Debt-Led Growth and Its Financial Fragility

Debt-led growth and its ﬁnancial fragility

Joana David Avritzer

December 30, 2020

Abstract

This paper discusses the financial sustainability of demand-led growth models. We assume a supermultiplier growth model in which household consumption is the autonomous component of demand that drives growth. We show that for positive rates of growth the model converges to an equilibrium where worker households are accumulating debt and not wealth. We also show that when the economy is growing at a rate that is positive, but not too high, the dynamics of the model also implies that households will not be able to service their debt at equilibrium. We then conclude that this household debt-financed consumption pattern of economic growth generates an internal dynamic that leads to financial instability.

Keywords: household wealth dynamics, debt-financed consumption, growth and financial fragility; JEL Classification: E11, E12, E21, O41.

1 1 Introduction

This paper discusses the financial sustainability of demand-led growth models, in which household debt-financed consumption is the autonomous component of demand that drives growth. It is composed of three sections, besides this introduction and a conclusion. In the first section we present a literature review of demand-led growth theories focusing on those models that incorporate household debt-financed consumption. We discuss the models presented both by the neo-Kaleckian and the supermultiplier theory. We see that while in the neo-Kaleckian model the debt-financed component of consumption must be determined by current income, as investment is autonomous, in the supermultiplier literature credit-financed consumption can be autonomous from current income. These models also allow us to think about the financial sustainability of these patterns of growth from the perspective of the working households as they accumulate debt so as to maintain consumption. In the second section of this paper we present our own model for the consumption function of the working household. We argue that workers consumption is determined by a wealth target, as is suggested in Dutt (2006). However, we also suggest that their wealth target cannot be fully explained by their current income. Conse- quently, their consumption becomes partially autonomous from current income. We then develop a surpermultiplier model of growth in which household consumption, which can be financed through debt or accumulated wealth, becomes the autonomous component of demand that drives growth. Finally, in the third section of this paper we discuss the long run dynamics of our model. We find that for positive rate of economic growth the long run stability of our model require households to accumulate debt. We also find that for positive, but not too high, rates of economic growth, the dynamics of our model leads households

2 to accumulate a level of debt is not ﬁnancially sustainable to them as their debt servicing becomes higher than their wages.

2 Demand-led growth and household debt-ﬁnanced consumption

The neo-Kaleckian approach assumes that capitalist investment is autonomous from current income as it follows the function described in Bhaduri and Marglin (1990). In these models household debt dynamics has been incorporated through the deﬁni- tion of a workers consumption function that allows them to consume beyond their income as they accumulate debt. However, since this approach assume an exogenous investment function, this accumulation of debt must always become, for some reason or another, determined by current income.

Dutt (2005, 2006) suggests that worker’s consumption, CW , and capitalists consump- 1 tion, CΠ, is given by the following two equations :

dD CW = (1 − π)Y − rD + ; dt (1) CΠ = (1 − s)(πY + rD)

Where Y is income, r is the interest rate, π is proﬁt share, s is the fraction of income that household save and D is the stock of debt. It is also assumed that household accumulate a stock of debt that is given by a desired level of new borrowing, which is then determined by current income:

dD (2) dt = B = Bd = β[(1 − π)Y − rD]

1The notation of the model has been slightly modiﬁed from the original contribution for consis- tency of notation through out the paper

3 Where β is the borrowing to net income ratio. Since the accumulation of new loan is a function of workers current income, total consumption becomes entirely determined

I by current income and we can then assume that capital accumulation, K , follows a typical Kaleckian function as developed by Bhaduri and Marglin (1990). Another attempt of incorporating household debt-ﬁnanced consumption into a neo-Kaleckian framework has been done by Setterﬁeld and Kim (2016, 2017, 2018). In their model total income Y is divided into:

Y = WP N + WSαN + Π (3)

Where WP is the real wage of production workers, N is the number of production workers employed, WS is the real wage of supervisory workers, α < 1 denotes the necessary ratio of managers to production workers and Π denotes total proﬁts. It is further assumed that WS = φWP , supervisor work get a constant portion of production workers. Additionally, total consumption, C, is given by2:

˙ C = CW + CR + D; ˙ T D = β(C − CW ), β > 0; T C = ηCR − ωs, ωs = tΠ; (4)

CW = cW WP N;

CR = cπ[φαWP N + (1 − t)Π + rDR]

Where CW is consumption by working households, CR is consumption by rentier households and D˙ is the borrowing by working households to ﬁnance additional consumption. Household borrowing is then determined by a targeted level of con-

2The notation of the model has been slightly modiﬁed from the original contribution for consis- tency of notation through out the paper.

