On impatience, temptation and Ramsey’s conjecture

Abstract

This article explores the implications of the introduction of self-control costs and temptation motives in intertemporal preferences in an elementary competitive equilibrium. We let heterogeneous agents differ in both their discount parameters and their temptation motives, and the degree of financial-market imperfections vary, and establish their implications for the long-run value of the capital stock and the underlying long-run distribution of consumption and wealth. The results differ from those obtained in a standard Ramsey benchmark model, in that long-run distributions are commonly non-degenerate.

Appendices

Appendix 1: Proof of Proposition 1

The satisfaction of $R= D _{K}F(K,{1})+{1}-\eta$ and $w= D_{{L}}F(K,{1})/n$ directly results from Eqs. (3h) and (3i) when they are considered at the long-run stationary competitive equilibrium. In parallel, the consideration of (3g) allows us to derive $h=w/(R-{1})$, whereas (3d) will hold in a stationary state. Equations (3a), (3b), (3c), (3e) and (3f) then allow us to obtain Propositions 1(i) and 1(ii) through the following statement:

Claim

Consider the optimal solution:

• (i) a stationary optimal solution for an unconstrained individual corresponds to stationary values $c^{i}$ and $x^{i}$ such that:

  $$\begin{aligned} R\delta _{i}&=\dfrac{ D u(c^{i})+\gamma _{i} D v(c^{i})}{ D u(c^{i})+\gamma _{i}\bigl ( D v(c^{i})- D v(x^{i}+\lambda h)\bigr )}, \end{aligned}$$

  where

  • a/ $x^{i}=(Rc^{i}-w)/(R-{1})$;

  • b/ the satisfaction of $c^{i}<x^{i}+\lambda h$ boils down to $c^{i}>w({1}-\lambda )$ or $\gamma _{i} D v[({ 1}-\lambda )w]<\bigl (\delta _{i}R-{1}\bigr ) D u[({1}-\lambda )w]$;

  • c/ the existence of such a solution requires $R\delta _{i}>{1}$ and implies $R>{1}$;

• (ii) a stationary optimal solution for a constrained individual corresponds to stationary values $c^{i}$ and $x^{i}$ such that

  • a/ $c^{i}=({1}-\lambda )w$;

  • b/ $x^{i}=[R({1}-\lambda )w-w]/({R-{1}})$;

  • c/ the existence of such a solution requires $\nu ^{i}>{0}$, a condition that can be restated as: $\gamma _{i} D v(({1}-\lambda )w)>(\delta _{i} R-{1}) D u(({ 1}-\lambda )w)$.

Proof

• (i) For an unconstrained individual, $\nu _{t}^{i}={0}$ and from (3b) and (3c):

  $$\begin{aligned} R_{t+{1}}\delta _{i}\bigl [ D u\bigl (c_{t+ { 1}}^{i}\bigr )+\gamma _{i} D v\bigl (c_{t+ { 1}}^{i}\bigr )-\gamma _{i} D v\bigl (x_{t+{1}}^{i}+\lambda h_{t+2}\bigr )\bigr ]= D u\bigl (c_{t}^{i}\bigr )+\gamma _{i} D v\bigl (c_{t}^{i}\bigr ). \end{aligned}$$

  (*)

  Rearranging and considering stationary trajectories in (*), we obtain the conditions of the statement.

• (ii) The basic form of the condition $\nu _{i}>{0}$ is:

  $$\begin{aligned} \nu ^{i}= D u(c^{i})+\gamma _{i} D v(c^{i})-\delta _{i}\mu ^{i} R>{ 0}, \end{aligned}$$

  for $\mu ^{i}= D u(c^{i})+\gamma _{i} D v(c^{i})-\gamma _{i} D v(x^{i}+\lambda h), c^{i}=x^{i}+\lambda h$, $\mu ^{i}= D u(c^{i})$, that yields, merging and rearranging, the above statement.

