The Spirit of Capitalism, Asset Pricing and Growth in a Small Open Economy

Turalay Kenc∗ Sel Dibooˇglu∗∗

March 2005

∗Kenc: Tanaka Business School, Imperial College London, Direct Line: +44-20-7594 9212, Fax: +44-20-7823 7685, Email: [email protected]. Dibooˇglu:Department of Economics, University of Mis- souri - St Louis, 408 SSB, 8001 Natural Bridge Rd., St Louis, MO, 63121, Direct Line: +1-314-516- 5530, Fax: +1-314-516 5352 and Email: [email protected]. Dibooˇglugratefully acknowledges ﬁnancial support from the Center for International Studies at the University of Missouri - St Louis.

1 The Spirit of Capitalism, Asset Pricing and Growth in a Small Open Economy

Abstract

Conventional models of economic behavior have failed to account for a number of observed empirical regularities in macroeconomics and international economics. This may be due to preference speciﬁcations in conventional models.

In this paper, we consider preferences with the “spirit of capitalism” (the desire to accumulate wealth as a way of acquiring status). We analyze a number of potential eﬀects of international catching-up and the spirit of capitalism on savings, growth, portfolio allocation and asset pricing. Moreover, we obtain a multi-factor Capital Asset Pricing Model (CAPM). Our results show that status concerns have non-trivial eﬀects on savings, growth, portfolio allocation and asset prices.

Keywords: International Catching-Up; Asset Pricing; Exchange Rates; Spirit

of Capitalism; Stochastic General Equilibrium Models. 1 Introduction

Conventional models of economic behavior assume that a person’s well-being depends on the absolute quantities of various goods and services he consumes and not on how these quantities compare with those consumed by others. Yet, from Max Weber’s pursuit of wealth as an end in itself (spirit of capitalism) to Duesenberry’s “demon- stration effect” in consumption, there is a long tradition in sociology and economics which acknowledges that people are concerned with their relative standing in soci- ety and individuals’ consumption decisions are influenced by others (Duesenberry (1949)).1 There is a growing recognition that modelling preferences to accommodate such elements enhances our understanding of saving/consumption, asset pricing, and economic growth. For example, several authors have recently shown that the spirit of capitalism has nontrivial consumption-saving, portfolio allocation, asset pricing and growth effects. Cole, Mailath, and Postlewaite (1992) examine how the desire to increase social status affects wealth accumulation and economic growth while Robson (1992) show that status concerns affect risk taking. Bakshi and Chen (1996) develop a model where the relative wealth status leads to a two factor capital asset pricing model (CAPM). Smith (2001) extends the model to recursive preferences and obtains a three factor CAPM. In another paper Smith (1999) also examines growth effects. In a series of papers, Gong and Zou examine long run growth, saving, policy effec- tiveness, and monetary issues in a cash-advance setting under the spirit of capitalism (Zou (1994), Zou (1995), Gong and Zou (2001), Gong and Zou (2002)). Following

1Another line of research in sociology such as “reference group theory” and the concept of “relative deprivation” incorporate a social frame of reference whereby an individual uses other groups for self- appraisals, comparisons, and in making choices. It is not unusual for an individual to use diﬀerent reference groups for diﬀerent situations. For reference group theory see, Dawson and Chatman (2001).

1 Bakshi and Chen (1996) and others, Carroll (2000) presents a model where wealth enters consumers’ utility functions directly and shows that the model yields results consistent with the available data on the saving behavior of the wealthy. Even though some research has shown that an individual’s well-being depends significantly on his relative standing in local rather than global hierarchies, the norms by which people make judgements tend to depend on close contact and available information.2 Thus, as the information revolution removes barriers and the world economy gets more integrated, one would expect distant reference groups to become more important and individuals to be less indifferent to wealth standards in the rest of the world. Indeed, research by Juliet Schor has emphasized that when the neighborhood declined in 1950s and 1960s so did the Joneses down the street as a focus of comparison. As such, the television became more important as a source of consumer cues and information.3 In this paper, we model preferences to accommodate status concerns by representative individuals irrespective of the reference group. In order to distinguish between domestic and international reference groups, we use “spirit of capitalism” to describe individuals’ concern about fluctuations in individual wealth relative to a domestic wealth standard while we use “international catching-up” to describe concerns about fluctuations in individual wealth relative to international wealth standards4. When individuals care about relative status and the associated risks of falling out of status, they will hedge against these risks. This can have potentially important portfolio allocation, consumption-saving and growth effects. Our model incorporates the “spirit of capitalism” and “international catching-up” to analyze consumption, growth, international portfolio allocation, and asset pricing

2See Frank (1985) and the references cited therein. 3See Schor (1999). 4Since we are not making any claim as to which concern is more important or empirically relevant, we use “spirit of capitalism” and “international catching-up” to describe individuals’ status concerns.

2 in a dynamic stochastic general equilibrium setting. The model builds on the rich equilibrium framework of Grinols and Turnovsky (1993, 1994) and well known to be capable of addressing a number of interesting issues (such as how means and variances of domestic government policy impact on the economy and isolating determinants of the real foreign exchange rate premium). Even though the capitalist spirit model of Bakshi and Chen (1996) captures the main features of the catching-up with the Jonesses features of Abel (1990, 1999), the idea has not been modelled in a stochastic general equilibrium setup in an open economy.5 There are several distinctive features of the model developed here (the ﬁrst two are familiar from Turnovsky and Grinols (1996)): First, the equilibrium involves the joint determination of the means and variances of the relevant economic variables (in terms of the ﬁrst two moments of the exogenous stochastic processes impinging on the economy).6 Second, the model incorporates portfolio choice - thereby giving rise to an integrated analysis of exchange rate determination with a risk adjusted PPP and portfolio equilibrium. Third, our use of a recursive utility function which disentan- gles the two preference parameters for risk aversion and intertemporal substitution ensures that the model is fully capable of assessing the importance of the distinct and separate roles played by agents’ attitudes towards risk and intertemporal substitution.7 Finally, the model has a utility function that includes the level of wealth and

5The empirical analysis of the catching-up with the Jonesses in an international asset pricing set-up can be found in Gomez, Priestley, and Zapatero (2003) and Lauterbach and Reisman (2004). 6It can be important to make proper allowance for interactions such as variances affecting means in a model, for example, this type of mean variance optimization framework has formed the basis for important empirical work relevant to interest parity relationships and the determination of exchange rate risk premia: see, for example Frankel (1986), Hodrick and Savastava (1986), Giovannini and Jorion (1987) and Lewis (1998). 7This type of General Isoelastic Preferences was introduced by Epstein and Zin (1989), and Weil (1990) and found a wide range of applications in macroeconomics and finance. For example, one can cite Svensson (1989), Epstein and Zin (1991), Duffie and Epstein (1992a), Obstfeld (1994b), Smith

3 a world wealth index - thereby capturing features of ‘catching up with the Joneses’ models such as Abel (1990), and exogenous habit formation models such as Campbell and Cochrane (1999)8. This latter feature is of particular interest because it captures elements of a popular approach to explaining a number of puzzles associated with the equity premium and the foreign exchange market. For example, Campbell and Cochrane (1999) proposed a preference specification in which there is both an aggregate consumption externality and utility is time-inseparable because of habit persistence; and this helped explain the US stylized facts of the equity premium puzzle. Because the habit persistence takes the form of an aggregate consumption externality, this preference specification embodies keeping up with the Joneses’ effects in the spirit of Duesenberry (1949) and Abel (1990). However, the particular way in which the utility function is specified here also draws on another recent and fast-developing literature which incorporates the spirit of capitalism (the desire to accumulate wealth as a way of acquiring status). Our approach runs along the lines of Bakshi and Chen (1996)) who find that the spirit of capitalism seems to be a driving force behind stock market volatility (and economic growth). Our paper is organized as follows. Section 2 outlines our development of exist- ing continuous time stochastic endogenous growth models and presents the solution. Section 3 presents the the effects of status concerns on international portfolio diversification, asset prices, and exchange rates. Section 4 concludes the paper.