4 sumption, CT , which, in its turn is determined by η, the emulation (keeping up with the Joneses) eﬀect, minus the ωs, social wages (social welfare system). Finally,

DR = D − DW is the part of workers’ debt that is owned by the rentiers households for which they must pay an interest rate r. Palley (2010) and Pariboni (2016) emphasize that both models above require that

B I D = K , in other words, that the stock of capital and the stock of debt grow at the same rate. This means that the pace of total consumption (induced plus credit ﬁnanced) must be determined by the rate of capital accumulation. In Dutt (2005,

2006) this is done through the assumption that the desired level of borrowing Bd is determined by current income, as Bd = b[(1 − σ)Y − rD]. In this way the demand B I for loans grow in line with income and output and condition D = K is satisﬁed. In the model of Setterﬁeld and Kim (2016, 2017, 2018) we must also have that the rate of accumulation determines the rate of growth of aggregate demand. This is done through assuming an endogenous consumption target.

2.1 The supermultiplier approach

Under the supermultiplier approach, since investment is not autonomous from current income, we can assume that consumption is split into an induced component and an autonomous component such that:3

a Ct = c(1 − t)Yt + Ct + Et (5)

a Where c is the marginal propensity to consume, t is the tax rate, Ct is worker’s autonomous consumption ﬁnanced out of endogenous credit money and Et is cap-

3The notation of the model has been slightly modiﬁed from the original contribution for consis- tency of notation through out the paper.

5 italists’ autonomous consumption expenditure. Consequently we can sum all the autonomous expenditures which are neither ﬁnanced by income nor aﬀect the production capacity of the capitalist sector as:

a Zt = Et + Ct + Gt (6)

Where Gt is government expenditure. In this model, we must have investment It being determined by current income, such that:

It = htYt; (7) ˙ with h = htγ(ut − un)

Where ht is the marginal propensity to invest, which is determined by the adjustment ˙ of the rate of capacity utilization, ut, to its normal level, un: h = htγ(ut − un). We can then see that, in this model, the short run equilibrium output is determined by the autonomous component of demand, Zt, such that:

Zt Yt = (8) s − ht

˙ with s = 1 − c(1 − t). This means that, under steady state, as ut = un, h =u ˙ = 0, the rate of growth is given by the rate of growth of the autonomous component of demand:

∗ g = gZ (9)

W Pariboni (2016) then suggests we assume that workers’ consumption, Ct , has an induced part and a credit-ﬁnance part, Bt, which is autonomous from current income Π and that capitalists’ consumption, Ct , is entirely determined by current income. In

6 this model we have that:

W Ct = cw[(1 − π)Yt − (r + φ)Dt] + Bt; Π C = cππYt; t (10) a Ct = Bt − cw(r + φ)Dt; a ⇒ Yt = Yt[cw(1 − Π) + cΠΠ + ht] + Ct

Where Dt is the stock of accumulated debt, r is the interest rate, φ is the percentage of the D that is paid oﬀ in each period. In this version of the supermultiplier

a model Zt = Ct , in other words, the autonomous component of demand is household debt-financed consumption. It is also possible to show that, in this case, we have that g∗ = gCa , where gCa is the exogenously given rate of growth of debt-financed consumption.“[T]his result implies that, given enough time, demand and output will tend to evolve at the rate of growth of the autonomous components of demand; in this case, workers’ autonomous consumption." (Pariboni, 2016, p. 224) Finally, it is also important to mention the work of Brochier and Silva (2018), which suggests a stock flow consistent Supermultiplier model where non-capacity creating autonomous component of demand is households’ consumption out of wealth. In their model households’ consumption out of wealth becomes the autonomous expenditure component of demand. However, "[d]espite being autonomous (in relation to current income), it is endogenous to the model, since it depends on household wealth, so we can analyze its dynamic through household wealth dynamics." (Brochier and Silva, 2018, p.423)

7 2.2 Steady state equilibria and household ﬁnancial stability

One important aspect of the Setterfield and Kim (2016, 2017, 2018) work has been to emphasize the issue of the financial sustainability of their model of growth from the perspective of the worker household. Given the equations of the model above, it is possible to find long run, steady state, equilibria. However, Setterfield and Kim (2018) emphasize that this is only a short-

