Synthetic demand functions for solar energy

Neil Manning and Ray Rees

There is as yet very little empirical data on solar energy demand with which to estimate demand functions econometrically. The approach of this paper is to use economic theory and engineen’ng information to construct demand functions for solar energy installations. We present a general model of consumer choice of energy-using durable goods under uncertainty and energy rationing. From this we derive demand functions for solar water heating equipment which we quantify using technological data and assumptions about future variables such as electn’city prices. Finally we conduct a sensitivity analysis which indicates that the main conclusions are robust to changes in the assumptions.

Keywords: Energy; Solar; Demand functions

Recent sharp increases in the prices of fossil fuels such as oil and coal have stimulated research into alternative renewable sources of energy. Most of the research into solar energy has been concerned with its technological aspects, and especially the optimization of the design characteristics of solar energy systems.

There has, of course, been an awareness of the import- ance of demand conditions, and a number of studies* have examined some aspects of the determinants of the demand for solar energy. However it is fair to say that the literature at present lacks a properly formulated model of consumer choice which can provide the basis for empirical work in this area. The approaches so far have been distinctly ad hoc.

A theoretical model is especially necessary in the case of solar energy because it is a new commodity. Empirical

*See: Boer,’ Landsberg’ and Sulock.’

Mr Manning is a Research Student, Economics Depart- ment and Solar Energy Unit, Mechanical Engineering Department, University College, PO Box 78, Cardiff, CFI 1 XL, UK; and Mr Rees is Professor of Economics, Economics Department, University College, Cardiff.

Mr Manning’s work is supported by a research-linked post- graduate training award from the Energy Panel of the Social Science Research Council.

Final manuscript received 8 February 1982.

data from which statistical demand functions could be estimated do not exist in anything like abundance. It is therefore necessary to synthesize demand functions using economic theory on the one hand and engineer- ing data on technological relationships on the other. We have to construct demand functions a priori, on the assumption that the consumer is appropriately modelled by more or less conventional economic theory, and using such engineering data as we have to impose restrictions on the forms of these functions and estimate their parameters.

If it were to capture the essential elements of the problem, the type of model required would also repre- sent an interesting extension of the theory of consumer choice. It should take account of the fact that the con- sumer combines energy inputs with a stock of consumer durables to produce amounts of final consumption services. Uncertainty concerning future energy prices and availability of energy inputs must also be central to the analysis. In the following section, we present a general model which could be used to analyse any problem of consumer choice in the energy sect0r.t In the remainder of the paper we apply this to the case of the demand for a particular kind of solar energy installation, that for water-heating.

The aim is to derive an expression for the demand function for this type of installation, involving parameters

tThis is an extension of the model first presented in Flees.’

0140-98B3/82/040225-07 303.00 0 1982 Butterworth & Co (Publishers) Ltd 225

which can readily be estimated and variables whose boeR values can reasonably be assumed. In this paper, we

= the consumer purchase (>O) or sale (<0) of bonds at period 0.

choose values of parameters and independent variables u = the interest factor, 1 +r, where T is the one- largely for illustrative purposes, but this is the first step period interest rate. in a programme of work which we hope will consider- wo,ws = the consumer’s income at period 0 and period ably improve these estimates. 1, state s = 1, . . . S, respectively.

X%R,m = the vector of energy inputs bought in state

The general model s=l,...S.

In this section we set out a model of consumer durable PSeR,m = the vector of prices of energy inputs in state

s=l As. choice which takes the following form. At time period 0, the present, the consumer must choose an amount of

X%R~ = the L&y of quantity rations on energy

current consumption, purchases and sales of energy- inputs existing in state s.

using durable goods, and an amount of borrowing or zs = the quantity of the consumption service

lending. He knows the current prices of the durable producedinstates=l,...S.

goods and the interest rate on bonds (the price of con- M%RZ = the vector of per unit maintenance costs in

sumption is identically unity). He must also formulate state s=l, . . . S.

plans of his choices for next period, at which time he wlll combine bought-in energy inputs with his durable

The consumer seeks to maximize the expected utility

goods to produce some amount of a consumption function :

service. He wilI divide his income next period between qco, c,, zs) = p&&o9 cs, zs) (1) these energy inputs and a consumption commodity.