(1996b), among others. 8Strictly speaking, the literature distinguishes between diﬀerent external references in the utility function: agents care about past aggregate consumption in the economy (“catching up with the Joneses” as in Abel (1990) and “external habit formation” as in Campbell and Cochrane (1999)); agents care about current per capita consumption levels in the economy (“keeping with the Joneses” as in ?)); agents care about absolute or relative wealth in the economy (“spirit of capitalism” as in Bakshi and Chen (1996)).

4 2 The Model

We consider a continuous-time, infinite-horizon small open economy with complete financial markets and a single production good. In our model the utility maximiz- ing and portfolio optimizing behavior of a representative household has a somewhat nonstandard element. We assume that the representative agent’s preferences exhibit the “the spirit of capitalism” feature of Bakshi and Chen (1996), Gong and Zou (2002) and Smith (2001). Specifically, we assume that household utility is a stochastic differential function of the multiplicative of individual consumption and relative wealth. The latter is defined as the ratio of the individual’s absolute wealth to the international standard of living. This form of preferences retains the property of the standard constant relative risk aversion (CRRA) utility function that individual risk aversion does not change over time. The model is a monetary one where money enters into the utility function.9 The production technology assumes a Rebelo (1991) ‘AK’ production function which leads to endogenous growth. The stochastic nature of the model is characterized by four exogenous stochastic shocks: (i) productivity shocks; (ii) monetary growth shocks; (iii) foreign price shocks and (iv) foreign wealth index shocks. Other shocks could be included but this set is characteristic of the most important exogenous stochastic influences in a small open economy. In financial markets there exist four assets: (i) equity; (ii) money; (iii) domestic bonds with zero net supply; and (iv) foreign bonds. We allow for a full range of interactions across shocks. The paper deals with only the steady-state stochastic equilibrium which is separated into deterministic and stochastic components. The remainder of this section sets out the key features of the model and its solution.10

9There are alternative ways of incorporating money: the most well-known alternative is transac- tion cost technologies such as a cash-in-advance constraint [Lucas (1982), Svensson (1985), Rebelo and Xie (1999)]. There are diﬃculties in imposing a cash-in-advance constraint in continuous-time. 10Further details for the recursive utility part of the model can be found in Kenc (2004).

5 2.1 Preferences

At each point in time the representative consumer chooses its consumption C and allocates its portfolio of wealth, W , across four assets: money M, capital K, domestic bonds B and foreign bonds B∗. Of these assets only capital and foreign bonds are internationally traded.11 The only source of income for the representative household is the capital income received from holding these assets. The representative agent’s intertemporal utility is deﬁned as in Duﬃe and Epstein (1992b):

Z ∞ Ut = f(Cs,Ms/Ps,Ss,Us)ds (1) t where

1 n 1− ζ−1 o θ 1−θ α λ ζ ζ(1−γ) [Cs (Ms/Ps) ] Ss ] − [(1 − γ)Us] f(Cs,Ms/Ps,Ss,Us) = δ n ζ−1 o (2) −1 1 ζ(1−γ) (1 − ζ )[(1 − γ)Us]

Current utility Ut depends upon current consumption Ct, real money balances mt, “status” St and expected values of future utility Us, s > t. f(c, U) is known as the normalized “aggregator” function generating ordinally equivalent utility functions. The parameter γ > 0 is the coeﬃcient of relative risk aversion with respect to timeless gambles over xt, while ζ > 0 is the elasticity of intertemporal substitution deﬁned

12 over riskless paths of xt, δ > 0 is the rate of time preference. λ measures the extent 11The assumption that some assets are nontraded does not pose any problem as long as the risk characteristics of these nontraded assets can be replicated with those of the traded assets, i.e., nontraded assets are spanned. In other words, markets are complete in the sense that the number of stochastic processes equals the number of traded assets. 12For the special case of γ = 1/ζ the above utility function reduces to the well known time- separable, constant relative risk-aversion case:

Z ∞ 1−γ δ(s−t) Cs Ut = e ds. t 1 − γ

6 to which the investor cares about status - the degree of spirit of capitalism. Relative wealth is assumed to be status: the consumer’s absolute wealth, Wt, and a world wealth index, Vt. Status is thus described by a function St = f[Wt,Vt], where fW > 0 and fV ≤ 0. More precisely, it is deﬁned as

Wt St = . Vt

The relative importance of money in composite consumption is measured by θ.

2.2 Prices and Asset Returns

There are three commodity prices in the model: the domestic price of the traded good (P ); the foreign price level of the traded good (P ∗); and the exchange rate (E). P ∗ is assumed to be exogenous and other two prices P and E are endogenously determined. The exchange rate (E) is measured in units of domestic currency per unit of foreign currency. Prices and returns are both generated by geometric Brownian motion (Wiener) processes. Each of the prices P , P ∗ and E evolves according to

dx = (drift term) dt + (diﬀusion term) dZ (3) x

∗ ∗ where x is either P , P or E; and π (σP ), π (σP ∗ ) and (σE) are respective drift(diﬀusion) terms of these price processes. Thus, for example, π dt is the expected mean rate of change of P and σP dt is the volatility of this rate of change.

∗ Zj is a Wiener process for j = P,P ,E. We use ρ to denote the “instantaneous” correlation coeﬃcient between any two Wiener processes: ρij = dZidZj. This small open economy is linked with the rest of the world through the law of one price. Formally, it means that the exchange rate E relates foreign prices P ∗ to domestic prices of traded goods P , which is referred to as the purchasing power parity (PPP) relationship. The domestic price of the imported good is then given by

P = EP ∗ (4)

7 which yields the following price process

dP ∗ = (π + + σ ∗ ) dt + σ ∗ dZ ∗ + σ dZ (5) P P E P P E E

As for asset returns, the real rate of return to equity holders is calculated from the ﬂow of new output dY per capital K as follows

dRK = rK dt + duK rK = Adt duK = AσY dZY . (6)

where A is the marginal physical product of capital and σY dZY a productivity shock with σY being the volatility of the shock. dY follows an AK type of the aggregate production function with endogenous growth13

dY = [Adt + σY dZY ]K. (7)

Returns to other assets can be described in terms of the interest rates they pay. Domestic and foreign bonds pay nominal rates of interest, i and i∗, respectively. Applying stochastic calculus and separating real returns from nominal returns, we obtain the real rates of return to domestic holders of money, domestic bonds, and foreign bonds as follows:

∗ dRj = rjdt − σjdZj, j = M,B,B

2 2 ∗ ∗ ∗ 2 where rM = −π + σP , rB = i − π + σP and rB = i − π + σP ∗ . We assume that the world-wealth index follows a diﬀusion process:

dV = [µV dt + σV dZV ]V. (8)

13We assume that output is produced from capital by means of the stochastic constant returns to scale technology; and the economy-wide capital stock is assumed to have a positive external eﬀect on the individual factor capital.

8 2.3 Other Features of the Model

The government pursues the following monetary policy rule:

dM/M = φdt + σX dZX (9)

where φ is the mean monetary growth rate. The stochastic term σX dZX may reﬂect exogenous stochastic failures to meet the monetary growth target set by the monetary authority. Correlation between monetary growth shock and other shocks is important.

∗ For example, correlation between dZX and dZB may reﬂect stochastic adjustments in the money supply as the authorities respond to exogenous stochastic movements in the intermediate target, the exchange rate. The proceeds from printing money are distributed as transfers to households. They are non-marketable assets. The real value of transfers is random because of the stochastic monetary rule described above. We assume that the household expects that its real transfer income dT will remain proportional to wealth, as follows:

dT = τ W dt + σT W dZT , 0 < τ < 1, (10)

where τ is the transfer rate and σT dZT a shock with σT being the volatility of the shock.

