Sunspot Fluctuations in Two-Sector Models with Variable Income Effects

Abstract

We analyze a version of the Benhabib and Farmer (1996) two-sector model with sector-specific externalities in which we consider a class of utility functions inspired from the one considered in Jaimovich and Rebelo (2009) which is flexible enough to encompass varying degrees of income effect. First, we show that local indeterminacy and sunspot fluctuations occur in 2-sector models under plausible configurations regarding all structural parameters—in particular regarding the intensity of income effects. Second, we prove that there even exist some configurations for which local indeterminacy arises under any degree of income effect. More precisely, for any given size of income effect, we show that there is a non-empty range of values for the Frisch elasticity of labor and the elasticity of intertemporal substitution in consumption such that indeterminacy occurs. This contrasts with the results obtained in one-sector models in both Nishimura et al. (2009), in which it is shown that indeterminacy cannot occur under either GHH and KPR preferences, and in Jaimovich (2008) in which local indeterminacy only arises for intermediary income effects.

This work has been carried out thanks to the support of the A*MIDEX project (no. ANR-11-IDEX-0001-02) funded by the “Investissements d’Avenir” French Government program, managed by the French National Research Agency (ANR). We would like to thank an anonymous referee together with X. Raurich and T. Seegmuller for useful comments and suggestions.

4.8 Appendix

1.1 4.8.1 Proof of Proposition 1

Consider the steady state with $Y_{I}=\delta k$ and $r=(\delta \,+\,\rho )p$. Since $r=p\alpha Y_{I}/K_{I}$, we get

$$\begin{aligned} \begin{array}{c} K_{I}=\dfrac{\alpha \delta }{\delta \,+\,\rho } k \end{array} \end{aligned}$$

(4.35)

Using the production function (4.19) for the investment good we derive

$$\begin{aligned} \begin{array}{l} Y_{I}=\left( \frac{k}{l}\right) ^{(\alpha \,-\,1) (1\,+\,\Theta )}\left( \frac{ \alpha \delta }{\delta \,+\,\rho } k\right) ^{1\,+\,\Theta }=\delta k \end{array} \end{aligned}$$

Solving this equation yields

$$\begin{aligned} \begin{array}{l} k^{*}=l^{\frac{(1\,-\,\alpha )(1\,+\,\Theta )}{1\,-\,\alpha (1\,+\,\Theta )}}\left( \frac{\alpha }{\delta \,+\,\rho }\right) ^{\frac{1\,+\,\Theta }{1\,-\,\alpha (1\,+\,\Theta )} }\delta ^{\frac{\Theta }{1\,-\,\alpha (1\,+\,\Theta )}}\equiv l^{\frac{ (1\,-\,\alpha )(1\,+\,\Theta )}{1\,-\,\alpha (1\,+\,\Theta )}}\kappa ^{*} \end{array} \end{aligned}$$

(4.36)

Substituting this expression into (4.22) we get

$$\begin{aligned} \begin{array}{l} c^{*}=l^{\frac{1\,-\,\alpha }{1\,-\,\alpha (1\,+\,\Theta )}}\frac{\delta (1-\alpha )+\rho }{ \delta +\rho }\kappa ^{*\alpha }\equiv l^{\frac{1\,-\,\alpha }{1\,-\,\alpha (1\,+\,\Theta )} }\psi ^{*} \end{array} \end{aligned}$$

(4.37)

Recall that the trade-off between consumption and leisure is described by

$$\begin{aligned} \begin{array}{l} \frac{(1+\chi )l^{\chi }c^{\gamma }}{1+\chi \,-\,\gamma l^{1+\chi }c^{\gamma \,-\,1}}=w \end{array} \end{aligned}$$

(4.38)

Using (4.23) with (4.36)–(4.37) we get

$$\begin{aligned} \begin{array}{l} (1\,+\,\chi )l^{\chi \,+\,\frac{\gamma (1\,-\,\alpha )}{1\,-\,\alpha (1\,+\,\Theta )} }\psi ^{*\gamma }=(1\,-\,\alpha )l^{\frac{\alpha \Theta }{1\,-\,\alpha (1\,+\,\Theta )} }\kappa ^{*\alpha }\left[ 1\,+\,\chi \,-\,\gamma l^{1\,+\,\chi \,-\,\frac{(1\,-\,\gamma )(1\,-\,\alpha )}{ 1\,-\,\alpha (1\,+\,\Theta )}}\psi ^{*\gamma \,-\,1}\right] \end{array} \end{aligned}$$

If $\chi [1\,-\,\alpha (1\,+\,\Theta )]\,+\,\gamma (1\,-\,\alpha )\,-\,\alpha \Theta \ne 0$, solving this equation yields

$$\begin{aligned} \begin{array}{l} l^{*}=\left\{ \frac{(1\,-\,\alpha )\kappa ^{*}}{\psi ^{*\gamma }}\left[ 1\,+\,\frac{ (1\,-\,\alpha )\kappa ^{*}\gamma }{(1\,+\,\chi )\psi ^{*}}\right] ^{-1}\right\} ^{\frac{ 1\,-\,\alpha (1\,+\,\Theta )}{\chi [1\,-\,\alpha (1\,+\,\Theta )]\,+\,\gamma (1\,-\,\alpha )\,-\,\alpha \Theta }} \end{array} \end{aligned}$$