The future period is however, subject to uncertainty: where each us, and therefore u, is strictly concave,

there is a set of possible states of the world across which increasing, and differentiable to any required order ln all

energy prices and his income may vary (though his future its arguments.

income or outgoings from bond purchases/sales ln the His budget constraints are:

current period are known with certainty). In each future co+q-ytb,=w, (2) state of the world, there are also fured quantity rations which place an upper bound on the amount of each

MS~(yoty)tcs+pS~xS=wstubo

energy input the consumer may buy. For example, in s=l,. . .s (3) the state of the world in which the consumer is subjected to an electricity power cut, his electricity quantity ration By substituting for b. in each constraint in Equation (3)

is zero. we replace the St 1 constraints in Equation (2), (3) by

Of course, the quantity ration may be made arbitrarily the S wealth constraints:

large to take account of states in which the consumer is effectively unrationed. The device of the quantity c,tq - yt&tMS- (y”t y)+ps-xs)=wo+2

ration is also particularly useful for handling solar energy. V V

This can be thought of as an energy input whose price is zero in each state of the world, but whose availability s=l,. . .s (4) varies -the consumer may or may not be able to acquire as much as he would llke of it, in a given state of the

These constraints reflect the assumption that the con-

worId. sumer can transfer income between time periods but not

Finally, he will incur, in the future, maintenance costs between states of the world so that, whichever state

of the consumer durables, which may also vary across occurs, his intertemporal budget must balance.

states of the world. In each state of the world, the consumer will combine

Thus the consumer makes current choices and plans energy inputs with durable goods to produce the con-

in the light of known current prices and income but sumption service. Initially, for purposes of developing

uncertain future prices, incomes, and availability of the general model, we shall assume that this is done

energy inputs. according to the production function:

We now set out the formal model. z, = f(yO + y, xS) (9

S = {s} = the set of future states of the world, with subject also to the quantity constraints:

cardinality S.

YO&R= = the vector of durable goods stocks at the XS=GXS (6)

beginning of period 0, the present. The production function f is assumed concave, increas-

yeR” = the vector of purchases/sales of durable goods ing and differentiable in all its arguments. Note that it is

inperiodO,withyi>-y/‘,i=l ,... n. assumed the same across all states of the world. We co,cs&+ = amounts of the consumption commodity at should also point out that we are adopting a simple but

period 0 and period 1, state s = 1, . . . S, not unreasonable treatment of depreciation and main- respectively.

46

tenance. The amount Mf must be spent per unit of = the vector of durable goods prices at period 0. durable i in state s to maintain the stock intact. If it is

Synthetic demand functions for solar energy: N. Manning and R. Rees

226 ENERGY ECONOMICS October 1982

Synthetic demand ftrnctions for solm energy: N. Manning and R. Rees

not spent the stock simply vanishes. We assume it is always spent.

It is useful to carry out the solution of the con- sumer’s choice problem in two stages.

The short-run cost function Take first the problem.

mineS=pS*xS XS

subject to the constraints in Equations (5) and (6) and given preassigned values of z, and Y. Assuming all energy inputs are positive at the optimum we have as necessary conditions:

pS- &vfS+/iS= 0 (7)

f>z, is>0 $[f -z,] = 0 (8)

stS<XS p”>0 ps. [is-‘;;“] = 0 (9)

where the symbol ^ denotes an optimal value, Vf’ E [af/ ax,!] is the vector of marginal products of the energy inputs, x, is a (scalar) Lagrange multiplier, and p” = [,clJ is a vector of Lagrange multipliers. We take it that household production is always efficient and so 5 > 0 always. When 2; < $, fiy = 0, and Equation (7) reduces to the standard optimal input condition. When 4; > 0 how- ever, z?,! = 3, the quantity constraint binds and:

(10)

ie the value of the marginal product of input j in state s exceeds its price, and fif is in fact the value of this excess.

