## Abstract

In this paper we propose a theory to study how the aggregate demand of energy responds to energy prices and technical innovations that affect the price of energy services. In our theory, energy use is determined by the interaction of the choice of Energy Saving Technical Change with energy prices and Investment Specific Technical Change (ISTC). The key mechanism is that the energy saving technology is embodied in capital vintages as a requirement that determines their energy intensity. We show that higher ISTC that *increases the quality of capital goods* is an energy saving device and, therefore, a substitute for Energy Saving Technical Change (ESTC). However, higher ISTC that *rises the efficiency in producing capital goods* is energy consuming and fosters ESTC to compensate for the amount of energy required by the new investment. A higher price of energy also induces a higher level of ESTC, but the aggregate amount of energy used may not be affected as investment does not change. These effects are amplified with rising prices of energy. Thus, our theory can be used to test when and how we should see a rebound effect in energy use at the aggregate level and to evaluate the aggregate effect of any policy aiming to reduce energy use.

**Funding source: **Spanish Ministerio de Economía, Industria y Competitividad

**Award Identifier / Grant number: **ECO2016-76818, MDM 2014-0431

**Funding source: **Consejería de Educación, Juventud y Deportes de la Comunidad de Madrid

**Award Identifier / Grant number: **S2015/HUM-3444

**Funding statement: **Financial support from the Spanish Ministerio de Economía, Industria y Competitividad (grant ECO2016-76818) is gratefully acknowledged. Antonia Díaz thanks the Ministerio de Economía, Industria y Competitividad, María de Maeztu grant (MDM 2014-0431), and the Consejería de Educación, Juventud y Deportes de la Comunidad de Madrid for MadEco-CM grant (S2015/HUM-3444).

## Acknowledgement

We thank Raouf Boucekkine, Omar Licandro, Gustavo Marrero and Jesús Ruiz for insightful discussions. We also thank seminar participants at the Barcelona GSE Summer Forum 2015, Bellaterra Seminar Macro, ESEM 2016 and RIDGE December Forum 2016, for useful comments, as well as the editor and two anonymous referees. Puch thanks the Department of Economics at Universitat Autónoma de Barcelona for its support while staying as a visiting professor, year 2015–16.

# Appendices

## A Data and calibration

In this Appendix we document the construction of the data series we use in the empirical part of the paper. We obtain data from two sources: the Annual Energy Review (2000) and National Income and Product Accounts. The data we use can be accessed in the addresses: http://www.eia.gov/ and http://www.bea.gov. From now on we will refer to each source as AER, and NIPA, respectively. Our data set is available upon request.

### A.1 Energy price, use, and expenditures series

Our energy data covers primary energy consumption of end-users and is obtained from the Annual Energy Review (AER, hereafter). We consider four forms of energy: coal, petroleum, natural gas and electricity. AER (Table 2.1a) gives data on total energy consumption by end users measured in British termal units (BTUs) disaggregated into the four forms of energy considered. We denote these data on energy consumption for each type of energy by *Q _{it}*, where the index

*i*denotes the form of energy.

This measure *Q _{it}* is already net of energy consumption of the electricity sector. We subtract from total primary energy consumption of the industrial sector that of four energy sectors: oil and gas extraction, electricity and gas services, petroleum and coal production, and pipeline transportation. The BEA gives information on the net stock of Fixed Assets by industry and we assume that the amount of BTUs consumed by those four sectors, as a proportion of BTUs consumed by the industrial sector, is the same that the amount of capital in those sectors as a proportion of assets in the industrial sector. We define total energy use as

*E*= ∑

_{t}_{i}

*Q*, where

_{it}*Q*is the amount of BTUs yielded by each type of energy consumed.

_{it}*P*is the price in dollars per BTUs of energy type, divided by the implicit price deflator of non durable consumption goods and services in NIPA (which is constructed as a weighted average of the two implicit price deflators). For coal, natural gas and petroleum we use the production price series (AER, Table 3.1). For electricity, we use the retail price of electricity sold by electric utilities (see AER, Table 8.10). In Table 8.10 the price for electricity is in cents per kilowatt-hour. We use AER Table A.6 to convert the price to cents per BTUs. All prices are in real terms; i.e. divided by the implicit price deflator of non durable consumption goods and services. We construct the energy price deflator as

_{it}Finally, energy expenditure is

### A.2 Output, consumption, investment, and the capital stock

We follow the method described by Cooley and Prescott (1995) to construct broad measures of output, consumption, investment, and the capital stock. Specifically, our measure of capital includes private stock of capital, the stock of inventories, the stock of consumer durable goods and the government stock. Consequently, the measured value of GDP is augmented with the imputed flow of services from the stock of durable goods and the government stock. We subtract from each of the series of output, investment and capital the corresponding series for the energy producing sectors: oil and gas extraction, electricity and gas services, petroleum and coal production, and pipeline transportation, as our theory cannot account for the behavior of the energy-producing sectors. We have information on the three variables for the last two sectors but about the first two sectors we only have information about the net stock of capital, and we use it to impute estimates of output and investment. Gross output is the sum of value added and the final expenditure on energy. Real variables are obtained by dividing the nominal variables by the implicit price deflator of non durable consumption goods and services.