We finally derive from (4.23)

$$\begin{aligned} \begin{array}{rcl} p^{*}= & {} \alpha (k^{*}/l^{*})^{\alpha -1} \end{array} \end{aligned}$$

$\square$

1.2 4.8.2 Proof of Lemma 1

Using (4.24) and the first order conditions (4.9)–(4.10), we get $\epsilon _{cl}=\epsilon _{lc}(c/wl)$. Using the expression of w given in (4.23) together with the values of $k^{*}$ and $l^{*}$ provided in Sect. “Proof of Proposition 1” we find $wl/c=(1-\alpha )(\delta \,+\,\rho )/[\delta (1- \alpha )\,+\,\rho ]$. Then at the steady state we get

$$\begin{aligned} \begin{array}{c} \epsilon _{cl}=\frac{\delta (1\,-\,\alpha )\,+\,\rho }{(1\,-\,\alpha )(\delta \,+\,\rho )}\epsilon _{lc} \end{array} \end{aligned}$$

(4.39)

Using (4.24), we compute for the utility function as given by (4.14) the following elasticities:

$$\begin{aligned} \begin{array}{ll} \frac{1}{\epsilon _{cc}}=\sigma \frac{c\,-\,\gamma \frac{ l^{1\,+\,\chi }}{1\,+\,\chi } c^{\gamma }}{c\,-\,\frac{ l^{1\,+\,\chi }}{1\,+\,\chi }c^{\gamma }}\,-\,\gamma (1\,-\,\gamma )\frac{ \frac{ l^{1\,+\,\chi }}{1\,+\,\chi }c^{\gamma }}{c\,-\,\gamma \frac{ l^{1\,+\,\chi }}{1\,+\,\chi } c^{\gamma }}, &{}\quad \frac{1}{\epsilon _{lc}} =\sigma \frac{c\,-\,\gamma \frac{ l^{1\,+\,\chi } }{1\,+\,\chi }c^{\gamma }}{c\,-\,\frac{ l^{1\,+\,\chi }}{1\,+\,\chi }c^{\gamma }}\,-\,\gamma \\[10pt] \frac{1}{\epsilon _{cl}}= \frac{ l^{1\,+\,\chi }c^{\gamma }}{c\,-\,\gamma \frac{ l^{1\,+\,\chi }}{1\,+\,\chi }c^{\gamma }}\left[ \sigma \frac{c\,-\,\gamma \frac{ l^{1\,+\,\chi }}{ 1\,+\,\chi }c^{\gamma }}{c\,-\,\frac{ l^{1\,+\,\chi }}{1\,+\,\chi }c^{\gamma }}\,-\,\gamma \right] , &{} \frac{1}{\epsilon _{ll}}= \sigma \frac{ l^{1\,+\,\chi }c^{\gamma }}{c\,-\,\frac{ l^{1\,+\,\chi }}{1\,+\,\chi }c^{\gamma }}\,+\,\chi \end{array} \end{aligned}$$

(4.40)

Obviously, normality holds as we derive from these expressions that

$$\begin{aligned} \begin{array}{c} \frac{1}{\epsilon _{cc}}\,-\,\frac{1}{\epsilon _{lc}}\ge 0 \text{ and } \frac{1}{ \epsilon _{cl}}\,-\,\frac{1}{\epsilon _{ll}}\ge 0 \end{array} \end{aligned}$$

(4.41)

Consider now Eq. (4.39) together with the expressions given by (4.40). We then derive that

$$\begin{aligned} \begin{array}{c} \frac{l^{1\,+\,\chi }c^{\gamma \,-\,1}}{1\,-\,\gamma \frac{l^{1\,+\,\chi }}{1\,+\,\chi } c^{\gamma \,-\,1}}=\frac{(1\,-\,\alpha )(\delta \,+\,\rho )}{\delta (1\,-\,\alpha )\,+\,\rho } \end{array} \end{aligned}$$

(4.42)

Denoting $\mathcal {C}=[(1\,-\,\alpha )(\delta \,+\,\rho )]/[\delta (1\,-\,\alpha )\,+\,\rho ]<1$, solving this equation yields

$$\begin{aligned} \begin{array}{c} l^{1\,+\,\chi }c^{\gamma \,-\,1}=\frac{\mathcal {C}(1\,+\,\chi )}{1\,+\,\chi \,+\,\gamma \mathcal { C}} \end{array} \end{aligned}$$

(4.43)

and thus

$$\begin{aligned} \begin{array}{rl} \frac{c\,-\,\gamma \frac{l^{1\,+\,\chi }}{1\,+\,\chi }c^{\gamma }}{c\,-\,\frac{l^{1\,+\,\chi }}{ 1\,+\,\chi }c^{\gamma }}=\frac{1\,+\,\chi }{1\,+\,\chi \,-\,\mathcal {C}(1\,-\,\gamma )},&\quad \frac{ l^{1\,+\,\chi }c^{\gamma }}{c\,-\,\frac{l^{1\,+\,\chi }}{1\,+\,\chi }c^{\gamma }}=\frac{ \mathcal {C}(1\,+\,\chi )}{1\,+\,\chi \,-\,\mathcal {C}(1\,-\,\gamma )} \end{array} \end{aligned}$$