Given that this system can be solved for the optimal x,!, we have the constant-output demandfunctions:

ki” = h;(p’, z,, Xs, y) s = 1, . _ . S (11) j=l,...m

and so we can derive the state-contingent cost functions:

e, = ps - hS = es(ps, z,, 2, u) (12)

It can be shown that the derivatives of e, are as follows:

aes - = -+Ye+Y,*‘) (13) aYi i

The first three of these derivatives are non-negative, the last is non-positive. We can then define the second stage of the problem:

Consumption and investment choices The problem is:

Subject to: e, MS w

cl,+4 *y+ ““+-+ -‘(yO+y)=wo+~ u v v V

s=l,...S

ENERGY ECONOMICS October 1982

(14)

where the cost function in Equation (12) has been used to derive the wealth constraints in (14) from those in Equation (4). Assuming no corner solutions, the necessary conditions are:

a+js=o 0

au bs - --=o s=l,. ..S ac, v

(15)

(16)

aii p, ae, aZ,

-,c=o s=l,...S (17)

SB,‘~~~‘~~+M;]=O i=l,...n (18) u dYi

together with the constraints in (14). Here, & > 0 is the Lagrange multiplier associated with the s’th wealth constraint and so has the interpretation of the marginal expected utility of wealth contingent on state s, and again A denotes an optimal value.

Writing

BS & = -

q& as the consumer’s time/risk discount rate for wealth contingent on state s: we have from Equations (15) and (16):

aiijac, -= - aii/aco

Ps

and from (18):

41’ ps As af ay_ (Y’ + Yi,2’) -Mf I 3

(19)

(20)

This last condition says that the optimal net purchase of the i’th durable good is that which equates the sum, across states of the world, of discounted marginal value products of the durable good, net of maintenance costs, to its price. Condition (19) gives us the standard condition on the marginal rate of substitution between current consumption and future consumption contingent on state s. Because of the absence of markets in state contin- gent claims, these will not in general be equal&d across states of the world.

Note finally that from Equations (16) and (17), and using Equation (13) we have:

aiqaz, - = iis aqac,

(21)

Now & can be interpreted as the marginal cost of the consumption service in state s, while the price of c, is of course unity. Hence this is a standard marginal rate of substitution = price ratio condition.

Given that from these conditions we can solve for the endogenous variables, we obtain, in particular, demand functions for the durable goods purchases:

pi =gi(q, WO, ws, 0, ~‘3 XSpMS) i=l,...n (22)

227

Synthetic demand functions for solat energy: N. Manning and R. Rees

Thus we find that in general, the demand for a durable good will depend on durable goods prices, current and future incomes, the interest rate, the vectors of future energy prices, the vectors of future quantity constraints on energy inputs, and maintenance costs.

A brief explanation of this formal model may be use- ful at this point. We are regarding the various forms of energy - coal, gas, electricity, solar - essentially as inputs into a production process conducted by the con- sumer. He combines these inputs with his stock of durable goods to produce consumption services such as warmth, light, hot water, etc.

The first step in the above analysis derives the relation- ships between the amount of the consumption service produced and its cost, given the durable good stock. It is assumed that the consumer seeks to minimize the produc- tion cost at each level of output of the consumption ser- vice -he acts rationally or efficiently. Because the prices and availabilities of the energy inputs are uncertain, there will be a different cost function for each possible set of assumptions about these, ie for each possible state of the world in the present terminology.

For example, we would expect the cost of producing a given degree of warmth to be higher when energy prices are assumed high and there is no electricity (because of a power-cut) than when energy prices are assumed low and all forms of energy are readily available. But of course these cost functions will also depend crucially on the stock of consumer durables possessed by the consumer, which is to be determined.

The second step in the analysis then considers the con- sumer’s choice of durable goods stock and present and future consumption levels. The particular scale and com- position of this stock will determine the cost of the con- sumption service for each set of assumptions about future energy prices and availabilities, and will in turn be influenced by them. On the one hand the higher the cost of producing the consumption service relative to the price of the consumption good in each future state of the world, the lower will tend to be the desired demand for the service and hence the smaller the durable good stock.

On the other hand the choice of particular durable goods will depend on the relative prices and availabilities of the energy inputs. An essential aspect of the model is the uncertainty about future prices and availabilities: the consumer’s demand for durable goods will depend on the probabilities he assigns to the levels of future prices and quantity rations, and on his attitude to risk. For example a risk-averse consumer may choose a durable goods stock which has a higher expected value of future ‘production cost’ than another, but a much lower prob- ability of being highly expensive (or, equivalently, of ‘allowing’ a very low level of the consumption service) in some state(s) of the world.

Our purpose here however is not to develop these general aspects of the model further, but to apply it to the particular case of the demand for solar installations. We now turn to this.

