Chapter 4: A stock-flow consistent model for the analysis of stranded assets and the transition to a
4.2 The model
4.2.1 The firm sectors
This section presents the equations relating to the firm sector. Section 4.2.1.1 presents some general accounting identities. Section 4.2.1.2 discusses the determination of aggregate demand and the limits to firm supply. Section 4.2.1.3 covers the output of consumer goods. Section 4.2.1.4 looks at the determination of firm investment. Section 4.2.1.5 looks at the financing of firm investment. Section 4.2.1.6 presents the equations that determine the output and consumption of intermediate goods (i.e. the input-output framework). Section 4.2.1.7 covers the equations that deal with
employment, labour productivity and wages. Section 4.2.1.8 describes the cost and price structures employed by firms. Finally, Section 4.2.1.9 presents the equations that determine how carbon taxes are set and how the revenues they raise are distributed to households.
4.2.1.1 Firms – accounting identities
Before detailing the equations that determine firm behaviour, it is worth setting out some of the accounting identities that apply to firms and how these relate to final demand.
Starting with the general accounting identities, total real output (st) is the sum of real output by
green (sg) brown (sb) and other (so) firms (equation 1). For each firm sector real output is the sum of
real consumption (cj), real gross investment (ij), and real output of intermediate goods (icij) (equation
2).33
1 st = sg + sb + so
2 sj = cj + ij + icjo + icjg + icjb
With the real values determined, the nominal value of a sector’s output (Sj) can be calculated by
multiplying real sectoral output by the sector’s price level (pj) (equation 3). Total nominal output (St)
is then equal to the sum of the nominal outputs of each firm sector (Sj) (equation 4).
3 Sj = sj ∙ pj
4 St = Sg + Sb + So
Because intermediate goods are used in the production process, real final demand – real
consumption expenditures by households and real investment expenditures by firms – is not equal to real total output (i.e. the sum of all real production by firms). Instead, real final demand (yt) is
arrived at by netting out intermediate consumption from total output. This is what equation 5 does.34 The (a
ij’s) represent the coefficients from the A-matrix (see Section 4.2.1.6). Total nominal
final demand (Yt) is then equal to real final demand multiplied by the general price level (p)
(equation 6). The general price level is determined by Equation 87. 5 yt = ([1 - agg] ∙ sg) + ([0 - abg] ∙ sg) + ([0 - aog] ∙ sg) +
([0 - agb] ∙ sb) + ([1 - abb] ∙ sb) + ([0 - aob] ∙ sb) +
([0 - ago] ∙ so) + ([0 - abo] ∙ so) + ([1 - aoo] ∙ so)
6 Yt = yt ∙ p
4.2.1.2 Firms – supply and demand
This section discusses the factors related to the demand and supply of each firm sector’s goods. In the model the supply of goods by green or brown firms will not necessarily be equal to the demand for these goods. Instead, green firms and brown firms have a maximum level of output that is determined by its capital stock. When the demand for an energy sector’s goods exceeds its capacity limit, firms and households are forced to buy from the other (non-capacity constrained) energy sector. As mentioned above, this prevents unrealistic elasticities emerging when prices change. The rest of this section proceeds as follows. Part (a) describes the equations that determine the aggregate demand for each sector’s goods. 35 Part (b) describes the equations that determine the
maximum supply of each sector’s goods. Finally, part (c) describes the equations that determine the actual quantity of goods supplied by each sector.
34 Normally, matrix notation would be used to represent these equations and the other equations associated with the input-output matrix. However, to aid transparency, the equations in this chapter are reproduced exactly as they appear in the computer program used to numerically simulate the model (Eviews). In addition, it should be noted that final demand is also equal to: Yt = cg + cb + co + ig + ib + io
35 The determinants of the demand for investment, consumption and intermediate goods from each household and firm sector are discussed in detail in their respective sections (e.g. household demand for consumption goods is discussed in detail in the section on consumption demand).
(a) Aggregate demand
In this section the demand for goods from each sector are summed together to give the aggregate demand for each firm sector’s output. Two sets of equations are specified. The first of these includes investment goods, whereas the second is net of investment goods. Both sets of equations are necessary due to the assumption that firms prioritise the demand for investment spending over all other types of output demand.
Equation 7 says that the demand for a sector’s total real output (sD
j) is equal to the demand for
consumption goods (cD
j x), investment goods (iDj), and intermediate goods (icDij) from that sector.