(4.44)

Using these expressions we then derive from (4.40):

(4.45)

Concavity of the utility function requires

$$\begin{aligned} \begin{array}{rcl} \frac{1}{\epsilon _{cc}\epsilon _{ll}}\,-\,\frac{1}{\epsilon _{lc}\epsilon _{cl}} \ge 0&\quad \text{ and }&\quad \frac{1}{\epsilon _{cc}}\ge 0 \end{array} \end{aligned}$$

Straightforward computations show that these two inequalities are satisfied if and only if

$$\begin{aligned} \begin{array}{l} \sigma \ge \sigma _{c}(\gamma )\equiv \frac{\gamma \mathcal {C}(\gamma +\chi )\left[ 1+\chi \,-\,(1\,-\,\gamma )\mathcal {C}\right] }{(1+\chi )^{2}\left[ \chi +\gamma \mathcal {C}\left( 2\,-\,\frac{\mathcal {C}(1\,-\,\gamma )}{1+\chi }\right) \right] } \end{array} \end{aligned}$$

$\square$

1.3 4.8.3 Proof of Proposition 2

We start by the computation of $\mathcal {D}$ and $\mathcal {T}$ using a general formulation for $U(c,\mathcal {L})$. Consider the consumption-labor trade-off as described by (4.9)–(4.10) together with the expressions of wage and consumption as given by (4.22) and (4.23). We get the following two equations

$$\begin{aligned} U_{2}(c,\ell -l)l^{\alpha } = (1-\alpha )k^{\alpha }U_{1}(c,\ell -l) \end{aligned}$$

(4.46)

$$\begin{aligned} cl^{\alpha \,-\,1} = k^{\alpha \,-\,1}\left[ k\,-\,\left( \frac{k}{l}\right) ^{1\,-\, \alpha }p^{-1/\Theta }\right] \end{aligned}$$

(4.47)

Total differentiation of (4.46) gives

$$\begin{aligned} \begin{array}{l} \frac{dc}{c}\left( \frac{1}{\epsilon _{cc}}\,-\,\frac{1}{\epsilon _{lc}}\right) + \frac{dl}{l}\left( \frac{1}{\epsilon _{ll}}\,-\,\frac{1}{\epsilon _{cl}} \,+\,\alpha \right) =\alpha \frac{dk}{k} \end{array} \end{aligned}$$

(4.48)

Total differentiation of (4.47) gives

$$\begin{aligned} \begin{array}{l} \frac{dc}{c}\,-\,(1-\alpha )\frac{dl}{l}=-(1\,-\,\alpha )\frac{dk}{k}\,+\,\frac{k^{*}}{ k^{*}\,-\,K_{I}^{*}}\frac{dk}{k}\,-\,\frac{K_{I}^{*}}{k^{*}\,-\,K_{I}^{*}}\left[ (1\,-\,\alpha )\left( \frac{dk}{k}\,-\,\frac{dl}{l}\right) \,-\,\frac{1}{\Theta }\frac{dp}{p} \right] \\ \end{array} \nonumber \\ \end{aligned}$$

(4.49)

At the steady state we know that $(\delta \,+\,\rho )p=r$ with $r=p\alpha Y_{I}/K_{I}=p\alpha \delta k/K_{I}$. We then derive $K_{I}^{*}=\alpha \delta k^{*}/(\delta +\rho )$ and thus

$$\begin{aligned} \begin{array}{rl} \frac{k^{*}}{k^{*}\,-\,K_{I}^{*}}=\frac{\delta \,+\,\rho }{\rho \,+\,\delta (1\,-\,\alpha )}, \frac{K_{I}^{*}}{k^{*}\,-\,K_{I}^{*}}=\frac{\alpha \delta }{\rho \,+\,\delta (1\,-\,\alpha )}&\end{array} \end{aligned}$$

Equation (4.48) then becomes:

$$\begin{aligned} \begin{array}{l} \left[ \rho \,+\,\delta (1\,-\,\alpha )\right] \frac{dc}{c}\,-\,(1\,-\,\alpha )(\delta \,+\,\rho )\frac{ dl}{l}=\alpha (\delta \,+\,\rho )\frac{dk}{k}\,+\,\frac{\alpha \delta }{\Theta }\frac{dp}{p } \end{array} \end{aligned}$$

(4.50)

From (4.48) we derive

$$\begin{aligned} \begin{array}{l} \frac{dl}{l}=\,-\,\frac{dc}{c}\frac{\frac{1}{\epsilon _{cc}}\,-\,\frac{1}{ \epsilon _{lc}}}{\frac{1}{\epsilon _{ll}}\,-\,\frac{1}{\epsilon _{cl}}+\alpha }+ \frac{dk}{k}\frac{\alpha }{\frac{1}{\epsilon _{ll}}\,-\,\frac{1}{\epsilon _{cl}} +\alpha } \end{array} \end{aligned}$$

(4.51)