## B The putty-clay economy

### B.1 Proof of Proposition 2

We proceed in three steps:

*Only capital that is not very intensive, *

### Proof

The first order condition of the firm’s problem with respect to *k _{t}*(

*z*) is

For a given vintage *z*,

Since, in equilibrium, the return to capital must be non-negative,

*Across all units of capital of the same vintage, *

*where *

### Proof

The net return at time *t* is

In Lemma 3 we defined *ϕ* < 1 we know that *z* has a unique maximum at

*The equilibrium net return of any capital not used is equal to *

Now we can turn to the proof of Proposition 2. Inspecting the problem solved by the firm producing new capita, (8), we see that new capital is produced whenever

where *ξ*_{t}(*z*, *e*) is the Lagrange multiplier associated to the nonnegativity constraint on investment in the household’s problem. Expression (33) is the first order condition with respect to *k*_{t+1}(*z*, *e*) in the household’s problem. If *ξ*_{t}(*z*, *e*) > 0, the household does not want to invest in class (*z*, *e*). Working backwards, it must be the case that

Any new class of capital that is produced must satisfy

Thus,

Expression (36) determines the set of intensity types *e*_{t+1}. Moreover, it must be the case that *e* to rise the net return after period *t* + 1. Since agents have rational expectations, competition implies that households know that there is one intensity type that yields the highest present value of future net return. As a result, only one type has a positive price, which is the type that receives investment in equilibrium and, therefore, is produced. Thus, all units of capital of the same vintage have the same energy intensity.

### B.2 Proofs of Proposition 3 and Proposition 4

We need to conjecture that the service life of capital is constant. Thus, *t*. Any *t* is

The evolution of this stock can be written as

At the balanced growth path, it must be the case that energy grows at a constant rate. Thus, the distribution of energy used across utilized vintages must be constant. Thus, we can define the depreciation rate of energy use as

For the energy share in gross output, *e*_{t+1} falls at the combined rate (1 + *θ*)(1 + *γ*_{p}).

In order to prove that the service life of capital is constant (and finite) we need to obtain the growth rate of the output to capital services ratio, *y _{t}*/

*𝒦*

_{t}. Recall the expression of capital services, shown in (4). The evolution of the stock can be written as

At the balanced growth path, it must be the case that capital services grow at a constant rate. Thus, the distribution of services across utilized vintages must be constant. Thus, we can define the depreciation rate of capital services as

where *ψ* measures the weight of services used in production in total capital services

For capital services to grow at a constant rate, *g* the growth rate of gross output, *y _{t}*, it must be that

It follows that the output to capital services ratio,

### B.3 Proof of Proposition 5

From the household problem we know that the amount of investment, *x _{t}*, satisfies:

The optimal intensity level, *e*_{t+1}, satisfies

That is, the intensity level *e*_{t+1} is chosen so that the present value of energy expenditures amount to the fraction *ϕ* of the present value of future gross returns. Finally, the service life of capital satisfies

These three equation together determine the investment level, the detrended intensity level of investment, *T*. Using the fact that the gross return to capital, *T*, only depends on the discount factor, and the growth rate of income and capital services and not on the value of

It follows from (46) that an acceleration in ISTC growth reduces *T*. Now combining (44) and (45) we find that the value of

This equation, detrended, can be written as

It follows from (49) that an acceleration in ISTC growth rises

### B.4 The price of capital in units of gross output

In the Proof of Proposition 2 we have proved that the relative of investment in units of value added is (1 + *θ*)^{−t}. The price of investment in units of gross output is equal to

It is easy to check that *z*, *e*) satisfies:

The fact that capital services (in units of gross output) are perfect substitutes implies that the ratio of prices must be equal to the ratio of gross returns:

### B.5 The depreciation rate of the capital stock

From the expression of the aggregate stock of capital we find that the depreciation rate of capital may vary out of the balanced growth path. Its expression is

where *ψ* is shown in equation (42). Along the balanced growth path, the depreciation rate is constant.

### B.6 The market value of capital

We have aggregated capital using the price *q _{t}*(

*z*,

*e*), shown in expression (18) from Lemma 2. This is the price of capital in units of gross output. It is not the market price of capital (in units of value added),

*p*(

_{t}*z*,

*e*), which is given by the present value of all future returns, net of energy expenditures, as shown in expression (36). The market value of the total economically active stock is

The market price of capital of class (*z*, *e _{z}*) has a complicated expression. Nevertheless, at the balanced growth path, using equation (13) and Proposition 5, it can be written as

where factors

for all *z* ≤ *t* + 1, and *T* − (*t* + 1 − *z*) ≥ 1. Notice that this expression collapses to (1 + *θ*)^{−t} for the latest vintage, *t* + 1. The divergence between the market price and the value in units of gross value added increases with the age of capital. Thus, the stock of capital in units of gross output is larger than the value of capital in units of value added. Figure 2 show graphically the difference between the market price and the price in units of gross output in our benchmark calibration.

## C The balanced growth path in the putty-putty economy

It is convenient to find the expression of the stock of capital in units of final good, **k _{t}**:

Thus, we need to find the expression for the market price of capital. Here all capital is used in equilibrium. Expression (11) shows the net return to capital of vintage *z* in the putty-putty economy. Given the demand of energy shown in (12), it follows that

Since the price of new capital is equal to (1 + *θ*)^{−t}, it follows that the relative price of capital of vintage *z* at time *t* is

It is easy to check that the aggregate stock of capital satisfies

Expression (12) shows the energy demand associated to utilization of capital of vintage *z*. Thus, aggregate energy use can be written as

Using the expression for aggregate capital services shown in (4) we can express

which implies that we can express aggregate capital services as a function of aggregate energy use:

Using the expression for energy demand, (61), we find that aggregate value added can be expressed as

It follows that the growth rate of value added, gross output and capital in units of final good is equal to

The growth rate of aggregate energy use must be equal to (1 + *g*)/(1 + *γ*_{p}). Using (63), it follows that the price of capital services (that is, the price of capital adjusted by quality) falls at the rate:

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**Published Online:**2019-01-12

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