Equation 8 says that the demand for a sector’s real output of consumption and intermediate goods (gdD
j) is equal to the demand for the output of a sector net of the investment goods it produces.
7 sD
j = cDj e + cDj n + icDjo + icDjg + icDjb + iDj
8 gdD
j = cDj e + cDj n + icDjo + icDjg + icDjb
(b) Maximum supply
We now turn to the capacity constraints of each sector. Equation 9 says that a sector’s maximum supply of consumption and intermediate consumption goods (gdS
j) (i.e. output net of investment) is
equal to the product of its real capital (k) and maximum output to capital ratio (ζj), minus its level of
desired investment. The principle here is that firms prioritise investment expenditures over all other forms of output, to ensure that they move as quickly as possible towards their target level of capital. This means that firm investment, and therefore the speed at which a sector increases its productive capacity, is not affected by the sector being at its capacity limit.
9 gdS
j = (kj ∙ ζj) - ij
Equations 10 and 11 determine the maximum supply of consumption (10) and intermediate consumption (11) goods by a sector. Equation 10 says that the maximum supply of consumption goods by firm sector (j) to a household sector (x) (cS
i x) is equal to the maximum amount of
consumption and intermediate consumption goods the sector can supply (gdS
i), multiplied by the
demand for that sector’s consumption goods from that household sector (cD
i x) divided by the total
demand for that sector’s consumption and intermediate consumption goods (gdD
i). Equation 11 says
that the maximum amount of intermediate consumption goods firm sector (i) can supply to firm sector (j) (icS
ij) is equal to the maximum amount of consumption and intermediate consumption
goods firm sector (i) can supply, multiplied by the demand for intermediate consumption goods from firm sector (i) by firm sector (j) (icD
ij), divided by the total demand for firm sector (i's) consumption
10 cS
i x = gdSi ∙ (cDi x / gdDi)
11 icS
ij = gdSi ∙ (icDij / gdDi)
(c) Logical functions
The final part of this section describes a series of logical functions (equations 12 to 15) which are set either equal to zero or one depending on whether each energy sector is within or in excess of its capacity limits. These logical functions (z10 to z14) appear in the equations that determine the actual
amount of goods supplied by each firm sector to each firm/household sector. They determine, along with the demand for a sector’s goods and that sector’s maximum supply, the quantity of goods produced by each sector (see Sections 4.2.1.6 and 4.2.2.3 for more details).
12 z10 = 1 if: gdDg < gdSg; and 0 otherwise.
13 z11 = 1 if: gdDg > gdSg; and 0 otherwise.
14 z12 = 1 if: gdDb > gdSb; and 0 otherwise.
15 z13 = 1 if: gdDb < gdSb; and 0 otherwise.
4.2.1.3 Firms – consumption output
This section discusses some identities related to the consumption goods produced by each firm sector (a full discussion of how consumption demand is determined will have to wait until the section on household behaviour).
Within the model, firms hold no inventories. The underlying assumption is therefore that firms produce all goods to order, or conversely, that firms are able to correctly estimate the demand for their goods and produce just enough to satisfy this demand (as long as this quantity is within the firm sector’s output capacity). The output of real consumption goods by a firm sector (cj) is therefore
equal to the sum of ethical (cj e) and normal households’ (cj n) real consumption (equation 16). Total
real consumption (ct) is then equal to the sum of the real consumption goods produced by each firm
sector (equation 17). 16 cj = cj e + cj n
17 ct = cg + cb + co
Nominal consumption sales (Cj) by each firm sector are equal to the product of each firm sector’s
real consumption output and its price level (equation 18). Total nominal consumption (Ct) is then
equal to the sum of the nominal consumption output of each firm sector (equation 19). 18 Cj = cj ∙ pj
19 Ct = Cg + Cb + Co
4.2.1.4 Firms – investment output
36This section discusses factors related to firm investment. Part (a) summaries the factors that influence each firm sector’s target level of capital. Part (b) explains how firms move towards their capital targets. Part (c) describes the outcomes of the investment process.