DR run equilibrium because it assumes a constant net debt to capital ratio, dR = K . "This net debt capital ratio will, however, vary endogenously over time, as workers accumulate debt and the economy grows."(Setterfield and Kim, 2018, p. 10) It then becomes important to understand these debt dynamics and their implications for growth, which is done, following Setterfield and Kim (2016, 2018), by analyzing the ˙ DR ˙ DR long run steady state behavior of dR = K , which is given by dR = K − gK dR. Setterfield and Kim (2018) then shows that from the dynamics above we can derive ˙ dR as a function of dR which results in a quadratic function, with two steady state equilibria. If we then define dRmax to mean worker’s maximum feasible debt servic-

(1−cW )ωP u ing payment, then we arrive at dRmax = i . Finally, it is then possible to make a discussion on the relationship between steady state dR that guarantees the ˙ Rmax equilibrium of dR = 0 and d and on the actual feasibility of the steady state for the worker household. The models discussed served as an important theoretical background to the model that will be developed in the next section. In it we suggest a consumption dynamics that is driven by a net worth target, similar to what is suggested in Dutt (2006). However, we will also suggest that this wealth target cannot be fully explained by current income such that is becomes partially autonomous from current income. Consequently, we will be developing a type of surpermultiplier model in which con-

8 sumption becomes partially autonomous from current income. We will also allow for this consumption to be financed through debt or accumulated wealth. However, we will focus on the dynamics where households accumulate debt, as it will allow us to discuss the financial sustainability of such dynamics from the perspective of the worker household in a similar fashion as is done in Setterfield and Kim (2018).

3 A model for household consumption

Following the previous contribution we suggest here that households’ savings decision is determined by a wealth target, such that:

˙ T St = H = β(H − Ht) (11)

Where St is households’ savings, Ht is households’ wealth, β is the speed of adjustment and HT is a targeted level of wealth. In other words, we suggest that households are making consumption and savings decisions so as to adjust their wealth to its targeted level. We also suggest that this target of wealth is determined by a targeted wealth to wage ratio, σ multiplied by households’ wage. Finally, we empirically es- timated this consumption function for US households on Chapter 2 and we found that, households when calculating this target take into account, not only current wage, but an average of wages over a period of eight past years. This then implies that households savings decision is driven by the following equation:

˙ St = H = β(σω¯ − Ht) (12)

9 Where ω¯ is an average of past wages. This then implies a consumption dynamics that is close to what is suggested in Dutt (2005, 2006), except that here we have an autonomous wealth target. In order to see what this implies in terms of growth dynamics we will compare this model to the existing literature on household debt- ﬁnanced consumption that was reviewed in ﬁrst section of this paper. Just like in Pariboni (2016) let us assume a closed economy without government such that the components of aggregate demand are investment (It) and consumption (Ct). Therefore, in equilibrium we have that:

Yt = Ct + It (13)

Under neo-Kaleckian theory we have that consumption must be a function of output since investment is exogenous, such that output growth is determined by the rate of capital accumulation. Alternatively, under the Supermultiplier approach, Pariboni (2016) and Brochier and Silva (2018) have suggested that household consumption is the autonomous component of demand such that investment becomes endogenously determined. More speciﬁcally, in Pariboni (2016) it is suggests that the autonomous component of demand is workers debt-ﬁnanced consumption, such that:

W Ct = cw[(1 − Π)Yt − (r + φ)Dt] + Bt; Π Ct = cΠπYt; a Ct = Bt − cw(r + φ)Dt; (14) a ⇒ Yt = Yt[cw(1 − Π) + cΠΠ + ht] + Ct a Ct ⇒ Yt = [1−(cw(1−Π)+cΠΠ+ht)]

W Where Ct is workers consumption, which has an induced part (cw(1 − Π)Yt) and ˙ Π an exogenously determined part Bt = D, which is households new borrowing. Ct is

10 capitalists consumption which is assumed to be fully endogenous to current income.

a We can then see that the autonomous component of demand is Ct = Bt−cw(r+φ)Dt. then we have that g∗ = gCa , where gCa is the exogenously given rate of growth of debt-financed consumption. It is important to emphasize that the model developed by Pariboni (2016) has assumed households’ debt-financed consumption to be autonomous (i.e.: not determined by current income) and exogenously given. However, if we assume a household wealth dynamics as described by equation 12, household consumption will continue to be autonomous, but no longer exogenously given, as we will be defining a specific dynamic for it. We then have that in our model total consumption will be determined by: W Π Ct = Ct + Ct Π C = cπΠt = cππYt; t (15) W W W Ct = Wt + rHt − St = (1 − π)Yt + rHt − St ; W ˙ T W ith St = H = β(H − Ht);