Substituting this expression into (4.50) gives

$$\begin{aligned} \begin{array}{rcl} \frac{dc}{c} &{} = &{} \frac{\alpha (\delta \,+\,\rho )\left( \frac{1}{\epsilon _{ll}}\,-\, \frac{1}{\epsilon _{cl}}\,+\,1\right) }{\left[ \rho \,+\,\delta (1\,-\,\alpha )\right] \left( \frac{1}{\epsilon _{ll}}\,-\,\frac{1}{\epsilon _{cl}}\,+\,\alpha \right) \,+\,(1\,-\,\alpha )( \delta \,+\,\rho )\left( \frac{1}{\epsilon _{cc}}\,-\,\frac{1}{\epsilon _{lc}}\right) } \frac{dk}{k} \\[12pt] &{} + &{} \frac{\alpha \delta \left( \frac{1}{\epsilon _{ll}}\,-\,\frac{1}{\epsilon _{cl}} +\alpha \right) }{\Theta \left[ \left[ \rho \,+\,\delta (1\,-\,\alpha )\right] \left( \frac{1}{ \epsilon _{ll}}\,-\,\frac{1}{\epsilon _{cl}}\,+\,\alpha \right) \,+\,(1\,-\,\alpha )(\delta \,+\,\rho ) \left( \frac{1}{\epsilon _{cc}}\,-\,\frac{1}{\epsilon _{lc}}\right) \right] }\frac{dp }{p} \end{array} \end{aligned}$$

(4.52)

Substituting (4.52) into (4.51) finally gives

$$\begin{aligned} \begin{array}{rcl} \frac{dl}{l} &{} = &{} \,-\,\frac{\alpha \left[ (\delta \,+\,\rho )\left( \frac{1}{ \epsilon _{cc}}\,-\,\frac{1}{\epsilon _{lc}}\right) \,-\,\left[ \rho \,+\,\delta (1\,-\,\alpha ) \right] \right] }{\left[ \rho \,+\,\delta (1\,-\,\alpha )\right] \left( \frac{1}{ \epsilon _{ll}}\,-\,\frac{1}{\epsilon _{cl}}\,+\,\alpha \right) \,+\,(1\,-\,\alpha )(\delta +\rho ) \left( \frac{1}{\epsilon _{cc}}\,-\,\frac{1}{\epsilon _{lc}}\right) }\frac{dk}{k} \\[12pt] &{} - &{} \frac{\alpha \delta \left( \frac{1}{\epsilon _{cc}}\,-\,\frac{1}{\epsilon _{lc}} \right) }{\Theta \left[ \left[ \rho +\delta (1\,-\,\alpha )\right] \left( \frac{1}{ \epsilon _{ll}}\,-\,\frac{1}{\epsilon _{cl}}+\alpha \right) +(1\,-\,\alpha )(\delta +\rho ) \left( \frac{1}{\epsilon _{cc}}\,-\,\frac{1}{\epsilon _{lc}}\right) \right] }\frac{dp }{p} \end{array} \end{aligned}$$

(4.53)

We then conclude from this

$$\begin{aligned} \begin{array}{rcl} \frac{\partial c}{\partial k}\frac{k^{*}}{c^{*}} &{} = &{} \frac{\alpha (\delta +\rho )\left( \frac{1 }{\epsilon _{ll}}\,-\,\frac{1}{\epsilon _{cl}}+1\right) }{\left[ \rho +\delta (1\,-\, \alpha )\right] \left( \frac{1}{\epsilon _{ll}}\,-\,\frac{1}{\epsilon _{cl}} +\alpha \right) +(1\,-\,\alpha )(\delta +\rho )\left( \frac{1}{\epsilon _{cc}}\,-\,\frac{1}{ \epsilon _{lc}}\right) } \\[10pt] \frac{\partial c}{\partial p}\frac{p^{*}}{c^{*}} &{} = &{} \frac{\alpha \delta \left( \frac{1}{ \epsilon _{ll}}\,-\,\frac{1}{\epsilon _{cl}}+\alpha \right) }{\Theta \left[ \left[ \rho +\delta (1\,-\,\alpha )\right] \left( \frac{1}{\epsilon _{ll}}\,-\,\frac{1}{ \epsilon _{cl}}+\alpha \right) +(1\,-\,\alpha )(\delta +\rho )\left( \frac{1}{ \epsilon _{cc}}\,-\,\frac{1}{\epsilon _{lc}}\right) \right] } \\[10pt] \frac{\partial l}{\partial k}\frac{k^{*}}{l^{*}} &{} = &{} \,-\,\frac{\alpha \left[ (\delta +\rho )\left( \frac{1}{\epsilon _{cc}}\,-\,\frac{1}{\epsilon _{lc}}\right) \,-\,\left[ \rho +\delta (1\,-\,\alpha )\right] \right] }{\left[ \rho +\delta (1\,-\,\alpha )\right] \left( \frac{1}{\epsilon _{ll}}\,-\,\frac{1}{\epsilon _{cl}}+\alpha \right) +(1\,-\,\alpha )( \delta +\rho )\left( \frac{1}{\epsilon _{cc}}\,-\,\frac{1}{\epsilon _{lc}}\right) } \\[10pt] \frac{\partial l}{\partial p}\frac{p^{*}}{l^{*}} &{} = &{} \,-\,\frac{\alpha \delta \left( \frac{1}{ \epsilon _{cc}}\,-\,\frac{1}{\epsilon _{lc}}\right) }{\Theta \left[ \left[ \rho +\delta (1\,-\,\alpha )\right] \left( \frac{1}{\epsilon _{ll}}\,-\,\frac{1}{ \epsilon _{cl}}+\alpha \right) +(1\,-\,\alpha )(\delta +\rho )\left( \frac{1}{ \epsilon _{cc}}\,-\,\frac{1}{\epsilon _{lc}}\right) \right] } \end{array} \end{aligned}$$