(a) The target level of capital
Each firm sector requires capital (along with labour and intermediate goods) in order to produce its output. Equation 20 says that each firm sector targets a stock of real capital (kT
j) that is equal to the
product of its target real capital-output ratio (κT
j) and its expected real output (sKT ej), with
expectations based on a simple extrapolation of the trend in (sKT
j) over the previous periods
(equation 21, which is similar to equation 5 in Jackson and Victor [2015]).37 The variable (sKT e j) is
equal to the expected level of (sKT
j) which is itself equal to the larger of either: i) a sector’s expected
demand; or ii) its actual sales (equation 22). Setting up firm output expectations as an extrapolation of the trend in output was felt to be important as the simulations in Chapter 6 look at transitions, which involve large changes in expected output. As such, when firms set their capital targets they take into account the trend in their output. If instead, output expectations were backwards looking (so that they depended on output in the last period), energy firms would constantly be producing too little capital (in the case of green firms) or too much capital (in the case of brown firms) during the transition period.
20 kT j = κTj ∙ sKT ej 21 sKT e j = sKTj(-1) ∙ ( 1 + [ ( sKTj(-1) - sKTj(-2) ) / sKTj(-1) ] ) 22 sKT j = (sDj ∙ z02 j) + ([1 - z02 j] ∙ sj)
23 z02 j = 1 if: sDj > sj and 0 otherwise.
Equations 22 and 23 determine the value of (sKT
j). Taken together, these equations say that (sKTj) will
equal a sector’s real output (sj) as long as the demand for a sector’s real output does not exceed the
sector’s maximum capacity. In this case, a sector’s real output (sj) will also be equal to the demand
for its real output (sD
j) (i.e. sKTj = sDj = sj). However, if the demand for an energy sector’s output
36 Many of the equations in Section 4.2.1.4 and 4.2.1.5 draw on Godley and Lavoie (2007a, Ch.7). 37 Here and elsewhere, the superscript (T) stands for target and the subscript (W) stands for weighted.
exceeds the amount it is able to supply then (sKT
j) will be equal to the demand for that sector’s
output.
An example should help to clarify why this is the case. Take the situation where the ex-ante demand for the green sector’s output exceeds its maximum level of output. In this situation, the green sector will target a quantity of capital that will (eventually) allow it to fulfil this level of demand. However, because the green sector is currently not able to fulfil this level of demand for its output, the brown sector’s actual output will exceed the ex-ante demand for its output (because consumers are forced to substitute brown goods for green ones due to the green sector being at its maximum capacity). Because the brown sector wants to be able to fulfil the actual level of demand for its output, it targets a level of capital that allows it to fulfil its actual level of sales.
Turning to the determination of (κT
j), equation 24 says that for each sector the target capital-output
ratio depends positively on that sector’s exogenously determined output elasticity of capital (αj), and
negatively on its exogenously determined risk premium (εj), endogenously determined expected real
weighted cost of capital (rre
W j), and exogenously determined depreciation rate (δj).38 The first two
terms in the denominator on the right hand side of the equation jointly make up the cost of capital: the expected real weighted cost of capital is a measure of each sector’s borrowing cost (to be explained in more detail subsequently), while (δj) is the depreciation rate of capital. The higher the
cost of capital, the lower the capital-output ratio.
The other variable that appears in the denominator on the right hand side of the equation is the risk premium. This measures the additional return, above the cost of capital, which firms require in order to undertake an investment project. As is the case with the cost of capital, the higher the risk
premium, the higher the return required on an investment, and so the lower the capital-output ratio.
24 κj = αj / (rre W j + δj + εj)
Equation 24 can be derived from a Cobb-Douglas production function (although this type of production function does not appear in the model).39
38 The argument that capital-output ratio depends negatively on the cost of capital is also made by Carreras et al. (2016), although the form of the equation which determines their capital-output ratio is different to the one presented here.
39 To see this, start with a Cobb-Douglas production function of the form: s = A ∙ kα ∙ n(1- α)
(b) The investment process
Once a target level of real capital has been determined, a partial adjustment function (equation 25) determines the speed at which each firm sector moves towards its target level of real capital. In each period, each firm sector invests (iD
j) just enough to cover part (γj) of the difference between the
target level of real capital and the level of real capital in the previous period, as well as that required to cover losses on capital due to depreciation (daj).
25 iD
j = z01 j ∙ [γj ∙ (kTj - kj(-1)) + daj]
Equation 25 also includes a logical function (z01 j) that stops real gross investment falling below zero.
