers in regard to land and associated compensation and delays. We assume that the use of clean technology is associated with zero emission whereas the use of dirty technology is associated with significant emissions. We also assume that emission is proportional to the scale of dirty output (output using the dirty technology) which enables us to define a unit of emission such that 1 unit of dirty output leads to 1 unit of emission. In short, the emission intensity of output associated with clean and dirty technologies is 0 and 1, respectively, and the average emission intensity of input in our stylized economy, e, would belong to $[0,1]$.

To keep matters simple, we assume that $A_i=1$ and ${\beta }_i=\beta$ for all i where i = c, d. The input requirement functions are thus given by ${L_i=y}_i^{(\frac{1}{\beta })}$and $z_i=y_i$. Thus, the cost of production of $y_i$ is given by

where $\frac{1}{\beta }=\alpha >1,r$ is the price of capital, and w is the price of labour. Note that $\frac{rK}{y_i}+w{y}_i^{\alpha -1}+t_i$ is the average total cost of production, which can be separated into average fixed cost, $\frac{rK}{y_i}$ and average variable cost, $w{y}_i^{\alpha -1}+t_i$. This has the standard U-shape because of a falling average fixed cost and rising average variable cost with respect to output. Average variable cost consists of a component $t_i$ which does not vary with the scale of production and another component $w{y}_i^{\alpha -1}$, the product of the wage rate w and $y_i^{\alpha -1}$, the labor-output ratio which increases with output because of diminishing marginal product of labor.

The firm level supply curves are obtained by solving for output using the equality between price and marginal cost, the condition for profit maximization. Marginal cost (MC) is given by $\alpha w{y}_i^{\alpha -1}+t_i$, which again has a component which increases with output and another which is invariant. Let p be the price of output. Then according to the price-marginal cost equality

The second equality gives the firm level supply function for power production based on input type i for $p\ge t_i$ Note that $y_i=0$ for $p\le t_i$ as it is not rational for the firm to produce any positive level of output if it spends more on resource input per unit of output than the price it gets for it. Therefore for ${p\le t}_d<t_c$, $y_c=y_d=0$; for $t_d<p\le t_c,y_d>y_c=0$; and for $t_d<t_c<p$, $y_d>y_c>0$ Given $t_d{<t}_c$, as assumed, the marginal cost curve for production of clean power is above that for production of dirty power, resulting obviously in a firm level supply curve for the former which is to the left of that for the latter and higher profits for the firm supplying dirty power at any price satisfying $p>t_d$.

Assuming $\alpha =2$ to make the algebra simpler yields the following firm level supply function :

We assume that the government gives $n_d$ licenses for production of dirty power which form the basis of an initial short run equilibrium in the industry. At a later point of time, the technology for production of clean power becomes available and therefore, in the long run, firms based on this technology (clean firms) have an opportunity to enter the industry. However, whether such entry actually happens or not depends on firms possessing this technology sensing an opportunity to make supernormal profits after entry.

We also assume a linear demand curve given by

The equation for the demand curve may be rewritten as follows : where $\bar{p}=\frac{C}{D}$ denotes the choke price.

Before we proceed further, we need to calculate the prices corresponding to normal profits for the dirty and clean technology (denoted by $p_{\beta }$ and $p_{\theta }$ respectively) and the firm level outputs corresponding to such prices ($y_{\beta }$ and $y_{\theta })$. We denote the quantity demanded in the market at these two prices as $Y_{\beta }$ and $Y_{\theta }$ respectively. Denoting marginal cost and average total cost by $\mathrm{\text{MC}}$ and $\mathrm{\text{ATC}}$ respectively, the price and output levels which bring about normal profits in equilibrium are given by

with the second equality occurring at the minimum point of the ATC curve (see Varian, 1992 etc.). This in turn implies that the price associated with normal profits is given by the level of minimized ATC. Now MC = ATC for the clean firm implies that Thus, $p_{\theta }$ is given by substituting $y_c=\sqrt{\frac{rK}{w}}$ in the expression for ATC, $w{y}_c+t_c+\frac{rK}{y_c}$. This equals $2\sqrt{rKw}+t_c$. Similarly, $p_{\beta }=2\sqrt{rKw}+t_d$ and $y_{\beta }=\sqrt{\frac{rK}{w}}$.