(4.54)

Recall now that $r=\alpha (k/l)^{\alpha \,-\,1}$ and $Y_{I}=p^{-(1+\Theta )/\Theta }$. Using again the steady state relationships $Y_{I}=\delta k$ and $(\delta +\rho )p=r$, we derive

$$\begin{aligned} \begin{array}{rcl} \frac{dY_{I}}{dp}\frac{p^{*}}{Y_{I}^{*}}=-\frac{1+\Theta }{\Theta },&\frac{dr }{dk}\frac{Y_{I}^{*}}{p^{*}}=-\delta (1-\alpha )(\delta +\rho )\left( 1-\frac{dl}{ dk}\frac{k^{*}}{l^{*}}\right) ,&\frac{dr}{dp}=(1-\alpha )(\delta +\rho )\frac{ dl}{dp}\frac{p^{*}}{l^{*}} \end{array} \end{aligned}$$

(4.55)

Linearizing the dynamical system (4.26) around the steady state leads to the following Jacobian matrix

$$\begin{aligned} \mathcal {J}=\left( \begin{array}{cc} -\delta &{} -\frac{1+\Theta }{\Theta }\frac{Y_{I}^{*}}{p^{*}} \\ \,-\,\frac{\frac{\partial r}{\partial k}+\delta \left[ \frac{1}{\epsilon _{cc}}\frac{\partial c}{\partial k}\frac{ p^{*}}{c^{*}}\,-\,\frac{1}{\epsilon _{lc}}\frac{\partial l}{\partial k}\frac{p^{*}}{l^{*}}\right] }{E(k^{*},p^{*})} &{}\quad \frac{\delta +\rho \,-\,\frac{\partial r}{\partial p}\,-\,\frac{1+\Theta }{\Theta } \frac{Y_{I}^{*}}{p^{*}}\left[ \frac{1}{\epsilon _{cc}}\frac{\partial c}{\partial k}\frac{p^{*} }{c^{*}}\,-\,\frac{1}{\epsilon _{lc}}\frac{\partial l}{\partial k}\frac{p^{*}}{l^{*}}\right] }{ E(k^{*},p^{*})} \end{array} \right) \end{aligned}$$

with E(k, p) as given by (4.27). The associated characteristic polynomial is then given by (4.32) with the Determinant and Trace of the Jacobian matrix as defined by (4.33). Using (4.31), (4.54) and (4.55) we finally derive after straightforward simplifications

$$\begin{aligned} \begin{array}{rcl} \mathcal {D}(\gamma ) &{} = &{} \frac{ \delta (\delta +\rho )(1+\chi +\gamma \mathcal {C})[\rho +\delta (1-\alpha )]\left[ \frac{(1\,-\,\alpha )(\gamma +\chi )}{1+\chi }\,-\,\alpha \Theta \right] }{\left[ \alpha +\chi +\gamma \mathcal {C}\left( 2\,-\,\frac{\mathcal {C}(1\,-\,\gamma )}{1+\chi } \right) \right] \left[ \Theta [\rho +\delta (1\,-\,\alpha )]\,-\,\frac{\sigma (1+\chi )\alpha \delta }{1+\chi \,-\,\mathcal {C}(1\,-\,\gamma )}\right] +\frac{\gamma \mathcal {C} \alpha \delta }{1+\chi }\left[ \alpha +\chi +\gamma (1\,-\,\alpha )\right] }\\[16pt] \mathcal {T}(\gamma ) &{} = &{} \frac{\left[ \alpha +\chi +\gamma \mathcal {C}\left( 2-\frac{\mathcal {C}(1\,-\,\gamma )}{1+\chi } \right) \right] \left[ \rho \Theta [\rho +\delta (1\,-\,\alpha )]\,-\,\frac{ \sigma (1+\chi )\alpha \delta }{1+\chi \,-\,\mathcal {C}(1\,-\,\gamma )}\left[ \rho +\Theta (\delta +\rho )\right] \right] +\gamma \mathcal {C}\alpha \delta \left[ \frac{\rho \left[ \alpha +\chi +\gamma (1\,-\,\alpha )\right] }{1+\chi } +\alpha \delta \Theta \right] }{\left[ \alpha +\chi +\gamma \mathcal {C}\left( 2\,-\,\frac{ \mathcal {C}(1\,-\,\gamma )}{1+\chi }\right) \right] \left[ \Theta [\rho +\delta (1\,-\, \alpha )]\,-\,\frac{\sigma (1+\chi )\alpha \delta }{1+\chi \,-\,\mathcal {C}(1\,-\,\gamma )} \right] +\frac{\gamma \mathcal {C}\alpha \delta }{1+\chi }\left[ \alpha +\chi +\gamma (1\,-\,\alpha )\right] } \end{array} \end{aligned}$$

(4.56)

Note that if $\Theta =0$ we conclude under the concavity condition $\sigma \ge \sigma _{c}(\gamma )$ that $\mathcal {D}<0$, and the steady state is always saddle-point stable, i.e. locally determinate.