This logical function is specified by equation 26. It says that (z01 j) takes a value of one as long as a
firm sector desires an amount of real gross investment greater than zero. When a firm sector desires a level of real gross investment that is less than zero, the variable (z01 j) takes a value of zero. Hence,
a sector’s real gross investment cannot fall below zero, and therefore its real capital can only decrease as fast as it depreciates.
26 z01 j = 1 if: [γj ∙ (kTj - kj(-1)) + daj] > 0; and 0 otherwise.
Where (s) is output, (A) is total factor productivity, (k) is capital, (n) is labour, and (α) takes a value between zero and one. Setting total factor productivity equal to one and then solving for the marginal product of capital (mpk) gives (i.e. taking the derivative of output, s, with respect to capital, k):
mpk = ∂s / ∂k = α ∙ k(α-1) ∙ n(1- α) This can be rewritten as: mpk = α ∙ (s / k)
Which can then be rearranged to get the capital-output ratio: k / s = α / mpk
Under the assumption of perfect foresight, we would expect the real cost of capital to be equal to the marginal product of capital in equilibrium, so that:
k / s = α / r
Where (r) is the real cost of capital. This cost of capital includes the cost of borrowing, which could be either a cost of borrowing or an opportunity cost on own funds, as well as the rate at which income must be set aside to cover depreciation.
Of course, in reality firms are unlikely to know with any certainty the returns to their investment expenditures. Hence they will require a return in excess of their cost of capital. This justifies adding an exogenous risk premium term (ε) to the denominator, so that:
It is assumed that firms prioritise investment expenditures over all other forms of output, so that in practice the demand for investment goods is equal to the actual amount of investment goods produced (ij) (equation 27). Real capital depreciates at a constant rate (δj) (equation 28).
27 iD j = ij
28 daj = δj ∙ kj(-1)
(c) Realised investment outcomes
The investment process results in firms accumulating real capital and making internal transfers between their current and capital accounts in line with their investment expenditures (as per the transactions matrix). The assumption here is that each sector produces its own capital. Equation 29 describes how investment and depreciation alter the capital stock. It says that the change in real capital (kj - kj(-1)) is equal to real investment minus real depreciation.
29 kj = kj(-1) + ij - daj
Equations 30 to 32 detail the nominal outcomes associated with the investment process. Equation 30 calculates the nominal capital stock (Kj), as a product of the real capital stock and the relevant
price level (pj). Equations 31 and 32 determine the nominal flows of gross investment (Ij) and
depreciation (DAj) between each sector’s capital and current account as the product of the relevant
real value and price level. 30 Kj = kj ∙ pj
31 Ij = ij ∙ pj
32 DAj = daj ∙ pj
4.2.1.5 Firms – investment financing
The previous section described the factors related to firm investment. This section describes the financial counterparts to those investment decisions. Part (a) discusses how firms chose between different forms of financing: i.e. retained earnings, bank loans and equities. Part (b) details the outcomes of each sector’s financing decisions. Part (c) describes how the cost of each form of financing can be converted into a single figure (the expected real weighted cost of capital) that can help to determine the investment decisions of firms.
(a) Financing decisions
Once firms have decided how much to invest they need to decide how to finance these investment expenditures. Gross investment includes the cost of capital depreciation. As in Godley and Lavoie (2007a, Ch.7), we assume that firms automatically set aside part of their income – the amortization
funds (AFj) – to cover these depreciation costs (equation 33). Subtracting amortisation funds from
gross investment therefore gives the net investment (INET j) required to adjust the level of capital
stock (equation 34). 33 AFj = DAj
34 INET j = Ij - AFj
Firms finance their net investment expenditures out of retained earnings (FRETj), by issuing new
equity (IEFj) and by borrowing from banks (ILFj). Following Ryoo (2010), it is assumed that firms
finance an exogenously determined proportion (ιj) of net investment through retained earnings
(equation 35), whereas bank and non-bank forms of financing are determined endogenously. 35 FRETj = INET j ∙ ιj
The part of net investment not financed by retained earnings is financed by equities and bank loans. The equations that determine the split between the two types of financing have some similarities to the portfolio allocation equations in Brainard and Tobin (1968), the Almost Ideal Demand System in Deaton and Muellbauer (1980), and the input-output equations in Hudson and Jorgenson (1974). From a purely mathematical perspective, each system of equations determines a series of
proportions. These proportions are then multiplied by either a stock or a flow variable to acquire the proportion of that stock (e.g. wealth) or flow (e.g. spending) that is allocated to the variable of