Equation (4a) then corresponds to the resource constraints of the economy. It is derived by combining Eqs. (3j), (3h), (3i) and (3a) when considered at the stationary state. Equation (3j) allows us to state $\sum _{i= {1}}^{n}x^{i}=RK +nw$ while, from Eq. (3a), $\sum _{i= {1}}^{n}x^{i}(R-{1})=R\sum _{i={1}}^{n}c^{i}-nw$. Finally, eliminating $\sum _{i={1}}^ {n}x^{i}$ between these equations yields:

$$\begin{aligned} (R-{1})K+nw=\sum _{i={1}}^{n}c^{i}. \end{aligned}$$

Making use of (3h) and (3i), we obtain Eq. (4a). $\square$

Appendix 2: Proof of Lemma 1

(i) From (5), the equation defining stationary competitive equilibrium consumption is:

$$\begin{aligned} D u(c^{i})+\gamma _{i} D v(c^{i})=\dfrac{\delta _{i} R}{\delta _{i} R-{1}}\gamma _{i} D v\biggl [\dfrac{Rc^{i}-({1}-\lambda )w}{R-{1}}\biggr ], \end{aligned}$$

We then require a solution to the above equation such that $c>({ 1}-\lambda )w$. It is worth noting that this can conveniently be restated as ${H}^{\ell }(c^{i}, \gamma _{i})={H}^{r}(c^{i}, \delta _{i}, \gamma _{i}, R,w,\lambda ),$ where the simultaneous holding of $D _{c^{i}}{H}^{\ell }(c^{i}, \gamma _{i})<{0}$ and $D_{c^{i}}{H}^{r}(c^{i},\delta _{i}, \gamma _{i}, R,w,\lambda )>{0}$ first ensures that there exists at most one solution. Turning to existence, and for $c^{i}\rightarrow ({1}-\lambda )w$, we have:

$$\begin{aligned}&{H}^{\ell }(c^{i},\gamma _{i})\rightarrow D u[({1}-\lambda )w]+\gamma _{i} D v[({1}-\lambda )w],\\&{H}^{r}(c^{i}, \delta _{i}, \gamma _{i}, R,w)\rightarrow \dfrac{\delta _{i} R}{\delta _{i} R-{1}}\gamma _{i} D v[({1}-\lambda )w]. \end{aligned}$$

From the previous assumption, $\gamma _{i} D v[({ 1}-\lambda )w]<(\delta _{i} R-{1}) D u[({1}-\lambda )w]$, and hence

$$\begin{aligned} {H}^{\ell }(({1}-\lambda )w, \gamma _{i})>{H}^{r}(({1}-\lambda )w, \delta _{i}, \gamma _{i}, R,w,\lambda ). \end{aligned}$$

Conversely, and for $c^{i}\rightarrow +\infty$, whereas $D u(c^{i})\rightarrow { 0}, [Rc^{i}-({1}-\lambda )w]/(R-{1})>c^{i}$ as $c^{i}>({ 1}-\lambda )w$. This gives us

$$\begin{aligned} {H}^{\ell }(+\infty , \gamma _{i})<{H}^{r}(+\infty , \delta _{i}, \gamma _{i}, R,w,\lambda ), \end{aligned}$$

so that we have the existence and uniqueness results for $c^{i}$.

(ii) Looking then at comparative static properties, it first emerges that $D _{{ R}}C(\delta _{i}, \gamma _{i}, R, w, \lambda )>{0}$ as $D _{{ R}}{H}^{r}(c^{i}, \delta _{i}, \gamma _{i}, R,w)<{0}$. We have

$$\begin{aligned} \text{ Sgn }\biggl [ D _{{ R}}\biggl (\dfrac{Rc^{i}-({1}-\lambda )w}{R-{1}} \biggr )\biggr ]&=\text{ Sgn }[c^{i}(R-{1})-Rc^{i} + ({1}-\lambda )w]\\&=\text{ Sgn }[({1}-\lambda )w-c^{i}]\\&<{0}. \end{aligned}$$