Assume first that

$$\begin{aligned} \begin{array}{l} \frac{(1\,-\,\alpha )(\gamma +\chi )}{1+\chi }\,-\,\alpha \Theta>0 \text{ or } \text{ equivalently } \chi >\frac{\alpha \Theta \,-\,\gamma (1\,-\,\alpha )}{1\,-\,\alpha \,-\,\alpha \Theta }\equiv \underline{\chi }(\gamma ) \end{array} \end{aligned}$$

(4.57)

To keep reasonable values for the external effect we assume from here that $\Theta <\bar{\Theta }\equiv (1-\alpha )/\alpha$ and thus $\underline{\chi } (\gamma )>0$. Then $\mathcal {D}>0$ if and only if its denominator is positive, namely if and only if

$$\begin{aligned} \begin{array}{l} \sigma <\sigma ^{sup}(\gamma )\equiv \frac{\left[ 1+\chi \,-\,\mathcal {C}(1\,-\,\gamma ) \right] \left\{ \left[ \alpha +\chi +\gamma \mathcal {C}\left( 2\,-\,\frac{\mathcal {C} (1\,-\,\gamma )}{1+\chi }\right) \right] \Theta [\rho +\delta (1\,-\,\alpha )]+\frac{\gamma \mathcal {C}\alpha \delta }{1+\chi }\left[ \alpha +\chi +\gamma (1\,-\,\alpha )\right] \right\} }{(1+\chi )\alpha \delta \left[ \alpha +\chi +\gamma \mathcal {C}\left( 2\,-\, \frac{\mathcal {C}(1\,-\,\gamma )}{1+\chi }\right) \right] } \end{array} \end{aligned}$$

(4.58)

But then local indeterminacy arises if and only if $\mathcal {T}(\gamma )<0$, namely if and only if its numerator is negative, i.e.

$$\begin{aligned} \begin{array}{l} \sigma >\sigma ^{H}(\gamma )\equiv \frac{\left[ 1+\chi \,-\,\mathcal {C}(1\,-\,\gamma ) \right] \left\{ \left[ \alpha +\chi +\gamma \mathcal {C}\left( 2\,-\,\frac{\mathcal {C} (1\,-\,\gamma )}{1+\chi }\right) \right] \rho \Theta [\rho +\delta (1\,-\,\alpha )]+\gamma \mathcal {C}\alpha \delta \left[ \frac{\rho \left[ \alpha +\chi +\gamma (1\,-\,\alpha ) \right] }{1+\chi }+\alpha \delta \Theta \right] \right\} }{(1+\chi )\alpha \delta \left[ \rho +\Theta (\delta +\rho )\right] \left[ \alpha +\chi +\gamma \mathcal {C} \left( 2\,-\,\frac{\mathcal {C}(1\,-\,\gamma )}{1+\chi }\right) \right] } \end{array} \end{aligned}$$

(4.59)

Obvious computations show that $\sigma ^{sup}(\gamma )>\sigma ^{H}(\gamma )$ for any $\gamma \in [0,1]$. We need however to check that $\sigma ^{sup}(\gamma ) >\sigma _{c}(\gamma )$ in order to be able to have a compatibility between the concavity property of the utility function at the steady state $\sigma \ge \sigma _{c}(\gamma )$ and the condition for local indeterminacy $\sigma <\sigma ^{sup}(\gamma )$. Tedious but straightforward computations yield $\sigma ^{sup}(\gamma )>\sigma _{c}(\gamma )$ if and only if

$$\begin{aligned} \begin{array}{l} \Theta >\tilde{\Theta }(\gamma )\equiv \frac{\gamma ^{2}\mathcal {C} \alpha ^{2}\delta \left[ 1\,-\,\frac{(1\,-\,\gamma )\mathcal {C}}{1+\chi }\left( 2\,-\,\frac{\mathcal {C} (1\,-\,\gamma )}{1+\chi }\right) \right] }{\left[ \chi +\gamma \mathcal {C}\left( 2\,-\,\frac{ \mathcal {C}(1\,-\,\gamma )}{1+\chi }\right) \right] \left[ \alpha +\chi +\gamma \mathcal { C}\left( 2\,-\,\frac{\mathcal {C}(1\,-\,\gamma )}{1+\chi }\right) \right] [\rho +\delta (1\,-\,\alpha )]} \end{array} \end{aligned}$$

(4.60)

Under Assumption 1 we have $\tilde{\Theta }^{\prime }(\gamma )>0$ and $\tilde{\Theta }(\gamma )<\bar{\Theta }$ for any $\gamma \in [0,1]$.