2.4 Household Optimization

Utility is maximized subject (i) to the following wealth W constraint, which in real terms is

M B EB∗ W = + + + K P P P

9 where E is the exchange rate and P equals the price level and (ii) to the stochastic wealth accumulation equation:

dW = ψdt + σ dZ (11a) W W W ∗ ∗ ψ =nM rM + nBrB + nK rK + nBrB − τ − Ct/Wt (11b)

∗ ∗ σW dZW = − nM σP dZP − nBσP dZP + nK AσY dZY − nBσP dZP ∗ − σT dZT (11c)

where ni is the share of portfolio held in asset i. More speciﬁcally, nM = (M/P )/W ,

∗ ∗ nB = (B/P )/W , nK = K/W and nB = (EB /P )/W . The representative consumer constructs an optimal portfolio of his total wealth subject to the adding up condition for portfolio shares:

∗ 1 = nM + nB + nK + nB (12)

Consumers are assumed to purchase output over the instant dt at the nonstochastic rate Cdt using the capital income generated from holding assets. To define each asset return, dRi, requires a description of the dynamics which generate asset prices and asset returns. The maximization of (1) subject to the stochastic differential equation (11a) and the adding up condition (12) represents a continuous-time stochastic dynamic optimization problem of the type pioneered by Merton (1969). The solution strategy is based on the dynamic programming approach of Duffie and Epstein (1992b). The household maximizes utility by choosing the optimal full (composite) consumption- wealth ratio and the optimal portfolio shares of assets, taking the rates of return on assets, and the relevant variances and covariances as given14.

14However, the general equilibrium conditions, i.e. market-clearing conditions, of the model will determine these rates of return, variances and covariances.

10 Solving the consumer’s problem yields the following optimality conditions:

C σ2 = Φ∆ + (1 − Φ)[r − τ − Γ W ] W Q 1 2 λ σ2 + (1 − Φ) [µ − Γ V ] + (1 − Φ)λ(1 − γ)σ (13) α + λ V 2 2 WV where Γ1 = 1 − (α + λ)(1 − γ) is the eﬀective coeﬃcient of relative risk aversion; Γ2 =

αθδ ζ 1 − λ(1 − γ); ∆ = α+λ is the eﬀective rate of time preference; and Φ = (ζ−(ζ−1)θα) is the eﬀective elasticity of intertemporal substitution. rQ is the return on the portfolio; rQ = nM rM + nBrB + nK rK + nB∗ rB∗ .

(1 − θ) C 1 n = , (14) M θ W i

[rK − rB]dt = Γ1(σYW + σPW ) + λ(1 − γ)(σYV + σPV ), (15)

∗ [rB − rB]dt = Γ1(−σP ∗W + σPW ) + λ(1 − γ)(−σP ∗V + σPV ). (16)

2.4.1 Goods Market Equilibrium and Balance of Payments

In our small open economy net exports in real terms are given by

Net Exports = dY − dC − dK.

The balance-of-payments equilibrium condition in real terms is

∗ ∗ ∗ d(EB /P ) = (EB /P )dRB + Net Exports (17)

Substituting and simplifying we derive the following expression for the rate of growth of the capital stock: dK 1 C ∗ ∗ ∗ = ω A − + (1 − ω)rB dt + ωσY dZY − (1 − ω)σBdZB. (18) K nK W

nK where for notational convenience we deﬁne ω ≡ ∗ to be the share of capital in nK +nB the traded portion of the consumer’s portfolio.

11 3 International Catching-Up and the Spirit of Cap-

italism

In this section, we will discuss how the spirit of capitalism or the concern for social status aﬀects asset pricing, international portfolio diversiﬁcation, exchanger rate determination and economic growth. To facilitate this analysis we will simplify the model described in Section 2.

3.1 International Portfolio Diversiﬁcation

First, we give the equilibrium asset-pricing relationships. To make our model comparable with that of Giuliano and Turnovsky (2003) we consider a real economy with only two assets, a domestic equity and foreign bonds. Applying the optimization method yields the following optimal portfolio share of domestic equity:

the Sharpe ratio nK = + home bias (19) Γ1 where

2 r − [r ∗ − Γ (σ ∗ − σ ∗ )] the Sharpe ratio = K B 1 P P Y , (20) Λ σ + σ ∗ home bias = H YV B V , (21) Λ

2 2 Λ = σY + σP ∗ − 2σP ∗Y .

In eq. (21), the expression H denotes the ‘propensity to hedge’ against unanticipated ﬂuctuations in the world wealth index, and deﬁned as

J V λ(1 − γ) H = − WV = − . JWW W Γ1

Notice that the optimal share of domestic equity in (19) diﬀers from that of Giuliano and Turnovsky (2003) in two ways: (i) the degree of the spirit of capitalism

12 and (ii) the international “catching-up” feature of our model. As pointed out by Smith (2001), an increase in the the degree of the spirit of capitalism, λ, leads to a rise in the eﬀective coeﬃcient of relative risk aversion, Γ1 = 1 − (α + λ)(1 − γ), when

γ > 1, otherwise Γ1 declines. In turn, assuming γ > 1 the rise in λ decreases the demand for equity, nK . The second part in the optimum equity share depends on the correlations of assets held in the portfolio and the world wealth index, V . If these correlations are positive and γ > 1 to make the home bias effect positive, then domestic investors will hold more of the domestic asset. As emphasized by Bakshi and Chen (1996) adding domestic equity serves to insure against a future decline in V . However, if the correlations are negative, holding too much domestic assets will reduce the investor’s status further when V rises. Therefore domestic investors will shift their portfolio to foreign assets in order to insure them against the uncertain declines in status. Note that the higher λ, the more intensive is the insurance effect. Moreover, unless γ = 1, optimal portfolio shares are convex functions of λ as shown by Smith (2001). Giuliano and Turnovsky (2003) point out that the size of the equilibrium portfolio shares, nK , nB∗ , relative to the growth variance-minimizing portfolio shares,n ˜K , n˜B∗ is a crucial determinant of the effects of structural changes on the equilibrium. Fol- lowing Giuliano and Turnovsky (2003) we definen ˜K , n˜B∗ to be the portfolio shares

2 minimizing the growth volatility σW and obtain them by minimizing (11c) with respect to nK , nB∗ . They are given by

2 2 σP ∗ − σP ∗Y σY − σP ∗Y n˜ = n˜ ∗ = . K Λ B Λ These are the same as those obtained in Grinols and Turnovsky (1994) apart from covariances. The equilibrium portfolio shares are now separated as follows

r − r ∗ n = K B + home bias +n ˜ (22) K Λ K rB∗ − rK n ∗ = + home bias +n ˜ ∗ (23) B Λ B

13 These expressions show that with the spirit of capitalism our model yields diﬀerent

15 ratio of nj ton ˜j for j = capital and foreign asset since the basen ˜j is the same .

1−γ 1−Γ1 1−Γ2 Following Gong and Zou (2002) we deﬁne χ/[(1 − Γ1)b W V ] as ‘risk- free’ return, rf , in this all risky world—both returns on domestic equity and foreign assets are uncertain. Now one can obtain the following equilibrium asset-pricing relationships and assets’ beta coeﬃcients f f ∗ ri − r = βi rQ − r for i = K,B ,

cov(σQdZQ,σidZi) ∗ βi = for i = K,B . var(σQdZQ) where rQ is the rate of return on the equilibrium (market) portfolio and σQdZQ is the stochastic term of this portfolio. Following Turnovsky (1995) we form the market portfolio as Q = nK W + nB∗ W and therefore rQ can be derived as h i rK − rB∗ rQ = rK nK + rB∗ nB∗ = + home bias +n ˜K (rK − rB∗ ) + rB∗ Γ1Λ

Giuliano and Turnovsky (2003) obtain the following expression h i GT rK − rB∗ GT r = +n ˜ (r − r ∗ ) + r ∗ Q GT GT K K B B Γ1 Λ where

GT 2 2 2 2 Λ = σY + σP ∗ Λ = σY + σP ∗ − 2σP ∗Y

GT Γ1 = γ Γ1 = 1 − (α + λ)(1 − γ)

2 2 ∗ GT σP ∗ σP ∗ −σP Y n˜K = 2 2 n˜K = Λ . σY +σP ∗ Comparing our results with theirs (ignoring cross correlations) reveals that the spirit of capitalism makes diﬀerence through two channels ; eﬀective risk aversion

15The variance minimizing portfolio was recognized by Branson and Henderson (1995), who

∗ (rk−rb ) dubbed it “hedging demand” as it is independent of preferences. They called Λ “speculative demand” so the portfolio shares can be expressed as speculative demand, hedging demand, and home bias eﬀects.