Note that the supply curve of a typical dirty firm is given by $y_d=(\frac{p-t_d}{2w})$. In order to achieve any output Y in short run demand-supply equilibrium, the needed price is given by solving D [$\bar{p}-p]=Y\Rightarrow \left(\bar{p}-\frac{Y}{D}\right)=p(Y)$, the inverse demand function. The number of firms needed to be given licenses, $n_d(Y)$, to achieve this output is got from the equation $n_d(Y)\left(\frac{p(Y)-t_d}{2w}\right)=Y\Rightarrow n_d(Y)=\frac{2wY}{p(Y)-t_d}=\frac{2wY}{\bar{p}-\frac{Y}{D}-t_d}=\frac{2w}{\frac{\bar{p}-t_d}{Y}-\frac{1}{D}}$. The above shows that $n_d(Y)$ is increasing in the target, Y given $p-t_d>0$, an inequality which should always hold true for a positive amount to be supplied. It is also increasing in w and decreasing in D. Further note that the government will not target a dirty output greater than $Y_{\beta }$ corresponding to $n_d(Y_{\beta })$ licenses by issuing even more licenses for dirty firms, given that once $n_d(Y_{\beta })$ licenses are taken up by firms, no more firms will consider entry because of anticipated losses.

See Figure 1. The industry supply curve is determined by the government giving enough licenses to firms such that it is a flat enough line joining the point (0, $t_d$) to the point $(Y,p(Y))$ on the demand curve, where Y is the amount of dirty output targeted in the short run. In this case, the firm level supply, a function of supply price, is multiplied by a natural number such that it is equal to the targeted supply/demand at $p(Y)$ This number $n_d(Y)$ increases as Y increases but along with increase in Y, we see that demand-supply equilibrium implies price equaling $p(Y)$ which keeps on decreasing as Y increases. With increase in $n_d(Y)$ the industry supply curve is obviously flatter.

Study Figure 1 and consider the pursuit in the short run, through the appropriate number of licenses, of two different levels of output, less than $Y_{\theta }$, given by $Y_0$ and $Y_1$ where $Y_0<Y_1$. Now consider the long run equilibrium corresponding to pursuit of each of these levels of output in the short run. In each long run equilibrium, the total output, clean plus dirty, is always given by $Y_{\theta }$ and price is given by $p_{\theta }$: this is because the short run equilibrium price is always greater than $p_{\theta }$ and the possibility for clean firms to earn supernormal profits by entering is exploited by them so that price actually falls to $p_{\theta }$ The ratio of dirty outputs or emission intensities in long run equilibrium corresponding to these short run targets is given by $\frac{n_d(Y_0)\left(\frac{p_{\theta }-t_c}{2w}\right)}{n_d(Y_1)\left(\frac{p_{\theta }-t_c}{2w}\right)}=\frac{n_d(Y_0)}{n_d(Y_1)}$. Given that $Y_0<Y_1$, $n_d(Y_0)<n_d(Y_1)$ i.e., ratio of emission intensities is less than 1. In other words, as the amount of dirty output pursued for production in the short run increases, the long run emissions and emission intensity increase. The figure illustrates this: $Y_0^L<Y_1^L$ where $Y_0^L$ is the long run emission corresponding to the target $Y_0$ and $Y_1^L$ corresponds to $Y_1$.

If the government provides a number of licenses which is greater than or equal to $n_d(Y_{\theta })$ this implies that the targeted output for the dirty firms in the short run is greater than or equal to $Y_{\theta }$ and therefore $p\le p_{\theta }$ which implies that clean firms will not enter the industry as there are no supernormal profits to attract them. With long run number of dirty firms equal to the number of licenses awarded in the short run, the long run industry supply curve of output is the same as the short run industry supply curve for dirty output which implies that the short run equilibrium is replicated in the long run. In other words, the whole output is dirtily produced and therefore the emission intensity of output is equal to 1. Thus, when we consider the short run equilibrium target for dirty output to be enhanced from low levels below $Y_{\theta }$ and then crossing $Y_{\theta }$ and going beyond it in magnitude, the long run emission intensity of output increases in the target and reaches 1 at target equaling $Y_{\theta }$, finally remaining at 1 for higher short run equilibrium targets.