Denoting $\sigma ^{inf}(\gamma )=\max \{\sigma ^{H}(\gamma ),\sigma _{c}(\gamma )\}$, we have proved that under condition (4.57), for any given $\gamma \in [0,1]$, local indeterminacy occurs if and only if $\Theta \in (\tilde{\Theta }(\gamma ),\bar{\Theta })$ and $\sigma \in (\sigma ^{inf}(\gamma ),\sigma ^{sup}(\gamma ))$. Obviously, recalling that Lemma 1 shows that the JR utility function is locally concave if and only if $\sigma \ge \sigma _{c}(\gamma )$, we derive from (4.58) and (4.60) that $\mathcal {D}$ is negative and the steady state $(k^{*},p^{*})$ is saddle-point stable in two cases: (i) when $\Theta \in (\tilde{\Theta }(\gamma ),\bar{\Theta })$ and $\sigma >\sigma ^{sup}(\gamma )$, or (ii) when $\Theta <\tilde{\Theta }(\gamma )$ which implies $\sigma (\ge \sigma _{c}(\gamma ))>\sigma ^{sup}(\gamma )$.

Let us consider now the case in which

$$\begin{aligned} \begin{array}{l} \frac{(1-\alpha )(\gamma +\chi )}{1+\chi }\,-\,\alpha \Theta<0 \text{ or } \text{ equivalently } \chi <\frac{\alpha \Theta \,-\,\gamma (1\,-\,\alpha )}{1\,-\,\alpha \,-\,\alpha \Theta }\equiv \underline{\chi }(\gamma ) \end{array} \end{aligned}$$

(4.61)

We need to assume here that $\Theta >\gamma (1\,-\,\alpha )/\alpha$ and thus that $\gamma <1$ to get a compatibility with the assumption $\Theta <\bar{\Theta }$. Following the same argument as previously, we conclude now that local indeterminacy arises if $\sigma >\sigma ^{sup}(\gamma )$ and $\sigma <\sigma ^{H}(\gamma )$. But such a configuration is not possible as $\sigma ^{sup}(\gamma )>\sigma ^{H}(\gamma )$ for any $\gamma \in [0,1]$. It follows that under condition (4.61), the steady state $(k^{*},p^{*})$ is saddle-point stable when $\sigma <\sigma ^{sup}(\gamma )$, totally unstable when $\sigma >\sigma ^{sup}(\gamma )$ and is ruled out.

We conclude therefore that for any given $\gamma \in [0,1]$, local indeterminacy arises if and only if $\chi >\underline{\chi }(\gamma )$, $\Theta \in (\tilde{\Theta }(\gamma ),\bar{\Theta })$ and $\sigma \in (\sigma ^{inf}(\gamma ),\sigma ^{sup}(\gamma ))$.$\square$

1.4 4.8.4 Proof of Corollary 1

Taking into account the concavity condition as given in Lemma 1, the existence of a Hopf bifurcation requires the bound $\sigma ^{H}(\gamma )$ as given in (4.34) to be larger than $\sigma _{c}(\gamma )$. We then get $\sigma ^{H}(\gamma )>\sigma _{c}(\gamma )$ if and only if

$$\begin{aligned} \begin{array}{l} \Theta g(\rho ,\gamma ,\chi )>\gamma ^{2}\mathcal {C}\alpha ^{2}\delta \rho \left[ 1\,-\,\frac{(1\,-\,\gamma )\mathcal {C}}{1+\chi }\left( 2\,-\,\frac{\mathcal {C}(1\,-\,\gamma )}{1+\chi }\right) \right] \end{array} \end{aligned}$$

with

$$\begin{aligned} \begin{array}{rcl} g(\rho ,\gamma ,\chi ) &{} = \left[ \alpha +\chi +\gamma \mathcal {C}\left( 2-\frac{\mathcal {C}(1\,-\,\gamma )}{1+\chi } \right) \right] \left[ \rho [\rho +\delta (1\,-\,\alpha )]\left[ \chi +\gamma \mathcal {C} \left( 2\,-\,\frac{\mathcal {C}(1\,-\,\gamma )}{1+\chi }\right) \right] \right. \\[8pt] \quad &{} - \left. \frac{\gamma \mathcal {C}\alpha \delta (\gamma +\chi )(\delta +\rho )}{1+\chi }\right] + \gamma \mathcal {C}(\alpha \delta )^{2} \left[ \chi +\gamma \mathcal {C}\left( 2-\frac{\mathcal {C}(1-\gamma )}{1+\chi } \right) \right] \end{array} \end{aligned}$$

Under Assumption 1 we have $g(\rho ,\gamma ,\chi )>0$ for any $\gamma \in [0,1]$. It follows that $\sigma ^{H}(\gamma )>\sigma _{c}(\gamma )$ if and only if

$$\begin{aligned} \begin{array}{rcl} \Theta >\hat{\Theta }(\gamma )\equiv \frac{\gamma ^{2}\mathcal {C} \alpha ^{2}\delta \rho \left[ 1\,-\,\frac{(1\,-\,\gamma )\mathcal {C}}{1+\chi }\left( 2\,-\,\frac{\mathcal {C} (1\,-\,\gamma )}{1+\chi }\right) \right] }{g(\rho ,\gamma ,\chi )}&\,&\end{array} \end{aligned}$$