Establishing that $D _{w}C(\delta _{i}, \gamma _{i}, R, w, \lambda )\geqslant {0}$ then comes from $D _{w}{H}^{r}(c^{i}, \delta _{i}, \gamma _{i}, R,w,\lambda )\,\leqslant \,0$, noting that, for $\lambda ={1}$, this simplifies to $D _{w}C(\delta _{i}, \gamma _{i}, R, w, \lambda )= 0$. In the same way, we can easily show that $D _{\delta _{i}}C(\delta _{i}, \gamma _{i}, R, w, \lambda )>{ 0}$ and $D _{\lambda }C(\delta _{i}, \gamma _{i}, R, w, \lambda )<{0}$ are direct by-products of $D _{\delta _{i}}{H}^{r}(c^{i}, \delta _{i}, \gamma _{i}, R,w,\lambda )< 0$ and $D _{\lambda }{H}^{r}(c^{i}, \delta _{i}, \gamma _{i}, R,w,\lambda )\geqslant {0}$. Finally, showing that $D _{\gamma _{i}}C(\delta _{i}, \gamma _{i}, R, w, \lambda )<{0}$ proceeds from the restatement of the defining equation of the stationary values for $c^{i}$ as:

$$\begin{aligned} \dfrac{R\delta _{i}}{R\delta _{i}-{1}}&=\dfrac{ D u(c^{i})+\gamma _{i} D v(c^{i})}{\gamma _{i} D v[(Rc^{i}-({ 1}-\lambda )w)/(R-{1})]}\equiv {G}(c^{i}, \gamma _{i}, w, R). \end{aligned}$$

(**)

As the function ${G}(c^{i}, \gamma _{i}, w, R)$ is decreasing in both $\gamma _{i}$ and $c^{i}$, we have $D _{\gamma _{i}}C(\delta _{i}, \gamma _{i}, R, w, \lambda )<{0}$.

The expression $C(\delta _{i}, \gamma _{i}, R,w,\lambda )$ being monotone increasing as a function of R, it has a limit that is either finite or infinite. First assuming that $\ell <+\infty$ and from equation (**), we have that:

$$\begin{aligned} 1=\dfrac{ D u(\ell )+\gamma _{i} D v(\ell )}{\gamma _{i} D v(\ell )}, \end{aligned}$$

which is not possible for $\ell <+\infty$, so that $\ell =+\infty$.

Appendix 3: Proof of Proposition 2

(i)–(ii) In order to save on notation, we consider the function $c^{i}(K)$ as defined by:

$$\begin{aligned} {\left\{ \begin{array}{ll} c^{i}(K)=\mathscr {C}\bigl (\delta _{i}, \gamma _{i},K,{1}\bigr )&{}\quad \text{ for }\quad K\in \left]{0},\widetilde{K}_{i}\right[;\\ c^{i}(K)={0}&{}\quad \text{ for }\quad K\geqslant \widetilde{K}_{i}. \end{array}\right. } \end{aligned}$$

The existence of an equilibrium is associated with a value of K such that:

$$\begin{aligned} F(K,{1})-\eta K=\sum _{i={1}}^{n}c^{i}(K). \end{aligned}$$

Denoting this equation by ${H}^{\ell }\left( K\right) ={H}^{r}\left( K\right)$, as ${1}/\delta _{{1}}>{1}$, we have that $\widetilde{K}_{{ 1}}<\widehat{K}$. The function ${H}^{\ell }(K)$ is then increasing over $]{0},\widetilde{K}_{{1}}[$. On the contrary, the function ${H}^{r}(K)$ is decreasing over $]{0},\widetilde{K}_{ {1}}[$, with ${H}^{r}(\widetilde{K}_{{1}})={0}$ and $\lim _{{K}\rightarrow {0}}{H}^{r}(K)=+\infty$. These properties ensure the existence and uniqueness of $K\in \,]{0}, \widetilde{K}_{{ 1}}[$ such that ${H}^{\ell }(K)={H}^{r}(K)$.

Appendix 4: Proof of Lemma 2

(i)–(ii) Uniqueness is established by noting that the L.H.S. of (7) is non-decreasing as a function of K, while the R.H.S. falls as a function of K.