Proposition 1. Emissions and emission intensity in long run equilibrium will be increasing in the level of short run equilibrium output targeted by the government as long as that target is below $Y_{\theta }$ but emission intensity will be less than 1 in magnitude. For targets in excess of or equal to $Y_{\theta }$, the long run emission intensity is constant at 1, while emissions will keep on increasing with the target.

Now consider the case of a higher w, K or r. This leads to a higher $p_{\theta }=t_c+2\sqrt{rKw}$. and a lower $Y_{\theta }$. At the same time, the industry level supply curve needed to attain a given output, Y as short run equilibrium output does not change. Given that the industry supply curve for targeting a certain level of dirty output in the short run, less than the initial level of $Y_{\theta }$, remains the same, the amount of the dirty output produced in long run equilibrium at the enhanced level of $p_{\theta }$ increases for a constant level of this target. Thus, the emissions and emission intensity in long run equilibrium, the ratio of emissions produced in long run equilibrium and $Y_{\theta }$ are higher.

Proposition 2. Assume emission-intensity in long run equilibrium to be less than 1 for a given target for production of dirty output in short run equilibrium. Consider a higher $r,K$ or w than before. This results in higher (lower) $p_{\theta }{(Y}_{\theta })$. The emissions and emission-intensity in the long run equilibrium will be higher.

A higher r, K or w basically implies that it is more costly to produce power. This limits the entry of clean firms in the long run and increases the price of power in long run equilibrium with the second outcome inducing dirty firms to emit more. Thus, emissions and emission intensity in the long run will be higher.

2.1. The case of taxation

Consider the imposition of a per unit tax, T on dirty output such that $0<T\le t_c-t_d$. Given that 1 unit of output results from 1 unit of resource input, we can consider the introduction of the tax to lead to an increase in the per unit cost of the input by T. Given that the normal profit inducing price for technology i is $2\sqrt{rKw}+t_i$, the introduction of such a tax implies that $p_{\beta }$ rises ($Y_{\beta }$ falls) but is less than or equal to $p_{\theta }$, which remains unchanged. The firm level supply functions are thus given by $\frac{p-t_d}{2w}$ and $\frac{p-t_d-T}{2w}$ for the cases of ‘no tax’ and ‘tax’ respectively, with the first supply curve corresponding to an intercept on the vertical axis of $t_d$ and the other supply curve corresponding to an intercept on the vertical axis of $t_d+T$. The short run industry level supply curve for each of the two cases ($S_0$ and $S_T$ respectively in Figure 2) has the same intercept as the corresponding firm level supply curve. However, the higher intercept for the case of ‘tax’ would imply that any targeted output for short run equilibrium (Y in Figure 2) lower than $Y_{\theta }$ has to be attained through a supply curve with a flatter slope i.e., a higher number of dirty firms given licenses.

Geometrically, looking at Figure 2, at $p_{\theta }$ we have the dirty firms allowed entry producing a higher amount of dirty output in the case of the ‘no tax option’ in long run equilibrium as opposed to these firms in the case of the ‘tax option’ ($Y^L>Y_T^L)$. Note that total output produced in long run equilibrium, if the government targets an output smaller than $Y_{\theta }$ in the short run as dirty output, does not vary and is equal to $Y_{\theta }$. This implies that the ratio of emission intensities of ‘no tax’ and ‘tax’ options is greater than 1.

As mentioned, targets for short run output equal to $Y_{\theta }$ or greater than that level result in the same output produced in the long run by the dirty firms given licenses in the short run as none of the clean firms find it profitable to enter the industry at the corresponding price less than $p_{\theta }$. Thus, short run equilibrium is replicated as long run equilibrium irrespective of the per unit cost of the resource input, (the only effect of the tax is to raise that per unit cost), and hence for both cases of ‘no tax’ and ‘tax’ the emission intensity of output is 1.

Note that to attain the same short run equilibrium output less than $Y_{\theta }$ the supply functions for the case with ‘no tax’ and ‘tax’ ($T\le t_c-t_d$) imply that

where $n_d^T$ represents the number of firms on which tax T is imposed which is consistent with attainment of the specified target and $n_d$ represents the number of firms consistent with attainment of the same target when tax T is not imposed. Given such notation, we can easily see that the ratio of dirty output in long run equilibrium in the case of ‘tax’ to that for the case of ‘no tax’, $\frac{n_d^T}{n_d}\frac{p_{\theta }-t_d-T}{p_{\theta }-t_d}$ equals $(\frac{p-t_d}{p-t_d-T})∕(\frac{p_{\theta }-t_d}{p_{\theta }-t_d-T})$. Note that this ratio is obviously equal to the ratio of emission intensities in long run equilibrium corresponding to the case of ‘tax’ and ‘no tax’. This ratio is decreasing in p and hence increasing in the targeted output, reaching a value of 1 at $Y_{\theta }$.