Assumption 1 also implies $\hat{\Theta }^{\prime }(\gamma )>0$ and $\hat{\Theta }(\gamma )<\bar{\Theta }$ for any $\gamma \in [0,1]$. The result follows from Proposition 2 considering $\underline{\Theta } (\gamma )=\max \{\hat{\Theta }(\gamma ),\tilde{\Theta }(\gamma )\}$.$\square$

1.5 4.8.5 Proof of Lemma 2

The maximal value of $\underline{\chi }(\gamma )$ is $\underline{\chi } (0)=\alpha \Theta /(1\,-\,\alpha \,-\,\alpha \Theta )$. We then assume $\chi >\underline{ \chi }(0)$ in order to ensure $\chi >\underline{\chi }(\gamma )$ for any $\gamma \in [0,1]$. Let us consider the following two critical values

$$\begin{aligned} \begin{array}{rcl} \sigma ^{sup}(0) &{} \equiv &{} \frac{\Theta \left\{ \alpha \rho +\chi [ \rho +\delta (1\,-\,\alpha )]\right\} }{(1+\chi )\alpha \delta } \\[6pt] \sigma ^{H}(1) &{} \equiv &{} \frac{\left( \alpha +\chi +2\mathcal {C}\right) \rho \Theta [\rho +\delta (1\,-\,\alpha )]+\mathcal {C}\alpha \delta \left( \rho +\alpha \delta \Theta \right) }{\alpha \delta \left[ \rho +\Theta (\delta +\rho )\right] \left( \alpha +\chi +2\mathcal {C}\right) } \end{array} \end{aligned}$$

(4.62)

We easily get

$$\begin{aligned} \begin{array}{rclclcl} \displaystyle \lim _{\chi \rightarrow +\infty }\sigma ^{sup}(0)= & {} \frac{\Theta [\rho +\delta (1\,-\,\alpha )]}{\alpha \delta }>\displaystyle \lim _{\chi \rightarrow + \infty }\sigma ^{H}(1)= & {} \frac{\rho \Theta [\rho +\delta (1\,-\,\alpha )]}{ \alpha \delta \left[ \rho +\Theta (\delta +\rho )\right] }&\,&\end{array} \end{aligned}$$

(4.63)

Similarly, we have

$$\begin{aligned} \begin{array}{rcl} \sigma ^{sup}(0)\vert _{\chi =\underline{\chi }(0)} &{} = &{} \frac{\Theta [ \rho +\Theta (\delta +\rho )]}{\delta } \\[6pt] \sigma ^{H}(1)\vert _{\chi =\underline{\chi }(0)} &{} \equiv &{} \frac{\rho \Theta [ \rho +\delta (1\,-\,\alpha )]}{\alpha \delta \left[ \rho +\Theta (\delta +\rho )\right] }+ \frac{\mathcal {C}(1\,-\,\alpha \,-\,\alpha \Theta )\left( \rho +\alpha \delta \Theta \right) }{\left[ \rho +\Theta (\delta +\rho )\right] \left[ \alpha (1\,-\,\alpha )(1+\Theta )+2 \mathcal {C}(1\,-\,\alpha \,-\,\alpha \Theta )\right] } \end{array} \end{aligned}$$

It follows obviously that

$$\begin{aligned} \begin{array}{rclclc} \displaystyle \lim _{\Theta \rightarrow 0}\sigma ^{sup}(0)\vert _{\chi =\underline{ \chi }(0)}= & {} 0<\displaystyle \lim _{\Theta \rightarrow 0}\sigma ^{H}(1) \vert _{\chi =\underline{\chi }(0)}\equiv & {} \frac{\mathcal {C}}{\alpha +2 \mathcal {C}}&\end{array} \end{aligned}$$

while

$$\begin{aligned} \begin{array}{rclclc} \displaystyle \lim _{\Theta \rightarrow \bar{\Theta }}\sigma ^{sup}(0)\vert _{\chi = \underline{\chi }(0)}= & {} \frac{(1\,-\,\alpha )\left[ \rho +\delta (1\,-\,\alpha )\right] }{\alpha ^{2}\delta }>\displaystyle \lim _{\Theta \rightarrow \bar{\Theta } }\sigma ^{H}(1)\vert _{\chi =\underline{\chi }(0)}\equiv & {} \frac{ (1\,-\,\alpha )\rho }{\alpha \delta }&\end{array} \end{aligned}$$

Therefore, there exists $\bar{\bar{\Theta }}\in (0,\bar{\Theta })$ such that if $\Theta \in (0,\bar{\bar{\Theta }})$, then $\sigma ^{sup}(0)\vert _{\chi = \underline{\chi }(0)}<\sigma ^{H}(1)\vert _{\chi =\underline{\chi }(0)}$. Based on this result and using (4.63), we conclude that there also exists $\bar{\chi }\in (\underline{\chi }(0), +\infty )$ such that when $\Theta \in (0,\bar{\bar{\Theta }})$, $\sigma ^{sup}(0)\,-\,\sigma ^{H}(1)\lessgtr 0$ if and only if $\chi \lessgtr \bar{\chi }$.$\square$