Specialization to the case of solar energy In applying the approach of the previous section to the case of solar energy we first need to be specific about the nature of the consumption service zs, the consumer durables yi and the energy inputs xi”. The case we take is that where the consumption service is water-heating. In this case, we assume that there are two main alternative technologies: a system involving only a tank with elec- trical immersion heater; and one which has in addition solar collector panels which pre-heat the water before it enters the tank where electrical heating may then be applied. The consumer durable inputs therefore are:

l Solar panels, measured in terms of the total collector area and denoted y 1 ;

l Electric immersion heater, measured by the electrical rating of the appliance and denoted y2;

l Water tank, measured in terms of volume and denoted

Y3.

y 1, y2 and y 3 are the major consumer durable inputs, and for practical purposes may be taken as proxies for other capital elements - such as pipework - which vary directly with them. The energy inputs are:

l Average solar radiation available at zero price and denoted xf;

l Electricity, denoted x;.

The output variable zs can be expressed as the total energy input in mega Joules (MJ) (net of tank losses) required to achieve a given temperature level of 6O”C, a figure commonly taken as an engineering standard. Thus, we take the volume of hot water demanded (the consumers’ desideratum) to be proportionately related to the net energy input, and this will in general vary across states of the world.

To apply the general model exactly would require information on consumers’ utility functions, probability beliefs, incomes, etc. We can however avoid these require- ments by making some assumptions which appear empirically defensible. It is usually argued by engineers that across a wide range of prices and incomes the consumer adopts a given size of hot water tank? denoted L3, and a given size of immersion heater denoted?,. Engineering studies have established the following rela- tionship among the output zs and the various inputs$:

*This relationship was derived from work on the technology of solar water heating systems reported in Kenna.s It is equivalent to whet economists would call a production function. It quanti- fies the relationship between the inputs of solar energy.electricitv, and scale of solar installation on the one hand, and energy delivered to the hot water tank on the other. Given the volume of the tank end temperature of the inlet and outlet water flows, the energy which has to be delivered to the tank varies directly with the total flow of hot water demanded. Thus the former can ba taken as the output level. Equation (23) is therefore the basic input-output relation. We are grateful to J. P. Kenna of the Solar Energy Unit, University College, Cardiff, for his help with this relationship.

228 ENERGY ECONOM ICS October 1982

Synthetic demand finctions for solar energy: N. Manning and R. Rees

ZS = z.[,,,3y(c2+ yg)] + cG*x:

- csy32’3 (23)

where

B I

= 1 - exp(-c&d~3)"3

[ 4x3 1

X3 is the peak solar flux which is taken as constant, and OL],... 0~s are experimentally determined parameters. From Equation (23) we can see that the electrical input required per unit of the given desired output zs is

4 2 -=- l- ZS WY2 [

OlBzSYl %Y3 +-

a2 +GY,P: I

- 7.13

WY2

(24)

where B; = oz3 y1 /z”. Now since the price of solar energy is zero, it follows that in any state of the world, the short-run cost of achieving the given output level is simply the expenditure on whatever residual electricity input is required to supplement the solar energy input. In other words, solar energy will be used up to the maximum available and then electricity will be used to complete the heating process. Hence the short-run cost function (denoted by e, in the preceding section) is simply :

where B: =p;zs/01,y2. Referring back to Equation (18), the condition which

relates to optimal purchase of the solar panels - the solar energy consumer durable y, - can be written:

% q, = -& -+MS

aY1

But, from Equation (25) we have in the present case that:

aes - = -A;@; +AS,y, +AS_y:) aYI

(27)

whereA: =cY,B~B:cY~; A;=2a2B;/B,; A<=(Bf/Bl)2. This can be interpreted as the reduction in the cost

of meeting the fixed output requirement resulting from a marginal increase in solar panel area y , ,given the avail- able solar energy in state s. By inserting this into Equation (26), we would obtain a relation which could then be solved to yield the demand for solar panel area as a function of its price 4,.