The existence argument in turn results from noting that, for $K\rightarrow \widetilde{K}_{i}$,

$$\begin{aligned}&\gamma _{i} D v\left[ ({1}-\lambda ) D _{ {L}}F(K,{1})/n\right] \rightarrow \gamma _{i} D v\left[ D _{ {L}}F(\widetilde{K}_{i},{1})/n\right] >{0},\\&\Bigl \{\delta _{i}\bigl [ D _{K}F(K,{1})+({1}-\eta )\bigr ]-{ 1}\Bigr \} D u\left[ ({1}-\lambda ) D _{ {L}}F(K,{1})/n\right] \rightarrow {0}. \end{aligned}$$

At the same time, and for $K\rightarrow {0}$,

$$\begin{aligned}&\gamma _{i} D v\left[ ({1}-\lambda ) D _{ {L}}F(K,{1})/n\right] \rightarrow \gamma _{i} D v\left[ ({1}-\lambda ) D _{ {L}}F({0},{1})/n\right] ,\\&\Bigl \{\delta _{i}\bigl [ D _{K}F(K,{1})+({1}-\eta )\bigr ]-{ 1}\Bigr \} D u\left[ ({1}-\lambda ) D _{ {L}}F(K,{1})/n\right] \rightarrow +\infty . \end{aligned}$$

This gives us the existence result.

Appendix 5: Proof of Lemma 3

• (i) By definition, $\xi ^{i}(K)$ is a solution to:

  $$\begin{aligned}&\dfrac{\delta _{i}R(K)\gamma _{i}}{\delta _{i}R(K)-{1}} =\dfrac{ D u\bigl (\xi ^{i}w(K)\bigr )+\gamma _{i} D v\bigl (\xi ^{i}w(K)\bigr )}{ D v\bigl (\bigl (R(K)\xi ^{i}-({1}-\lambda )\bigr )w(K)/(R(K)-{ 1})\bigr )} \end{aligned}$$

  The L.H.S. of the above equation increases as a function of K, while the R.H.S. may conveniently be reformulated as ${H}^{u}(K, \xi ^{i})/{H}^{\ell }(K,\xi ^{i})$. We then see that ${H}^{u}$ falls with K while ${H}^{\ell }$ increases with both K and $\xi ^{i}, \left[ R\xi _{i}-({1}-\lambda )\right] /\left( R-{ 1}\right)$ being a decreasing function of R, which yields the monotonicity property for $\xi ^{i}$.

• (ii) As the function $\xi ^{i}(K)$ is decreasing monotone in K, it has a limit at zero that is either $\ell >{0}$ or $+\infty$. Assuming first that this is finite and $\lim _{{K}\rightarrow {0}}\xi ^{i}(K)=\ell$, the above equation yields:

  $$\begin{aligned} \gamma _{i}=\dfrac{ D u\bigl (w({0})\ell \bigr )+\gamma _{i} D v\bigl (w({0})\ell \bigr )}{ D v\bigl (w({0})\ell \bigr )}, \end{aligned}$$

  which is impossible (a contradiction), so that $\ell =+\infty$.

Appendix 6: Proof of Proposition 3

(i) For a constrained individual i, it is convenient to extend the function $\xi ^{i}(K)$ over the interval $[\overline{K}_{i},+\infty [$ with $\xi ^{i}(K)={1}-\lambda$. The existence of an equilibrium is hence ensured by a value of K such that:

$$\begin{aligned} F(K,{1})-\eta K=\sum _{i={1}}^{n}\xi ^{i}(K)\dfrac{ D _{ {L}}F(K,{1})}{n}, \end{aligned}$$

hence, making use of the Euler relationship,

$$\begin{aligned} \dfrac{ D _{K}F(K,{1})K -\eta K}{ D _{ {L}}F(K,{ 1})}=\dfrac{1}{n}\sum _{i={1}}^{n}\bigl [\xi ^{i}(K)-{1}\bigr ]. \end{aligned}$$

or ${H}^{\ell }(K)={H}^{r}(K)$. First, and for $K\rightarrow {0}, {H}^{\ell }(K)<{H}^{r}(K)$. While ${H}^{\ell }(K)$ is bounded under Assumption T.2, we have that ${H}^{r}(K)\rightarrow \infty$ as a direct corollary of Lemma 3(ii).

Now, for K such that:

$$\begin{aligned} K=\overline{K}_{{1}}>\overline{K}_{{ 2}}>\cdots >\overline{K}_{n}, \end{aligned}$$

$\xi ^{i}(K)={1}-\lambda$ for any $i={1}, \ldots ,n$ and ${H}^{r}(\overline{K}_{{1}})=-\lambda$. Further, $D _{K}F(\overline{K}_{{1}},{1})\overline{K}_{{1}}-\eta \overline{K}_{{1}}>{0}$, since $\overline{K}_{{ 1}}<\widehat{K}$, for $\widehat{K}$ that satisfies $D _{K}F(\widehat{K},{1})=\eta$. This implies that ${H}^{\ell }(\overline{K}_{{1}})>{H}^{r}(\overline{K}_{{ 1}})$. The existence of a $K\in \,]{0}, \overline{K}_{{ 1}}[$ such that ${H}^{\ell }(K_{{1}})={H}^{r}(K_{{1}})$ is then established.