However, when the common targeted short run output is greater than $Y_{\theta }$ for the cases of ‘no tax’ and ‘tax’ the short run equilibrium output is replicated in the long run, resulting in the mentioned ratio of equal to 1.

Proposition 3. Consider a tax T such that $0<T\le t_c-t_d$. Also assume that the same output is targeted for the cases of ‘tax’ and ‘no tax’ in the short run. In that case, the ratio of emission-intensities in the long run between the cases of ‘tax’ and ‘no tax’ is increasing in the short run output targeted if that targeted output is less than $Y_{\theta }$, reaches 1 at $Y_{\theta }$ and remains at the same value for further increases in target.

In Figure 3 below we graphically illustrate the results of Propositions 1 and 3. First, it exhibits that in long run equilibrium, emission intensity for the case of ‘tax’ as well as ‘no tax’ increases as short run equilibrium output increases but is below $Y_{\theta }$ i.e., the slope of a ray from the origin to the emission trajectory in Figure 3 increases as the short run equilibrium output increases, thus implying that this emission trajectory is convex. Further note that the percentage gap in emission intensity between the ‘no tax’ as well as ‘tax’ cases, the gap in emission intensity as a percentage of the emission intensity for the ‘no tax’ case, decreases in the short run equilibrium output (target), reaching 0 as that output reaches $Y_{\theta }$.

Note that, as already mentioned, the effect of per unit tax on output is to increase the per unit cost of resource input by that per unit tax. For any targeted output less than $Y_{\theta }$ therefore a higher tax would increase the intercept on the vertical axis by a higher margin for the firm level supply curve for dirty output as well as the industry level supply curve for such output, given that industry level supply at any given price is the product of firm level supply and the number of firms. This implies an even flatter supply industry supply curve for dirty output for a given level of output less than $Y_{\theta }$targeted in short run equilibrium and therefore even lower emission and emission intensity of output in long run equilibrium. Given the previous proposition, we have the following proposition.

Proposition 4. Consider a per unit tax T on dirty output such that $0<T\le t_c-t_d$. A higher per unit tax within this range on dirty output implies that the ratio of emission intensity in the case of ‘tax’ to that for the case of ‘no tax’ in long run equilibrium, for a given level of short run equilibrium output, is lower. However, this ratio is increasing in short run equilibrium output in the interval $(0,Y_{\theta }]$ and reaches 1 at $Y_{\theta }$, remaining constant at 1 thereafter.

Now consider the case of tax $T>t_c-t_d$. $p_{\beta }$, lower than $p_{\theta }$ before the tax, rises ($Y_{\beta }$ falls) as a result of the tax and becomes greater than $p_{\theta }$, which is not altered by the tax. After the imposition of the tax, the short run output has to lie in the range $(0,Y_{\beta }]$. But this is consistent with supernormal or normal profits for the dirty firms given licenses and supernormal profits for the clean firms. In the long run, dirty firms cannot enter the industry by assumption but clean firms can and will because of anticipation of supernormal profits. Such entry will drive the price in the long run to below $p_{\beta }$ which is still consistent with supernormal profits for the clean firm. But as price is driven below $p_{\beta }$ the exit of dirty firms will start and price will keep moving up and down in the interval $[p_{\beta },p_{\theta }]$ depending on the relative strength of the mentioned entry and exit. If entry overwhelms exit and takes the price to $p_{\theta }$, it will momentarily stop because of normal profits for the clean firm, but losses to the remaining dirty firms will take the price to the interior of the interval so that the mentioned entry will again start. If exit is stronger than entry and takes the price to $p_{\beta }$, exit will momentarily stop because of normal profits for the dirty firm but entry of clean firms will continue. Thus, price will again move into the interior of the mentioned interval to facilitate the mentioned entry as well as exit. This will happen till all the dirty firms have left the industry. If this coincides with the occurrence of a price in $(p_{\beta },p_{\theta }]$ then more of the mentioned entry will take place and take the price to $p_{\theta }$; if the price occurring is $p_{\theta }$ then the system will stay at that price. Equilibrium will take place with a total industry output of $Y_{\theta }$ being produced, all through clean means and zero emissions. Of course, the dirty firms might foresee the entire dynamics and therefore all exit as soon as they see the clean firms entering in the long run, thus leading to the same equilibrium.