In our model then total consumption, Ct, is divided into workers consumption and W Π capitalist consumption, such that Ct = Ct +Ct , following the literature. We assume that capitalists households own ﬁrms and banks and get their income from proﬁts. Therefore, they do not follow the dynamics described in equation 12, we assume that they just consume a constant fraction of their income, Π. Finally, workers consumption is determined by the dynamics in equation 12, which implies in the workers consumption function written above as can be seen in the Transaction Flow Matrix below. In the equation and matrix, Wt is wages and salaries received by households, r is the interest rate they get (or pay) from the wealth they accumulate (or debt), Ht.

11 Households Firms Banks Workers Capitalists Current Capital Current Capital Sum Consumption −CW −CΠ +C 0 Investment +I −I 0 Wages +W −W 0 Proﬁts +Π −Π 0 ∆ Wealth4 −H˙ +H˙ 0 Deposit Flows +H˙ −H˙ 0 Interest on Wealth +rH −rH 0 Interest on Deposits −rH +rH 0 Issue of equities −E˙ +E˙ 0 Sum 0 0 0 0 0 0 0

Table 1: Transaction Flow Matrix of the Model

It is important to notice here that we are assuming banks are just passive intermediaries and that capitalists either pay out or take receipt of interest income, depending on whether worker households are net debtors or creditors. Since banks are passive intermediaries, if workers are accumulating wealth, in other words if H˙ > 0, then the liabilities of capitalists rise, whereas if workers are acquiring new loans, in other words if H˙ < 0 then capitalists’ assets rise. To put it diﬀerently, capitalists receipts are boosted by worker saving H˙ > 0 whereas their outgoings are boosted by worker borrowing H˙ < 0, so that we might think of the equation describing capitalists transaction ﬂows in each period as Π + H˙ = ˙ Cπ +E +rH. The reason for that is that since banks are not creating money in equi-

4This can be a new loan (ﬂow of debt), deposits made on savings account or the purchase of ﬁnancial assets by the household

12 librium, total saving must fund total spending that isn’t directly funded by current income. That means that capitalists plus workers saving are being directed into new capital formation (investment) when workers are saving, with workers savings ﬁnding their way into the corporate sector via bank deposits and, subsequently, capitalists. Meanwhile, when workers are borrowing, they are being funded by capitalists savings - the remainder of capitalists’ saving being used to fund investment.

3.1 The short-run equilibrium of the model

As described above, we assume a closed economy without government such that the following equations hold:

Yt = Ct + It

It = htYt; ˙ h = htγ(ut − un) W Π Ct = Ct + Ct (16) Π Ct = cπΠt = cππYt; W W W Ct = Wt + rHt − St = (1 − π)Yt + rHt − St ; W ˙ T W ith St = H = β(H − Ht);

Total output, Yt is determined by consumption, Ct, and investment, It. Investment follows the Supermultiplier approach and is determined by current income and the marginal propensity to invest, ht , which is determined by the adjustment of the rate of capacity utilization, ut, to its normal level, un. Additionally, consumption, W Π Ct is divided into workers consumption, Ct and capitalists consumption, Ct which W follow the dynamics described above. Finally, if we deﬁne κ = St and d = Ht , t Yt t Kt

13 then we can arrive at the following short run equilibrium equilibrium:

u = vrdt ; t [(1−cπ)π+κt−ht] (17) W Y = rHt−St t π(1−cπ)−ht

P Yt Yt Where ut = P and v = . From the keynesian stability condition we also have Yt Kt that: W Π W St = St + St = St + (1 − c )π; Yt Yt Yt Yt π

It = ht; Yt (18) St > It ; Yt Yt W ⇒ St + (1 − c )π > h ; Yt π t

Which shows that u > 0, as long as r Ht > 0 in the short run equilibrium. Given Kt the short run equilibrium described above we can also derive a long run steady state rate of growth for our model which is done in the following section.

4 Long run equilibrium: estimating the steady state rate of growth

Following Pariboni (2016) and Freitas and Serrano (2015), we assume that in steady ˙ state ut = un such that h = 0, ht = h and u˙ = 0. As is shown in the Supermultiplier literature this will then imply that:

hun ∗ ∗ v = gK = gY = gZ ; (19) W ∗ d/dt[rHt−St ] gK = gZ = W rHt−St

14 W As in our model Zt = rHt − St . At this point, we can turn our attention to the two variables defined in the previous section d and κ . The first one, d = Ht , is the ratio t t t Kt W of the stock of net worth over the stock of capital and the second one, κ = St , is t Yt the ratio of the flow of savings over the flow of income. This means that in a scenario where worker households are actually accumulating debt, dt will represent the ratio of the stock of debt over the stock of capital, while κt will represent the flow of new loans over the flow of income.