Appendix 7: Proof of Proposition 4

• (i) This is straightforward as for $i>\xi , c^{i}=0$ and the parameter $\gamma _{i}$ has no impact on the condition $\delta _{i}R\leqslant 1$, that eventually implies $c^{i}=0$.

• (ii) Consider the equation $\sum _{i=1}^{n}\mathscr {C}(\delta _{i}, \gamma _{i}, K,1)=F(K,1)-\eta K$, that determines the value of K. $\mathscr {C}$ is a decreasing function of K when $F(K,1)-\eta K$ is increasing. The result follows from the features of $\mathscr {C}$, that increases with $\delta _{i}$ but decreases with $\gamma _{i}$.

Appendix 8: Proof of Proposition 5

By definition, K is the solution to the following equation:

$$\begin{aligned} F(K, {1})-\eta K=\sum _{i={1}}^{\zeta }c^{i}(K)+(n-\zeta )({ 1}-\lambda )w(K), \end{aligned}$$

or ${H}^{\ell }(K)={H}^{r}(K,\gamma _{i}, \delta _{i}, \lambda )$, for $i={1},{2},\ldots , k$. The uniqueness of the stationary competitive equilibrium ensures that

$$\begin{aligned} D _{K}{H}^{\ell }\bigl (K\bigr )- D _{K}{H}^{r}\bigl (K, \gamma _{i}, \delta _{i}, \lambda \bigr )>{0}. \end{aligned}$$

Noting then that ${H}^{r}(K,\gamma _{i}, \delta _{i}, \lambda )$ is independent of $\gamma _{i}$ and $\delta _{i}$ for any $i\leqslant \zeta$ establishes (i).

Result (ii) follows from $D _{\delta _{i}}C(\delta _{i}, \gamma _{i}, R, w, \lambda )>{0}, D _{\gamma _{i}}C(\delta _{i}, \gamma _{i}, R, w, \lambda )<{0}$ from Lemma 1.

Result (iii) in turn follows from $D _{\lambda }C(\delta _{i}, \gamma _{i}, R, w, \lambda )\leqslant {0}$ from Lemma 1 and $D_{\lambda }(1-\lambda )w<0$.

Appendix 9: Proof of Corollary 1

The argument of the proof replicates that in Proposition 5(iii), adding the property that, with $D^{2}v>0, D_{\lambda }C(\delta , \gamma , R, w,\lambda )<0$.

Appendix 10: Proof of Lemma 4

The consumer’s program is given by:

$$\begin{aligned} \max _{\{c_{t}^{i}\}}&\sum _{t={0}}^{+\infty } \bigl (\delta _{i}\bigr )^{t}\Bigl \{u\bigl (c_{t}^{i}\bigr )+\gamma _{i}\bigl [v\bigl (c_{t}^{i}\bigr )-v\bigl (c_{t}^{i*}\bigr )\bigr ]\Bigr \}\\ {s.t}\quad&a_{t+{1}}^{i}= R_{t}a_{t}^{i}+w_{t}-c_{t}^{i},\\&\lim _{t\rightarrow +\infty }a_{t}^{i}\Bigm /\prod _{\tau ={0}}^{t}R_{\tau }\geqslant {0},\\&c_{t}^{i}\leqslant c_{t}^{i*}, \quad a_{{0}}^{i}\ \text{ given }. \end{aligned}$$

The optimality conditions are obtained by first listing the Lagrangian:

$$\begin{aligned} {L}_{t}&=\bigl (\delta _{i}\bigr )^{t}\Bigl \{u\bigl (c_{t}^{i}\bigr ) +\gamma _{i}\bigl [v\bigl (c_{t}^{i}\bigr )-v\bigl (c_{t}^{i*}\bigr )\bigr ]\Bigr \} +\mu _{t+{1}}^{i}\bigl [R_{t}a_{t}^{i}+w_{t}-c_{t}^{i}\bigr ]\\&\quad -\mu _{t}^{i}a_{t}^{i}+\nu _{t}^{i}\bigl (c_{t}^{i*}-c_{t}^{i}\bigr ), \end{aligned}$$