14 and home bias. When γ > 1 the home bias (the risk aversion channel) increases

GT (decreases) the rate of return on the market portfolio (rQ > rQ ). Given the deﬁnition of the risk-free rate one can obtain the following expressions:

f 1−γ 1−Γ1 1−Γ2 f 1−γ 1−γ r = χ/[δ(1 − Γ1)b W V ]r ¯ = χ/[δ(1 − γ)b W ] (24) where a bar over a variable denotes a benchmark without the spirit of capitalism.

GT Since rQ > rQ it is evident that the excess return on asset i is enhanced by the spirit of capitalism

f f ri − r > r¯i − r¯ .

This is in line with the results reported in Gong and Zou (2002) who use absolute- wealth-is-status model of the spirit of capitalism. Notice that our model produces higher excess returns relative to that of Gong and Zou (2002) for two reasons; (i) international catching-up through the home bias term and (ii) lower risk-free rate [see eq. (24)]. As pointed out by Bakshi and Chen (1996) this improves the equity premium puzzle in Mehra and Prescott (1985). Moreover, the lower risk-free rate result alleviates the risk-free rate puzzle in Weil (1989). In this type of representative agent setting the equilibrium requires that

σQdZQ = σW dZW . (25)

Using (25) we obtain the beta coeﬃcients as

2 nK σY βK = 2 2 2 2 (26) nK σY + nB∗ σP ∗ + 2nK nB∗ σYP ∗ 2 nB∗ σP ∗ βB∗ = 2 2 2 2 (27) nK σY + nB∗ σP ∗ + 2nK nB∗ σYP ∗ Substituting (22) and (23) into (26) and (23) respectively and ignoring the cross correlations appearing in Λ yield h i 2 rK −rB∗ 2 2 σ + σ ∗ + σ h Y Γ1 P Y βK = (28) 2 σ2 −σ2 σ2 σ2 h rK −rB∗ 2 2 2 rK −rB∗ Y P ∗ Y P ∗ Γ + σP ∗ σY + h + 2h Γ 2 2 + 4 2 2 1 1 σY +σP ∗ σY +σP ∗

15 h i 2 rB∗ −rK 2 2 σ ∗ + σ + σ ∗ h P Γ1 Y P βP ∗ = (29) 2 σ2 −σ2 σ2 σ2 h rK −rB∗ 2 2 2 rK −rB∗ Y P ∗ Y P ∗ Γ + σP ∗ σY + h + 2h Γ 2 2 + 4 2 2 1 1 σY +σP ∗ σY +σP ∗ where h denotes the home bias. Setting h = 0 (i.e., without the international catching- up feature) and Γ1 = γ (i.e., without the spirit of capitalism) asset beta coeﬃcients reduce the expressions obtained by Giuliano and Turnovsky (2003). However, equations (28) and (29) do not give us a clear-cut comparison to those in Giuliano and Turnovsky (2003). Ultimately, numerical analysis can be used to assess any eﬀects for a reasonable set of parameter values.

3.2 Growth and Consumption Eﬀects

The expressions for the equilibrium growth rate and the consumption wealth ratio can be obtained by substituting the equilibrium portfolio shares implied by equation (19) into (18):

ψ = ψGT i Φ (1 − Φ)hΓ1 (rK − rB∗ ) 2 2 2 2 ∗ 2 2 h(rK − rB ) − 2 2 2 (σY − σP ∗ ) + 2σP ∗ σY h σ ∗ + σ (σ ∗ + σ ) Γ + | P Y P Y{z 1 } home bias effect n 2 (1 − Φ) λ σV 2 2 λ(1 − γ) (σYV + σP ∗V )h (1 − Φ) [µV − Γ2 ] σ ∗ + σ α + λ 2 + | P Y {z } + | {z } home bias effect direct catching-up effect n h i (1 − Φ) rB∗ − rK 2 2 λ(1 − γ) (σYV σY + σP ∗V ) σ ∗ + σ Γ + | P Y {z 1 } indirect (covariance) o 2 2 (σYV σP ∗ − σP ∗V σY ) + | {z }, (30) catching-up effect where

h i GT 1 + Φ 2 2 2 Γ1 2 2 1 ψ = (rK −rB∗ ) +Φ[rK σP ∗ +rB∗ σY ]+(1−Φ) σP ∗ σY 2 2 −Φ∆. 2Γ1 2 σY + σP ∗

16 GT C C (rK − rB∗ )h = + 2 2 − ψ (31) W W σP ∗ + σY where

GT 2 2 2 C (rK − rB∗ ) Γ1(rK σP ∗ − rB∗ σY ) GT = 2 2 + 2 2 − ψ (32) W Γ1(σY + σP ∗ ) σY + σP ∗ Similarly, the variance of the equilibrium growth rate is given by

2 n o 2 2 GT h 2h rK − rB∗ 2 2 2 2 σψ = (σψ) + 2 2 + 2 2 2 (σY −σP ∗ )+2σP ∗ σY h , (33) σP ∗ + σY (σP ∗ + σY ) Γ1 where rK −rB∗ 2 2 + σP ∗ σY 2 GT Γ1 (σψ) = 2 2 . σP ∗ + σY These expressions highlight the additional channels whereby the home bias and international catching-up influence the equilibrium growth and consumption rates and their variability relative to those found by Giuliano and Turnovsky (2003). How- ever, the qualitative effects of these additional terms cannot be determined a priori. For example, a calibration analysis for the UK economy based on the guesstimated values for the “deep parameters” such as the risk aversion and intertemporal elasticity substitution parameters and the estimated values for the parameters governing the dynamics of the state variables suggests that the growth effect of the spirit of capitalism is in the region of a 30% rise in the equilibrium growth rate.

3.3 A Multi-Factor CAPM

Following Duffie and Epstein (1992a) [cf. eq. (35)] in terms of the normalized aggregator f we define a stochastic discount factor (state-pricing process) as follows: Z h t i ξt = exp fuds fc + fm/p + fs (34) 0 where fi stands for the partial differentiation of the function f(C, M/P, S, U) with respect to i = C, M/P, S, U. As in Duffie and Epstein (1992a) applying Ito’s formula

17 to ξ yields

dξt = µξdt + σξdZ (35) ξt where

DJW µξ = fu + JW X WJ VJ σ = WW n σ + VW σ for i = M,K,B,B∗ ξ J i i J V i W W

The ﬁrst order condition for the Bellman equation for optimal interior c is fc =

JW . Assuming that the optimal consumption policy is given by a smooth function

C of states [i.e., ct = C(Wt,St, t)], we can diﬀerentiate JW = fc with respect to W and obtain JWW = fccCW + fcuJW . Substituting these deﬁnitions together with the

∗ deﬁnitions of σc, σi for i = M,K,B,B and σV in expression for σξ we obtain a new expression for σξ in term of key volatility terms such as σc as follows:

(η − 1) η 1 X σ = σ − ( − 1) [σ ] + λσ (36) ξ c c ε W i V i i = P,P,Y,P ∗. The derivation of Eq. (36) can be found in Appendix. The process for the stochastic discount factor in (35) is well-deﬁned provided that each of the underlying variable can be characterized as a diﬀusion process:

dX(j) = µ dt + σ dZ, j = M, P, W, V. X(j) j j

Having obtained the stochastic discount factor we can use the fundamental asset pricing expression given in Cochrane (2001) [cf. eq. (1.35)] h i dPt D f dξt dPt (APE) Et + dt = rt − Et Pt Pt ξt Pt to derive a multi-factor version of the Capital Asset Pricing Model (CAPM). Here,

Dt is the dividend at time t and Pt is the price of common stock at time t.