Therefore, in steady state it is interesting to look at the behavior of dt which will allow us to do a similar analysis Setterﬁeld and Kim (2018) of the ﬁnancial stability of worker household in this model. First, it is worth mentioning that we can arrive at the following steady state equations for κt and dt:

un[(1−cπ)π+κt−h] dt = vr (20) κ = h − (1 − c )π + dtvr t π un

W At this point it is important to emphasize that Zt = rHt − St is our autonomous component of demand. All of the other variables, which in our model will include dt and κt, will have to adjust to its rate of growth in the steady state. This means that dt will change according to gZ , which allows us to analyze the ﬁnancial stability of the worker household similar to Setterﬁeld and Kim (2018) as is done in the following section.

15 4.1 The ﬁnancial stability of the worker households

Following Setterfield and Kim (2018) we can departure from the definition of d = Ht t Kt and look at the steady state dynamics of the ratio. We find that:

dˆ= Hˆ − Kˆ ˙ H˙ d = K − gK d W ˙ St Yt d = − gK d Yt Kt (21) ˙ κtun d = v − gK d W ith κ = h − (1 − c )π + dtvr t π un W ∗ d/dt[rHt−St ] And gK = gZ = W rHt−St

˙ Which can be represented by a linear function between d and dt. If we assume

π = 0.4, cπ = 0.2, r = 0.01, un = 0.8, v = 2.5 and gZ = 0.03, then we get that the ˙ behavior of d as a function of dt will be given by:

Figure 1: The dynamics of dt when the economy is growing at 3%

The negative slope of the function tells us that the system will converge to equilib-

16 rium when d˙ = 0 if it starts oﬀ from out of equilibrium. Additionally, we also have that the point where d˙ = 0 is given by Ht = −3.62. This means that, the steady Kt state in this case will converge to a long run equilibrium point where households are in total accumulating a stock debt that is signiﬁcantly larger than the stock of capital.

If we now assume again that π = 0.4, cπ = 0.2, r = 0.01, un = 0.8, v = 2.5, but that ˙ gZ = 0.10, then we will get that the behavior of d as a function of dt is given by:

Figure 2: The dynamics of dt when the economy is growing at 10%

It is interesting to observe that at a higher rate of growth, the relationship is still negative, which means the system converges to the equilibrium at d˙ = 0, that is now at the point where Ht = −0.02. In this case the long run equilibrium point still Kt means households are accumulating debt, but less debt proportionally to the stock of capital, Kt.

Finally, if we assume, once again that π = 0.4, cπ = 0.2, r = 0.01, un = 0.8 and v = 2.5, but that gZ = −0.01, a negative rate of growth, then we get that the be-

17 ˙ havior of d as a function of dt is given by:

Figure 3: The dynamics of dt when the economy is facing a recession and it is decreasing its GDP by 1%

It is interesting to observe that the relationship now becomes positively inclined, meaning the dynamics is unstable and that it will not converge to an equilibrium point where d˙ = 0. We also get that the point of equilibrium is given by a positive dt equals to 5.62, which means households are accumulating wealth. However, this is not a stable equilibrium, which means households will not converge to it if they start off out of equilibrium. As a final exercise for this section we present below an analysis of how the dynamics of the ratio d = Ht changes with the different rates of growth. We assume once t Kt ˙ again that π = 0.4, cπ = 0.2, r = 0.01, un = 0.8, v = 2.5 and we plot below d as a function of dt and gZ such that:

18 Figure 4

In the graph above gZ varies from −0.10 to 0.10 and dt varies from −10 and 10. For ˙ negative rates of growth we get a positively inclined relationship between d and dt. For positive rates of growth we get a negatively inclined relationship, which means that the dynamics is stable and the system converges to equilibrium. However, as we have seen the equilibrium is obtained at a negative ratio of dt, which means households are accumulating debt, as opposed to wealth. In the ﬁnal graph below

5 we present the equilibrium value of dt for the diﬀerent rates of growth.