However, a difficulty here is that the time/risk dis- count rates /i, depend in general on the consumer’s utility function and state-contingent incomes, and so would be extremely difficult to estimate. To carry the analysis further, we make a risk neutrality assumption: we assume that the consumer’s marginal utilities of income are equal across states of the world, their common value denoted 0. It follows that the marginal

expected utility of income in state s is n,fi and so we have I

8s $6 =s bs =-= -=- ups UPp, lJ

Using Equations (27) and (28), Equation (26) then becomes:

q1 = F ?r” 4

-MS v LX; + A;yl + AS,y: 1

(28)

The right hand side of Equation (29) is effectively the ‘demand price’ of the solar installation, as a function of the area of panel installed, and given the state-distri- bution of the solar energy input.

In the following section, we present estimates of the (inverse) demand function for solar energy installations implied by Equation (29).

Numerical estimates

The expression in Equation (29) effectively tells us the maximum price which would be paid for a solar instal- lation of panel area y 1. In obtaining numerical estimates of the demand price - panel area function, we adopt parameter values some of which are available from hard technical data and some of which are assumptions. We detail these below. In addition, we have to generalize slightly. The previous analysis assumed only one future time period whereas in fact solar installations have a life of 1 O-20 years.

Therefore we have to adjust Equation (29) to take account of the extended time horizon. This adjustment is, however, quite straightforward. A ‘state of the world’ is now a complete history of the environment from period 1 to Ts, the future planning horizon, which depends on s. We let pf, xf, and My denote respectively the electricity price, solar energy input and per unit maintenance cost at time t = 1, . . _ Ts in state s. The generalization of Equation (29) to P periods is then:

s

q1 = E k u-‘(AT&Y; + A:yy, + A$:) - Ms? (30) t=1 I

where we assume the interest rate constant over time. It is this expression which we now estimate numerically.

The ultimate purpose of the present analysis is to provide forecasts of the probable demands for solar installations. Equation (30) can be regarded as giving a succinct statement of the relation between solar panel area and demand price for a rational consumer with as yet unspecified price expectations, expectations about the availability of solar energy, and probability beliefs. To yield the desired forecasts we would have to know the values of these expectations and beliefs which con- sumers typically have. But for the time being, in the absence of this information, we insert our own estimates into the equation and calculate the results.

This is partly to illustrate our theoretical approach and test whether its results appear reasonable. But also,

ENERGY ECONOMICS October 1982 229

Synthetic demand functions for solar energy: N. Mmningand R. Rees

in part, it suggests a normative interpretation of the analysis. If our chosen values are the best available, then this suggests what ought to be the demand for solar installations, and this may have future policy signifi- cance. We now set out therefore the assumptions we acquire to yield numerical estimates of the synthetic demand function in Equation (30).

Experimental engineering data provide us with the numerical values of the parameters of the basic ‘produc- tion function’ relationships in Equation (23). These are:

(IL1 = 0.74 a3 = 0.74 0~s = 1.80 ol = 3.60

os = 0.80 o4 = 540.00 (Ye = 0.50 X3 = 500.00 W/M2

ratings of immersion heater: 3 kW volume of tank: 0.12 m3

In the case of future electricity prices, maintenance costs, etc, we simply use our own best guesses.5 To assess the robustness of the results to these guesses we also carry out a sensitivity analysis. Thus we have:

Case I

u = 1.03 (real interest rate of 3%)

MS0 = S2

I

Annual maintenance cost per m* of

Me= (1.04)‘M”’ solar panel is $2 in real terms and grows at 4% pa

St Xl =33003630\

The probability distribution,

prob(x?) = OS71 o’29 j

each year, of annual solar radiation

, in MJ/m2

TS =5 10 20 The probability distribution prob(TS)= 0.26 0.52 0.22 of solar collector lifetimes

zSr = 10000 12000 14000 ’ The probability distri- prob(z’p bution of annual hot

1981-85 0.84 0.16 0.00 s water demand by the 1986-90 0.77 0.19 0.04 consumer, shifting 1991-2000 0.73 0.18 0.09 upward through time

@; 2% 4%

prob0-S ) 1981_85 o 31 o 69

1986-90 0107 0:93 1991_2000 o.12 0.88

I

The probability distribution of the annual real rate of increase in electricity price,

shifting through time

Given this basic case we then conduct a sensitivity analysis by defining:

Gzse 2 As for Case 1 except:

u= 1.02 M=‘=SO

This will therefore be more favourable to the solar instal-

0 Together with those of the Solar Energy Unit, University I I I I I I College, Cardiff, for whose help on these assumptions we are I 2 3 4 5 6 extremely grateful. Area m2

lation because the interest rate is lower and maintenance cost is zero.