For a country with low demand imposing a tax $T>t_c-t_d$, which is large, when the clean technology is not available would imply a very low per firm supply for a target necessarily less than $Y_{\beta }$. Thus, producer surplus per firm would be very low and undesirable. Hence the government may start with a much lower level of tax or no tax and then introduce tax $T>t_c-t_d$ just before it anticipates the emergence of clean technology to attain the long run equilibrium discussed in the last paragraph. Even here note that if $t_c$ is large and/or the demand low, the total amount of power supplied, consisting of just power based on clean input, would be low. Thus, the government will think of imposing such a large tax even in the long run only if these conditions are not satisfied.

Proposition 5. A per unit tax on dirty output in the short run, $T>t_c-t_d$ will always result in zero emissions and a total output level of $Y_{\theta }$, entirely clean, in long run equilibrium.

2.2. The impact of the magnitude of demand on emissions and emission intensity

Let us consider a country, hereafter called the first country, whose demand curve is given by $D_0$ and whose short run production, in which only the dirty technology is used, is given by $Y_0$ with corresponding price given by $p_0$. The government achieves this short run output by allowing the number of dirty firms in the short run that results in the industry supply curve, $S_0$ (see Figure 4). Now consider another country with higher demand, $D_1$. This country is hereafter called the second country. Note that $Y_0$ is demanded in the second country at price denoted by $p_0^{\prime }>p_0$.

Let $Y_{\theta }$ ($Y_{\beta })$ refer to demand in the second country corresponding to normal profit inducing price, $p_{\theta }$ ($p_{\beta })$ in the case of the clean (dirty technology). Further we denote $Y_1$ as the level of short run equilibrium output for the second country which results from the number of licenses granted to dirty firms in the first country; and $Y_2$ the level of quantity demanded in the second country at $p_0$. Finally, we consider $Y_{\alpha }$ and $Y_{\gamma }$ to be the lowest levels of short run equilibrium output which correspond, respectively, to the same emissions and emission intensity in the second country in long run equilibrium as that in the long run equilibrium of the first country. The casewise results are as follows:

(i)	$p_0>p_{\theta }$: Refer to Figure 4. Note that for a target of $Y_1$ the emissions in the second country in long run equilibrium are the same as that in the first country $(Y_{\alpha 1}=Y_1)$ but given higher demand the emission intensity in the former is lower. Hence, the target needs to be raised beyond $Y_1$ so that it reaches the emission intensity corresponding to the first country. Given that the emission intensity in the case of the first country is less than 1, it follows that $Y_{\gamma }\in (Y_1,Y_{\theta })$.
(ii)	$p_0<p_{\theta }<p_0^{\prime }$: Refer to Figure 5. In this case, emissions and output for the first country are equal to $Y_0$ in both short and long run equilibrium and therefore emission intensity is equal to 1. In order to determine the industry supply curve S which yields emission equal to $Y_0$ in the second country in long run equilibrium we need to draw a horizontal line from a vertical intercept of $p_{\theta }$, a vertical line from a horizontal intercept of $Y_0$ and then join the point of intersection of the vertical and horizontal lines with the point marking the vertical intercept equal to $t_d$. The output at which this supply curve intersects the demand curve for the second country is $Y_{\alpha }$ which is less than both $Y_{\theta }$ and $Y_1$ but greater than $Y_0$. In other words, $Y_{\alpha }\in (Y_{0}min[Y_1,Y_{\theta }])$. By definition, $Y_{\gamma }=Y_{\theta }$, as $Y_{\theta }$ is the lowest output at which emission intensity in long run equilibrium for the second country is 1, the emission intensity for the first country in long run equilibrium.
(iii)	$p_0^{\prime }\le p_{\theta }<$ choke price for second country: Again, in this case, emissions for the first country are equal to $Y_0$ in both short and long run equilibrium and therefore emission intensity is equal to 1. In this case, it is easy to see that $Y_{\gamma }=Y_{\theta }$ and $Y_{\alpha }=Y_0$.
(iv)	$p_0^{\prime }<$ choke price for second country $\le p_{\theta }$: Emission/dirty output in long run equilibrium in both countries is always equal to short run equilibrium output for all levels of such output, thus yielding an emission intensity of output in long run equilibrium of 1.