18 We now exclude money in this setup to render the results comparable to the ex- isting literature e.g., Epstein and Zin (1991), Bakshi and Chen (1996), Smith (2001).

In our case applying (APE) to the expected rate of return on asset Xj yields h i dX f dξt dXj Et − rt = −Et X ξt Xj

Simplifying it gives h f (η − 1) η 1 µ − r = − ρ σ σ + ( − 1) ρ σ σ + ρ σ σ + ρ ∗ σ ∗ σ x t c cx c x ε W wx w x px p x p x p x i + λρvxσvσx (37)

1 where η = ζ −1 and ε = 1−γ. This is a multi (five) factor CAPM model—an extension of equation 22 in Duffie and Epstein (1992b) to include domestic and foreign inflation uncertainties and the relative reference wealth uncertainty. The first factor (first term on the right hand side) is a “premium” due to the variation in consumption, as the agent must be compensated for bearing consumption risk. The second term represents compensation due to variation in wealth: this matters for the risk premium because in recursive utility, wealth is a proxy for future utility. Moreover, in the spirit of capitalism framework, the investor cares about wealth induced status, as such the investor must be compensated for risks associated with the variation in wealth. Similarly, agents need to be compensated for bearing domestic and foreign inflation risks in equilibrium, which is evident in the third and fourth terms in equation (37). Finally, as in Bakshi and Chen (1996), and Smith (2001), the risk premium depends on the covariance of an asset’s return with reference wealth and the investor must be compensated for risks associated with the variations in the reference wealth index.

3.4 Exchange Rate Determination

In this section, we derive the stochastic process generating the equilibrium exchange rate. To this end a full version of the model is used but with logarithmic preferences

19 in order for explicit expressions.16 In particular, we obtain the equilibrium drift and diﬀusion terms of the following stochastic process:

dE = ˜ dt +σ ˜ dZ = ˜ dt + σ dZ − ω˜ [σ dZ + σ ∗ dZ ∗ ] E X X Y Y P P where ˜ is the equilibrium depreciation of the exchange rate. This will be determined by two equilibrium loci governing domestic and foreign real rates of return (RR) and portfolio balance (PP) as is done by Grinols and Turnovsky (1994). With logarithmic preferences the optimum share of capital in the traded component of the portfolio, ω, derived from the portfolio optimality conditions (15) and (16) is given by

∗ ∗ rK − (i − π ) + σP ∗Y ω = 2 2 (38) σY + 2σP ∗Y + σP ∗

This expression is the same as that obtained in Grinols and Turnovsky (1994) apart from covariances [cf eq. (25)]. This is not surprising because logarithmic preferences eliminate tangency portfolio effects, hence no international portfolio diversification effects. The relationship between and π that maintains equilibrium between the real rates of return is

∗ 2 (RR) π = i + − rK + σY ω

Similarly, the relationship between and π that maintains the portfolio balance is given by

∗ ∗ ∗ ∗ 2 2 2 2 1 αθδ (PP ) π = φ − (i − π ) + (i − π − σ ∗ − r )ω + (σ + σ ∗ )ω + , P K Y P Υ α + λ 16In this case the intertemporal utility function is given by

Z ∞ h i −δs Ut = e α[θ ln Cs + (1 − θ) ln(Ms/Ps)] + λ ln Ss ds. t

20 where

(1 − θ)αδ Υ = 1 − ∗ 2 . (α + λ)(i + − σX )

The equilibrium values of π and are determined by the intersection of (RR), which is a 45 degree line in the π − space, and (PP), which is a rectangular hyperbola. The effects of various exogenous shocks on the equilibrium values of the inflation rate and the exchange rate depreciation can be analyzed as in Grinols and Turnovsky (1994). Here we restrict our focus to the effects of the spirit of capitalism. As seen from

(RR) none of the parameters representing the spirit of capitalism such as λ, µV or

σV appear in the equation. However, the degree of the spirit of capitalism λ aﬀects the (PP) schedule causing it to shift down; this results in lower equilibrium values for inﬂation and exchange rate depreciation. As is well known from Smith (1999) and Gong and Zou (2002) the presence of the spirit of capitalism unambiguously reduces consumption and hence increases savings. This is evident from equation (13) where imposing logarithmic preferences yields the following optimal consumption-wealth ratio:

C αθδ = ∆ = W α + λ

The resulting increase in saving reduces the equilibrium interest rate. Given perfect capital mobility uncovered interest parity necessitates a domestic currency depreciation. Similarly, with the Fisher effect the decrease in i is fully reflected in π. A numerical analysis based on a full version of the model suggests a depreciation in the region of a 30% in the equilibrium exchange rate reflecting the similar drop in the domestic inflation rate. We also observe a decline of 5%-7% in the equilibrium interest rate.

21 4 Conclusions

We considered preferences with the “spirit of capitalism” in a continuous-time, infinite- horizon, small open economy model with complete financial markets and a single production good. We analyzed the effects of international catching-up and the spirit of capitalism on savings, growth, portfolio allocation and asset pricing. Our results show that status concerns affect equilibrium consumption, growth and portfolio allocation through “home bias” and the catching-up effects. The former effect arises due to hedging against future fluctuations in the external wealth reference index. When assets held in the portfolio are positively correlated with the reference world wealth index, the home bias effect is positive and individuals hold more of the domestic equity. The degree of the spirit of capitalism acts to amplify this effect. Moreover, we obtain higher excess returns due to international catching up through home bias and a lower risk free rate. These improve on the risk premium and low risk free rate puzzles. We also obtain a five-factor Capital Asset Pricing Model: in addition to the variation in domestic consumption and wealth, the representative investor needs compensation to bear risks associated with domestic and foreign inflation and the relative reference wealth uncertainty. Preliminary numerical exercises show that an increase in the spirit of capitalism gives rise to a substantial increase in the equilibrium growth rate. Similarly the concern for status leads to non-trivial price effects. All the interest rate, the inflation rate, and the exchange rate decline in response to the degree of the spirit of capitalism. Even though ”the capitalist spirit” matters, proper care should be taken to account for government expenditure and finance, and market imperfections such as those in credit, insurance, input and output markets. These are avenues for future research.

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28 Appendix: Derivation of Eq. (36)

Reproduce expression σξ: X WJ VJ σ = WW n σ + VW σ . ξ J i i J V i W W

The ﬁrst order condition for the Bellman equation for optimal interior c is fc = JW . Assuming that the optimal consumption policy is given by a smooth function C of states [i.e., ct = C(Wt,St, t)], we can diﬀerentiate JW = fc with respect to W and obtain JWW = fccCW + fcuJW . P h i W [fccCw+fcuJW ] σξ = niσi + λσV i JW P h i W [fccCwniσi+fcuJW niσi] = + λσV i JW P h i (A. 1) fcc fcuWJW niσi = CwniW σi + + λσV i fc JW P = fcc σ + [f W n σ ] + λσ fc c i cu i i V

Taking ﬁrst and second diﬀerentiation of the normalized “aggregator” function f with with respect to c and u we obtain

η−1 η−2 c c fcc (η − 1) fc = β η/ε−1 fcc = (η − 1)β η/ε−1 = (εu) (εu) fc c

η−1 η/ε−2 η−2 η/ε−2 η c (εu) η cc (εu) η cfcc fcu = −βε( ε − 1) (εu)2(η/ε−1) = −βε( ε − 1) (εu)(η/ε−1)(εu)(η/ε−1) = −( ε − 1) (η−1)u

η cfcc η cfcc = −( − 1) AW ε = −( − 1) ε ε (η−1)fcW (η−1) ε η 1 = −( ε − 1) W , 1 where η = ζ − 1 and ε = 1 − γ. Substituting these partial diﬀerentials in expression (A. 1) yields

(η − 1) η 1 X σ = σ − ( − 1) [σ ] + λσ for i = P,P,Y,P ∗ ξ c c ε W i V i which is equation (36) in the text.