From which:

$$\begin{aligned}&\bigl (\delta _{i}\bigr )^{t}\bigl [ D u\bigl (c_{t}^{i}\bigr )+\gamma _{i} D v\bigl (c_{t}^{i}\bigr )\bigr ]=\mu _{t+{1}}^{i}+\nu _{t}^{i},\\&\mu _{t+{1}}^{i}=\mu _{t}^{i}/R_{t},\\&\lim _{t\rightarrow +\infty }\mu _{t}^{i}a_{t}^{i}={0}. \end{aligned}$$

At the same time, the optimal behavior of the competitive firm yields the equilibrium values of real wages and the rental rate for the capital stock as $w_{t}= D _{ {L}}F(K_{t},{1})/n$ and $R_{t}= D _{K}F(K_{t},{1})+{1}-\eta$. The equilibrium on the capital market giving $\sum _{i={1}}^{n}a_{t}^{i}=K_{t}$ for any $t\geqslant {0}$ produces the characterization.

Appendix 11: Proof of Proposition 6

The argument proceeds by proving that the sequence $\{c_{t}^{i*}, K_{t}^{*}\}, i={1},\ldots , n$, satisfies the whole set of conditions that characterize a competitive equilibrium with constraints upon consumption for appropriately chosen values of $\mu _{t}^{i}$ and $\nu _{t}^{i}$ and for a suitable distribution of initial wealth $a_{{0}}^{i}$. Let $\mu _{t+ { 1}}^{i}=\chi _{t+{1}}/\zeta _{i}$ and $\nu _{t}^{i}=(\delta _{i})^{t}\gamma _{i} D v(c_{t}^{i})$. Defining then the prices of the factors as $w_{t}^{*}= D _{ {L}}F(K_{t}^{*},{1})/n$ and $R_{t}^{*}= D _{K}F(K_{t}^{*},{ 1})+{1}-\eta$, the initial wealth of agents is selected so as to satisfy:

$$\begin{aligned} a_{{0}}^{i}=\sum _{t= { 0}}^{+\infty }\dfrac{c_{t}^{i*}}{\prod _{\tau = { 0}}^{t}R_{\tau }^{*}}-\sum _{t= { 0}}^{+\infty }\dfrac{w_{t}^{*}}{\prod _{\tau = {0}}^{t}R_{\tau }^{*}}, \end{aligned}$$

which is possible since

$$\begin{aligned} \sum _{i={1}}^{n}a_{{0}}^{i}&=\sum _{t={0}}^{+\infty }\dfrac{\sum _{i={1}}^{n}c_{t}^{i*}-nw_{t}^{*}}{\Pi _{\tau = {0}}^{t}R_{\tau }^{*}}\\&=\sum _{t={0}}^{+\infty }\dfrac{F\left( K_{t}^{*},{1}\right) -K_{t+{1}}^{*}+({1}-\eta )K_{t}^{*}-F_{ {L}}\left( K_{t}^{*},{1}\right) }{\Pi _{\tau = {0}}^{t}R_{\tau }^{*}}\\&=\sum _{t={0}}^{+\infty }\dfrac{R_{t}^{*}K_{t}^{*}-K_{t+{1}}^{*}}{\Pi _{\tau = {0}}^{t}R_{\tau }^{*}}\\&=K_{{0}}^{*}. \end{aligned}$$

The sequence $\bigl \{a_{t}^{i}\bigr \}$ is then recursively defined by

$$\begin{aligned} a_{t+{1}}^{i}= R_{t}a_{t}^{i}+w_{t}-c_{t}^{i*}. \end{aligned}$$

Finally, and by the definition of $a_{{0}}^{i}$, we have:

$$\begin{aligned} \lim _{t\rightarrow +\infty }a_{t}^{i}\Bigm /\prod _{\tau ={0}}^{t}R_{\tau }={0},\\ \end{aligned}$$

which establishes Proposition 6.