Case 3 As for Case 1 except:

u= 1.06 Mm=56

This raises the interest rate to 6% and annual maintenance cost to %6/m*, and so is less favourable to solar installation, since the electricity cost savings are discounted more heavily and the installation is costlier to maintain.

Given these sets of assumptions, and choosing values of yr increasing in ?4m* steps from 1.5 m* to 6.0m2, we obtain the demand prices of solar collector installations as a function of this area. These are shown in Figure 1.

As we would expect, in each case the price per m* of solar installation a consumer (who possesses the expec- tations and beliefs and faces the interest rates we have

4c

3c

2G

IC

5

: C

._ Ii

-10

-20

-30

-40

I-

)-

I -

I-

)‘-

I-

,-

230 ENERGY ECONOMICS October 1982

Synthetic demand functions for solar enemy: N. Manning and R. Rees.

assumed) would pay decreases as the scale of installation increases. The results are not favourable to solar energy. In the central Case 1, the consumer would actually have to be paid to install more than about 3.5 m* of solar panel (the demand price is negative) and would be pre- pared to pay only about E25 per ma for the smallest feasible installation. This compares with a current price of 2140 per m*.

Even the more favourable case 2 produces demand prices which are well below current price levels, while the least favourable case 3 would require payments to the consumer ranging from %I 2 to MO per m*. The results are also quite sensitive to the particular assump- tions on interest rate and maintenance cost which are adopted. Even so, solar installations do not appear viable at current price levels.

Of course a sensitivity analysis could be conducted on a much larger scale, by changing the assumed distri- butions of equipment life, future electricity prices and sunshine. We are ourselves pursuing these. Indeed, by using Equation (30) and the technical parameter values in Case 1 the reader could also derive his own estimates.+ It is hard to escape the conclusion however that reason- able synthetic demand functions will lie well below the current supply price.

Conclusions

This paper starts with the observation that, in the absence of a reasonable body of existing data on demand for solar installations from which econometric demand functions could be estimated, it is necessary to synthe- size demand functions a priori. We first construct a general model which shows how a rational consumer chooses stocks of consumer durables and planned energy inputs in the presence of uncertainty about future energy prices and availabilities.

This model could be applied to the demand for any and all forms of energy, but here we specialize it to the

*We would be happy to supply any interested reader with the details of our programme for computing Equation (30).

case of solar installations and the output of hot water. We then use a ‘production function’ derived from experi- mental engineering work, and a set of assumptions about the key economic variables, to quantify the relationship between scale of solar installation and demand price.

The results suggest that the supply price of solar installations would have to fall from its present level by at least 75% if solar energy is to be attractive to a con- sumer with the beliefs and expectations adopted here. Even then only quite smalI installations would appear to be warranted.

Apart from exploring further the sensitivity of these results to changes in the assumptions, the main direction of further work would appear to be on the supply side of the market. The first question is: are there sufficient economies of scale in production of solar installations to make it probable that prices could be brought down to the levels suggested by this analysis as necessary for viability? And secondly, may there be external benefits of the environmental and ecological type, not captured in the calculations given here, which could justify subsidization of solar installations? Opinions on these questions already exist, but we hope that further research will yield more firmly based answers.

References

K. W. Boer, ‘Payback of solar systems’, Solar Energy, Vol20,1978, pp 225-232. P. T. Landsberg, ‘A simple model for solar energy economics in the United Kingdom’. Enerm. Vol 2. 1977,pp 149-159. - ’ -_ ’ ’ J. M. Sulock. ‘The economics of solar heatine: a note on the apprdpriate discount rate and the use-bf the g;;,,,t;,period’, Solar Energy, Vol 24, 1980, pp

R. Rees, ‘Consumer choice and non-price rationing in public utility pricing’, in B. M. Mitchell and P. R. Kleindorfer, eds, Regulated Industries and Public Enterprise: European and US Perspectives, Lexington Books, Lexington, MA, USA, 1980. J. P. Kenna, ‘A parametric study of open loop solar heating systems’, Proceedings of the Solar World Forum, Brighton, Sussex, UK, August 1981.

ENERGY ECONOMICS October 1982 231