29 Appendix:

Guidance for referees A Solution to the Consumer’s Optimization Prob-

lem

Consumption and portfolio rebalancing by the household takes place at discrete in- tervals of length ∆t. The household’s ﬂow budget constraint over the interval ∆t is then

dWt = ψWtdt + σW WtdZW t (A. 1a)

∗ ∗ ψ =nM rM + nBrB + nK rK + nBrB − τ − Ct/Wt (A. 1b)

σW dZW t = − nM σP dZP t − nBσP dZP t + nK AσY dZY t (A. 1c)

∗ − nBσQdZP ∗t − σT dZT t (A. 1d)

The household maximizes the utility functional given by Equation (1) in the text, subject to (A. 1a) and given an initial wealth and an initial world wealth. The value function for this problem can be written as

Z ∞ J(Wt,Vt) = f[Cs,Ms/Ps,Ss,J(Wt,Vt)]ds (A. 2) t where

1 n 1− ζ−1 o θ 1−θ α λ ζ ζ(1−γ) [Cs (Ms/Ps) ] Ss ] − [(1 − γ)Us] f(Cs,Us) = δ n ζ−1 o (A. 3) −1 1 ζ(1−γ) (1 − ζ )[(1 − γ)Us]

Duﬃe and Epstein (1992b) [cf. their equations (44) and (45)] show that the Bellman equation for optimal control is

sup DJ(Wt,Vt) + f[Cs,Ms/Ps,Ss,J(Wt,Vt)] = 0 (A. 4) {C,~n ∈ C×RN } where

1 DJ(Wt,Vt) = JwψW + JV µV V + 2 tr(Σ)

1 with  T     T T W~n ~σ − σT W JWW JWV W~n ~σ − σT W Σ =       σV V JVW JVV σV V and

T ~n ~σ = −nM σP − nBσP + nK σY − nP ∗ σP ∗ .

We conjecture that the value function and consumption function are b1−γ J(W ,V ) = W (α+λ)(1−γ)V λ(1−γ) (A. 5) t t 1 − γ t t

Ct = κWt (A. 6) and use the following deﬁnition

M/P = nM W where b and κ are constants to be determined. Using these conjectures, and setting η = 1 − 1/ζ and % = 1 − γ, the expression in brackets in (A. 4) can be expressed as n % 1−Γ1 1−Γ2 % 1−Γ1 1−Γ2 1 sup (1 − Γ1)b W V ψ + (1 − Γ2)b W V µV + 2 tr(Σ)+ κ,~n η η θ 1−θ α λ % 1−Γ1 1−Γ2 % o [κ nM W ] (W/V ) ] − %b W V δ η = 0 (A. 7) −1 η %b%W 1−Γ1 V 1−Γ2 % with 0 1T 0 1 0 1 T % −Γ −1 1−Γ % −Γ −Γ T W~n ~σ − σT W −(1 − Γ1)Γ1b W 1 V 2 (1 − Γ1)(1 − Γ2)b W 1 V 2 W~n ~σ − σT W Σ = @ A @ A @ A % −Γ −Γ % 1−Γ −Γ −1 σV V (1 − Γ2)(1 − Γ1)b W 1 V 2 −Γ2(1 − Γ2)b W 1 V 2 σV V

where Γ1 = 1 − (α + λ)(1 − γ) is the eﬀective coeﬃcient of relative risk aversion and

Γ2 = 1 − λ(1 − γ). Simplifying (A. 7) yields:

n 2 σW sup η(α + λ)[ψ − Γ1 ] κ,~n 2 σ2 o h(κθn1−θ)α iη o + ηλ[µ − Γ V ] + η(α + λ)λ(1 − γ)σ + δ M − 1 = 0. V 2 2 WV b (A. 8)

2 Subject to the portfolio adding-up condition (12) the Bellman equation gives opti-

∗ mality conditions for κ, nK , nM nB and nB as follows

∂ [(κθn1−θ)α/b]η : αθ M − (α + λ) = 0 (A. 9a) ∂κ κ ∂ χ :(α + λ)rK − (α + λ)Γ1σYW + (α + λ)λ(1 − γ)σYV − = 0, (A. 9b) ∂nK δ θ 1−θ η ∂ [(κ nM )/b] : α(1 − θ) + (α + λ)rM (A. 9c) ∂nM nM χ + (α + λ)Γ σ − (α + λ)λ(1 − γ)σ − = 0, (A. 9d) 1 PW PV δ ∂ χ :(α + λ)rB + (α + λ)Γ1σPW − (α + λ)λ(1 − γ)σPV − = 0, (A. 9e) ∂nB δ ∂ ∗ χ ∗ : + (α + λ)rB + (α + λ)Γ1σP ∗W − (α + λ)λ(1 − γ)σP ∗V − = 0. ∂nB δ (A. 9f) where χ is the Lagrangian multiplier associated with the portfolio adding-up condition. Substituting (A. 9a) into (A. 8) yields:

σ2 κ = Φ∆ + (1 − Φ)[r − τ − Γ W ] Q 1 2 λ σ2 + (1 − Φ) [µ − Γ V ] + (1 − Φ)λ(1 − γ)σ (A. 10) α + λ V 2 2 WV

αθδ where ∆ = α+λ is the eﬀective rate of time preference and Φ = (−(−1)θα) is the eﬀective elasticity of intertemporal substitution. As for portfolio shares we obtain them (i) by subtracting (A. 9e) from (A. 9b) (ii) by subtracting (A. 9f) from (A. 9b) and (iii) by subtracting (A. 9e) from (A. 9c) together with (A. 9a):

[rK − rB]dt = Γ1(σYW + σPW ) + λ(1 − γ)(σYV + σPV ), (A. 11)

∗ [rB − rB]dt = Γ1(−σP ∗W + σPW ) + λ(1 − γ)(−σP ∗V + σPV ), (A. 12) (1 − θ) C 1 n = . (A. 13) M θ W i

3 B Equilibrium

The assumption of constant drift (mean) and diﬀusion (variance) parameters in the geometric Brownian motion which describes the model variables ensures that risks and returns on assets are unchanging through time. This feature of the model, together with the constant elasticity utility function, generates a recurring equilib-

∗ rium, implying that the consumer chooses the same portfolio shares nM , nB, nK and consumption-wealth ratio, C/W , at each instant of time. Since domestic bonds are in zero ne supply they will not be held in equilibrium. Moreover, the multiplicity of all shocks (meaning that stochastic disturbances are proportional to the current state variables such as the capital stock and wealth), leads to an equilibrium in which means and variances of the relevant endogenous variables are jointly and consistently determined – a mean-variance equilibrium. The exogenous factors, apart from the four stochastic shocks explained above, include (i) the preference and technology parameters γ, λ, δ, θ, A (ii) the monetary policy parameter φ (monetary), and (iii) the mean foreign inﬂation rate π∗. The endogenous variables include (i) the stochastic adjustments in the economy σP dZP

(the stochastic adjustment in the domestic price level), σEdZE (the stochastic PPP relationship), σT dZT (the stochastic adjustment in transfers), σW dZW (the stochastic component of wealth), (ii) the transfer rate τ, (iii) the optimal consumption-wealth ratio and optimal portfolio shares, (iv) the equilibrium prices π (the expected domestic inﬂation rate), i (the nominal domestic interest rate), (the expected exchange rate depreciation), and (v) the equilibrium growth rate ψ. The determination of endogenous variables involves several stages. By using the constancy assumption of portfolio shares we ﬁrst solve the model for the price level and thereby π and σP dZP . The next stage is to determine stochastic adjustments. With the stochastic adjustments obtained, one can then calculate the endogenous variances

4 and covariances that appear in the optimality conditions for the consumption-wealth ratio, portfolio shares etc. The ﬁnal stage is to substitute these variances and covariances into the deterministic components of the equilibrium.