Our exploration shows that if the normal profit inducing price for the clean technology is low enough due to low cost of the clean input and other factors of production (cases i and ii), a higher demand makes it possible to support a lower level of emissions (emission intensity) in long run equilibrium through the same or higher short run equilibrium output, given Proposition 1. This is because of the boost that higher demand gives to short run equilibrium price which results in greater entry by clean firms in the long run.

3. Conclusion and Policy Implications

In this paper, we consider a framework which mirrors real experiences — a country which only has access to dirty technology for producing electric power in the short run but both clean and dirty power in the long run. This is quite typical of the experiences of present-day developing countries such as China and India. We assume that the government sets a target for power production in the short run. Setting such targets is a practice which has been widely adopted, especially in developing countries, the reason being that power generation constitutes the engine for economic growth. Targets are also being set nowadays for the share of power produced from clean and renewable sources in total power production.

Efficient technologies for producing power using renewable sources such as solar and wind energy are a recent phenomenon. As mentioned in the introduction, countries such as China and India have traditionally used energy sources such as coal — which contribute majorly to pollution and therefore to morbidity, mortality and climate change — to meet their targets for power production. Now that cleaner sources are becoming available, this paper has sought to analyze how targets for power production in the era predating the arrival of clean and efficient technology will affect future outcomes in regard to the amount of emission and the average emission intensity of power production.

It is possible that policy makers in the past, if they were not myopic, were aware of the shadow cast by targets in regard to production of dirty power on subsequent pollution scenarios. Still, they were faced with a dilemma as growth considerations dictated that they set ambitious targets for the production of dirty power. Finally, even though the use of clean and renewable sources has made considerable headway in India and China this policy dilemma is still present: thermal power plants continue to get commissioned in these countries and other developing ones because of growth considerations, even though to a more limited extent.

It is easy to see the nature of this dilemma: stiffer targets for production of dirty power necessitates the entry of more firms; but given the demand for power, more of such entry depresses price and discourages entry by firms producing clean power in the long run. Thus, the mentioned stiffer targets cast their shadow on future emission scenarios: higher emissions and emission intensity.

The question then arises whether the policy maker can eat her cake and have it too: set ambitious targets in the short run and yet avoid the mentioned shadow by using additional policy instruments. Our analysis shows that taxation of firms producing dirty power works: when clean competitors emerge, they get a bigger share of the market on account of taxation depressing the competitiveness of dirty firms.

However, there are significant constraints on the leveraging of taxation for making large short run targets compatible with low emissions in the long run. If the tax on dirty firms is not too high, a high enough target and therefore entry by dirty firms might result in a price so low that clean firms will not even consider entry into the market. Of course, the alternative would be to levy such a high tax that clean firms have a clear competitive advantage on entry but such a tax would limit production in the short run when only dirty firms are operating.

It is possible, however, to make the timing of tax imposition roughly coincident with the entry of clean technologies if the policy maker is well informed. There would thus be a smooth enough transition to clean firms producing the economy’s requirement of power.

Finally, the demand facing the power sector is very important as illustrated by our comparison of India and China, the former with a much lower per capita income. Higher demand means that a targeted production of dirty power can be consumed at a much higher price. This creates more favorable conditions for entry of clean firms when such technologies become available. Thus, countries with higher demand can set stiffer targets when clean technologies are not available and yet hope for cleaner outcomes in the long run.

It is possible that the policy maker might not think in terms of the described framework and commission capacity for dirty power without considering the size of the market and the technological efficiency of firms. In that case, she risks ignoring the fact that her decisions would have long term implications for the state of pollution in the economy: unbridled entry by dirty firms into the power sector can enable high power consumption at low prices and rapid economic growth but does not augur well for the cleaning up of this sector in the long run.

ORCID

Partha Sen  https://orcid.org/0000-0003-2090-2194

Siddhartha Mitra  https://orcid.org/0000-0003-2173-0515

Notes

¹ The arguments presented here are the same as that in Dvorak (2020). It seems that the statements we have made in regard to solar power also hold for the comparison between ‘wind power’ and power from coal.