5 ˙ This graph is obtained from equations 21 and 20, from which we get that: d = 0 ⇒ dt = κtun κtun un[(1−cπ )π+κt−h] [(1−cπ )π−h]gK [(1−cπ )π−h]un ⇒ = ⇒ κt = ⇒ dt = vgK vgK vr r−gK v(r−gK )

19 Figure 5

At this point it then becomes important to know what is the dmax, the maximum amount of debt with which worker households can actually operate. For that we follow Setterﬁeld and Kim (2018) and we turn back to the initial consumption function of the worker household which was:

W W Ct = Wt + rHt − St (22)

If we assume that workers are not consuming or taking out new loans and are using all their income to service their debt, then we can ﬁnd a maximum ratio of stock of

max (1−π)un 6 debt over stock of capital which is given by: d = vr . Assuming, once again max that π = 0.4, r = 0.01, v = 2.5 and un = 0.8, we can calculate d = 4.8, which is lower than the equilibrium point of d˙ = 0 for positive rates of growth of Z below 2.6%. dmax is also represented by the grey horizontal line of Figure 5. Consequently, we can infer from the discussion above that if the economy is following the dynamics of growth described above, where household debt-ﬁnanced consumption

6 W W Wt Ht If we assume that C = 0 and S = 0, such that Wt = rHt, then = r ⇒ (1 − t t Kt Kt π) Yt = rdmax ⇒ (1−π)un = dmax Kt vr

20 is the autonomous component of demand that drives a positive, but not too high growth, households will not be able to service their debt. In other words, if the economy is following the dynamics of growth described above, then the economy will necessarily tend to a point of full equilibrium at which households can not even ﬁnancially sustain debt servicing. Finally, it is interesting to observe what will happen if we assume that households actually reach their wealth target eventually in the long run. In this case our steady state will be described by:

∗ ∗ u˙ = 0 as g = gK ; ˙ ∗ h = 0 as u = un; (23) W ˙ T St = H = 0 as Ht = H ;

Additionally, we will observe a κt = 0 and an economy that has no growth as Z = H˙ = 0.

5 Conclusion

This paper suggested a demand-led growth model in which household consumption is the autonomous component of demand that drives growth. In our model, working household consumption becomes partially autonomous from current income as it can be financed through credit or accumulated wealth. We followed a supermultiplier approach as it allowed us to think of credit-financed consumption as the autonomous component of demand that drives growth. This model also allowed us to think about the financial sustainability of these patterns of growth from the perspective of the working households as they accumulate debt so as to maintain consumption.

21 Once we derived our model we then turned to its long run equilibrium properties, focusing on the financial sustainability of the model from the perspective of the working households. We first found that for positive rates of economic growth our model requires that the worker household accumulates debt, and not wealth, in the long run. Secondly, we also find that for positive, but not too high, rates of economic growth, the dynamics of our model leads households to accumulate a level of debt that is not financially sustainable to them as their debt servicing becomes higher than their wages. In conclusion, this paper argues that this household debt-financed consumption pattern of economic growth generates an internal dynamic that leads to financial instability. In our model positive rates of economic growth require working households to reach a level of indebtedness that is not sustainable as their wages will eventually become lower than required to service their debt.

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23 Appendix A - Further Derivation of the Model

Derivation of the equilibrium rate of capacity utilization, ut:

Y C I t = t + t Kt Kt Kt Y CW CΠ h Y t = t + t + t t Kt Kt Kt Kt u W rH SW c πY h u t = t + t − t + π t + t t v Kt Kt Kt Kt v

W ut (1 − π)Yt St Yt cππYt htut = + rdt − + + v Kt Yt Kt Kt v u (1 − π)u u c πu h u t = t + rd − κ t + π t + t t v v t t v v v u t [π + κ − c π − h ] = rd v t π t t u rd t = t v [π(1 − cπ) + κt − ht]

Derivation of the steady state rate of growth g:

W Π Yt = Ct + Ct + It

W Yt = (1 − π)Yt + rHt − St + cππYt + htYt

W rHt − St Yt = [π(1 − cπ) − ht]

" W #" # " #" W # dY d(rHt − St ) 1 d[π(1 − cπ) − ht] rHt − St = − 2 dt dt [π(1 − cπ) − ht] dt [π(1 − cπ) − ht]

24 W ˙ If we deﬁne Zt = rHt − St and since h = 0 in steady state, then we have that:

Z˙ Y˙ = [π(1 − cπ) − ht]

Y˙ Z˙ Z˙ g = = = = gZ Y Y [π(1 − cπ) − ht] Z