B.1 Price level

Some of the expressions set out in the remainder of this Section are familiar from the literature developed by Grinols and Turnovsky. However, there are some important diﬀerences, resulting from particular characteristics of the model developed here - most notably, these can be seen in the expressions for core variables in Section 3.4 (equations 30-35).

M/P nM The constant portfolio share requirement implies ∗ ∗ = ∗ . The price B /P +K nK +nB level can then be written as

∗ nK + nB M P = ∗ nM EB + K

Calculating dP/P and equating its deterministic and stochastic parts yields

ω C π − = R1, (B. 14a) nK W

∗ ∗ σP dZP = σX dZX − ωσY dZY + (1 − ω)σP dZP , (B. 14b) where

∗ ∗ 2 ∗ ∗ R1 = φ − ω − (1 − ω)(i − π + σP ∗ ) − cov σX dZX , ωσY dZY − (1 − ω)σP dZP + var ωσY dZY − (1 − ω)σP ∗ dZP ∗ . (B. 14c)

B.2 Determination of transfers

Real transfers dT depend on the money supply rule and the price level. That is,

dT = (1/P )dM = φ(M/P )dt + σX (M/P )dZX

5 Using the deﬁnition of nM (portfolio share of money) and substituting (10) for dT we obtain

τ =φnM , (B. 15)

σT dZT =σX nM dZX . (B. 16)

B.3 Endogenous stochastic processes

Since random shocks that impinge on the economy remain constant through time, the investment opportunity set is constant over time. Stochastic diﬀerentiation of

∗ the equations for portfolio shares nM , nK and nB (following equation [σW dZW ]) then implies

σX dZX − σP dZP − σW dZW = 0, (B. 17a)

σK dZK − σW dZW = 0 (B. 17b)

∗ ∗ ∗ ∗ σBdZB − σP dZP − σW dZW = 0. (B. 17c)

Equations for σP dZP (B. 14b), σW dZW (11c), σT dZT (B. 16) and (B. 17a)—

(B. 17c) form a six-equation linear system in the six unknowns (σP dZP , σEdZE,

∗ ∗ σK dZK , σW dZW , σT dZT , σBdZB). Solving implies

∗ ∗ σP dZP = σX dZX − ωσY dZY + (1 − ω)σP dZP , (B. 18a)

∗ ∗ σEdZE = σX dZX + −ωσY dZY − ωσP dZP , (B. 18b)

∗ ∗ σK dZK = ωσY dZY + (1 − ω)σP dZP , (B. 18c)

∗ ∗ σW dZW = ωσY dZY + (1 − ω)σP dZP , (B. 18d)

σT dZT = σX nM dZX , (B. 18e)

∗ ∗ ∗ ∗ σBdZB = ω (σY dZY + σP dZP ) . (B. 18f)

6 B.4 Expressions for core variables

• Portfolio shares:

The portfolio share for real balances nM is obtained from the FOC: (1 − θ) C 1 n = . (B. 19) M θ W i The optimal portfolio share of each of the remaining three assets (equity, for-

∗ eign bonds and domestic bonds), nK and nB are determined from adding up

condition for portfolio shares (12) with nB = 0 and the deﬁnition of ω:

nK = ω(1 − nM ) (B. 20a) 1 − ω n∗ = n (B. 20b) B ω K

• ω can now be calculated from the optimality conditions (15) and (16)

∗ ∗ 2 2 A − (i − π + σ ∗ ) + Γ [Aσ ∗ + σ ∗ ] ω = P 1 P Y P + bias (B. 21) Γ1Λ where the term (home) bias can be written as σ + σ ∗ bias = H YV P V , (B. 22) Λ and the Λ is deﬁned as

2 2 2 Λ = A σY + 2σP ∗Y + σP ∗ .

In eq. (B. 22), the expression H denotes the ‘propensity to hedge’ against unanticipated ﬂuctuations in the world wealth index, and deﬁned as J V λ(1 − γ) H = − WV = − . JWW W Γ1

• The optimality condition (15) yields

i − π = R2 (B. 23)

where

2 R2 = A − σP − Γ1(σWY + σWP ) + λ(1 − γ)(σPV − σYV )

7 • The optimality condition (16) yields

i − π = R3 (B. 24)

∗ ∗ 2 2 R3 = i − π + σP ∗ − σP − Γ1(σPW − σP ∗W ) − λ(1 − γ)(σPV − σP ∗V )

• The deterministic part of the PPP condition (5) yields

π − = R4 (B. 25)

where

∗ R4 = π + σP ∗E

• The optimality condition for consumption (13) yields

a R5 C/W + 1 ψ = (B. 26) b1 b1

where

σ2 λ σ2 R5 = Φ∆ − (1 − Φ)Γ W + (1 − Φ) [µ − Γ V ] + (1 − Φ)λ(1 − γ)σ 1 2 α + λ V 2 2 WV

a1 = Φ − 1,

b1 = Φ.

• Setting the deterministic part of the capital accumulation expression (18) to the equilibrium growth rate ψ yields

ω ψ + C/W = R6 (B. 27) nK

where

∗ 2 R6 = ω + (1 − ω)(i − + σP ∗ )

As in Grinols (1996) there is now a seven-equation system ((B. 14a), (B. 20a),

(B. 23)-(B. 26)) plus (14) in the seven unknowns (C/W, π, nK , i, , nM , ψ).

8 C Numerical Analysis

In this section we undertake a full numerical analysis of the model. The generation of numerical estimates requires the speciﬁcation of a number of baseline parameters and variables. Tables 2 and 3 set out the values used in the numerical exercises carried out here.

[ Tables 2 and 3 approximately here.]

In an attempt to utilize plausible values, a number of the parameter values have been set by reference to quarterly data relevant for the UK economy from the period 1979Q1 to 1999Q2. All data were obtained from Datastream. For the UK, we have used series averages and standard deviations for: industrial production, government expenditure, the M1 money supply, and the inﬂation rate derived from the consumer price index. To capture ‘foreign’ price and interest rate variables we have used US data, in particular: series averages and standard deviations for the consumer price index and the 3-month Treasury bill rate. The variance and covariance parameters were calculated using the discrete time version of geometric Brownian motion

~ ~ 0 ~ ∆ ln X(ts) = ~η ∆t + Γ ∆Z(ts) (C. 28)

~ 0 where X(ts) = (x1(ts), . . . , x4(ts)) is the (four) exogenous stochastic shock vector, ~ 1 2 Z(ts) is the independent normal variates vector N(0,I∆t), ~η ≡ (µ1 − 2 σ1), ..., µ4 − 1 2 ~~ 0 ~ 2 σ4)) is the mean vector, and the variance-covariance matrix Σ ≡ ΓΓ with Γ =

(σ1, ..., σ4). Finally, time evolves ts+1 ≡ ts + ∆t, s = 0,...,S − 1 with tS = T . Because the regressors in (C. 28) are identical equation by equation, the least squares estimator ~ηˆ of ~η becomes

1 ~ηˆ = [ln X(T ) − X(0)] T

9 for S∆t. Similarly, the sampling estimator Σˆ of Σ is (" # 1 S XS−1 1 Σˆ = ∆ ln X~ (t ) − (ln X~ (T ) − X~ (0)) T S − 1 s S s=0 " #0) 1 × ∆ ln X~ (t ) − (ln X~ (T ) − X~ (0)) . (C. 29) s S

Particular mention should be made of the values assigned to the parameters for risk aversion and the intertemporal substitution elasticity: the values 4 and 0.5 are those used in Obstfeld (1994b). The former is the mid point of the range of conventional estimates (2 - 6) referred to in Obstfeld (1994a) although we are mindful that some authors suggest that values of unity or values as high as 30 cannot be ruled out (see Epstein and Zin (1991); and Kandel and Stambough (1991), respectively). The intertemporal substitution elasticity set to 0.5 is consistent with what Epstein and Zin (1991) describe as ‘a reasonable inference’. However, smaller values cannot be ruled out, for example Hall (1988) and Campbell and Mankiw (1989) suggest an intertemporal substitution elasticity of 0.10; and Ogaki and Reinhart (1998) refer to the range 0.32 - 0.45. Later in this paper we explore the sensitivity of our results to different values for the key risk aversion parameter and for a range of correlation coefficients. Table 4 presents numerical exercises to ascertain the effects of international catching up and the spirit of capitalism. The column labelled Model 0 presents the baseline model values whereas Model 1 gives the results from the absolute-wealth-is-status specification. Results from the Model 2 column are obtained from the relative-wealth- is status specification with a degree of the spirit of capitalism parameter λ set to 0.1. All numbers in Models 1 and 2 represent percentage changes relative to the baseline. It can be seen that the model generates significant growth and price effects. The consumption wealth ratio stays the same in Model 1 and declines in Model 2. This gives rise to a substantial increase in the equilibrium growth rate where the growth

10 rate rises by 27% and 38% in models 1 and 2 respectively. Similarly the decline in the interest rate, the inﬂation rate and the exchange rate are also substantial - ranging from 5% to 39%. Notice also that the variance of the equilibrium growth rate and of the price level decline in Model 2.

[Table 4 approximately here.]

C.1 Sensitivity analysis

In this section we run a sensitivity analysis in order to gauge the robustness of our results with respect to the key parameters; (i) intertemporal elasticity of substitution ζ, (ii) risk aversion γ and (iii) degree of the spirit of capitalism λ. However, values should be chosen in such way that they satisfy both feasibility ( ≤ 1) and transversality (γ ≥ 1) conditions as demonstrated in Smith (1996a). Figure 1 plots the sensitivity of the equilibrium growth rate, the portfolio share of foreign bonds, the exchange rate and the beta of foreign bonds with respect to ζ and γ under the benchmark value of λ. Figure 1 reveals that the growth rates and exchange rate are sensitive whereas the beta and portfolio share of foreign bonds are robust. However, γ plays no role in the determination of the equilibrium exchange rate as a result of the small-open economy representative agent nature of the model [see Sibert (1996)]. Figures 2 and 3 also reveal the sensitivity of the results with respect to λ.

C.2 Determination of nominal interest rate

Substituting (B. 14a) in (B. 23) we obtain an expression for i:

ωκ i = R1 + R2 + (C. 30a) nK

We use (A. 13) and (B. 20a) to eliminate nM

(1 + ν)(1 − θ)κ n + K = 1 (C. 30b) iθ ω

11 We use (B. 26) and (B. 27) to eliminate ψ

a ωκ a R5 κ − 1 + 1 R6 = (C. 30c) a1 + b1 nK a1 + b1 a1 + b1

Solve (C. 30c) for nK

c1ωκ nK = (C. 31a) κ + c1R6 − d1R5

a1 1 where c1 = and d1 = . b1 b1 Substitute (C. 31a) into (C. 30a) and (C. 30b) and rearranging gives

[κ + c R6 − d R5] i − R1 − R2 − 1 1 = 0 (C. 31b) c1 (1 + ν)(1 − θ) i −i + κ + κ = 0 (C. 31c) θ i − R1 − R2 Solve (C. 31b) for κ

κ = c1(i − R1 − R2) − [c1R6 − d1R5] (C. 32)

Finally, by placing (C. 32) in (C. 31c) and simplifying we obtain a quadratic expression for the interest rate:

2 −a0i + b0i + c0 = 0. (C. 33)

a0 = (1 − θ) − (1 + ν)θc1 − (1 − θ)c1

b0 = R12(1 − θ) − (1 + ν)θc1R12 − (1 + ν)θR65

− (1 + ν)θc1R12 − (1 − θ)c1R12 − (1 − θ)R65

2 c0 = (1 + ν)θc1R12 + (1 + ν)θR12R65 where

R12 = R1 + R2 and R65 = c1R6 + d1R5.

12 Table 1: Descriptive Statistics and Variance-Covariance Matrix

Panel A: Moments

Foreign Foreign Money inﬂation wealth Production supply (M1) rate index Variables YMP ∗ V

Mean 0.02500 0.03500 0.05000 0.03500 Std. Deviation 0.04000 0.06000 0.05000 0.03873 Variance 0.00160 0.00360 0.00250 0.00150

Panel B: Covariances (normal text) and Correlations (italicized text)

Variables Y M P ∗ V

Y 0.00160 0.20833 0.70000 0.11667 M 0.00050 0.00360 0.27778 0.27778 P ∗ 0.00140 0.00083 0.00250 0.66667 V 0.00018 0.00065 0.00129 0.00150 Table 2: Baseline Parameters and Variables

Variable Symbol Values Parameters Marginal product of capital A 0.080 Risk aversion parameter γ 4.000 Intertemporal substitution elasticity ζ 0.250 Degree of the spirit of capitalism λ 0.000 Rate of time preference ρ 0.040 Consumption intensity θ 0.850 Money growth rate φ 0.035 Foreign interest rate i∗ 0.080 Foreign inﬂation rate π∗ 0.050 Variables C Consumption–wealth ratio W 0.004 Rate of return on portfolio rQ 0.073 Mean equilibrium growth rate ψ 0.017 2 Variance of growth rate σw 0.001 2 Variance of price level σp 0.005 Inﬂation rate π 0.019 Interest rate i 0.094 Exchange rate 0.017 Rate of return on capital rK 0.080 Rate of return on foreign bond rF 0.079 1 2 Risk adjusted rate of return rQ − 2 γσw 0.071 Equity beta βK 0.129 ∗ Foreign bonds beta βB 0.161 Portfolio share of equity in tradeable ω 0.650 Portfolio share of money nM 0.105 Portfolio share of equity nK 0.582 Portfolio share of foreign bonds nF 0.313

14 Table 3: International Catching-up and the Spirit of Capitalism Eﬀects

Variable Symbol Model 0 Model 1 Model 2 C Consumption–wealth ratio W 0.004 0.000 -0.270 Rate of return on portfolio rQ 0.073 0.890 1.268 Mean equilibrium growth rate ψ 0.017 27.037 38.250 2 Variance of growth rate σw 0.001 0.000 -0.115 2 Variance of price level σp 0.005 0.000 -0.024 Inﬂation rate π 0.019 -25.140 -35.571 Interest rate i 0.094 -4.977 -7.030 Exchange rate 0.017 -27.513 -38.779 Rate of return on capital rK 0.080 0.000 0.000 Rate of return on foreign bond rF 0.079 0.063 0.132 1 2 Risk adjusted rate of return rQ − 2 γσw 0.071 0.915 1.306 Equity beta βK 0.129 0.140 0.121 ∗ Foreign bonds beta βB 0.161 0.140 0.121 Portfolio share of equity in tradeable ω 0.650 0.000 -0.548 Portfolio share of money nM 0.105 -2.364 -3.417 Portfolio share of equity nK 0.582 0.278 -0.149 Portfolio share of foreign bonds nF 0.313 0.278 1.422 Figure 1: Model 3 with Benchmark Degree of the SC Figure 2: Model 3 with High Degree of the SC Figure 3: Model 3 with Low Degree of the SC