Appendix
The time subscript has been removed in the proofs
Proof of Proposition 1
The individuals of the majority win the vote on taxes. The manager of the administration decides on the level efficiency and the size of the administration (e,α) with the objective of maximizing the total payoff of the bureaucracy: \(\tau \text {Hqe}\alpha (1-\alpha )-\alpha \frac {\beta e^{2}}{2}\), subject to (1 − q) ≥ α. Let us denote this function as \( f=\tau \text {Hqe}\alpha (1-\alpha )-\alpha \frac {\beta e^{2}}{2}\). The first-order conditions for an interior solution are follows:
$$\begin{array}{@{}rcl@{}} f_{e} &=&0\rightarrow \tau qH\left( 1-\alpha^{\ast }\right) \alpha^{\ast}-\alpha^{\ast}\beta e^{\ast }= 0\rightarrow e^{\ast}=\tau \tfrac{qH\left( 1-\alpha^{\ast }\right)}{\beta}, \\ f_{\alpha} &=&0\rightarrow \tau qHe\left( 1-2\alpha^{\ast}\right) -\frac{\beta e^{\ast}}{2}= 0\rightarrow \alpha^{\ast}=\tfrac{\tau 2qH-\beta e^{\ast}}{4\tau H\beta}. \end{array} $$
Solving for the optimal levels (e∗,α∗), we obtain the following:
$$\begin{array}{@{}rcl@{}} e^{\ast}(\tau ) &=&\tau \frac{2qH}{3\beta}, \end{array} $$
(17)
$$\begin{array}{@{}rcl@{}} \alpha^{\ast} &=&\frac{1}{3}. \end{array} $$
(18)
And, it is easy to check that it is a maximum since: fee = −αβ < 0, fαα = − 2τHqe∗ < 0. Furthermore, fee × fαα > (feα)2 since
$$\begin{array}{@{}rcl@{}} 2\alpha^{\ast }\beta \tau {Hqe}^{\ast} &>&\left( \tau {Hq}-2\tau {Hq}\alpha^{\ast }-\beta e^{\ast }\right)^{2}\\ &&\rightarrow \frac{8}{9}\left( \tau {Hq}\right)^{2} >\frac{5}{9}\left( \tau {Hq}\right)^{2}. \end{array} $$
If 1 − q < 1/3, then the maximum is in the boundary of the set of feasible solutions, α∗ = 1 − q and the effort level is \(e^{\ast }(\tau )=\tau \frac {qH^{2}}{\beta }\). Substituting the tax rate (τ = 1 or τ = 1/2), we obtain the administration efficiency in each case. The payoffs for both types of agents are determined by equations (1) and (3). There are two possible equilibria within a generation:
- 1.
Equilibrium if non-entrepreneurs are the majority, q < 1/2, characterized by the following:
$$\begin{array}{@{}rcl@{}} \tau &=&1,\\ e&=&\frac{2qH}{3\beta}, \\ y_{C}&=&\frac{2q^{2}H^{2}}{9\beta},\\ y_{E}&=&0. \end{array} $$
- 2.
Equilibrium if entrepreneurs are the majority, q ≥ 1/2, characterized by the following:
$$\begin{array}{@{}rcl@{}} \tau &=&1/2,\\ e&=&\left\{ \begin{array}{cc} \frac{qH}{3\beta} & \text{if } q\leq 2/3 \\ \frac{qH^{2}}{2\beta} & \text{if } q>2/3, \end{array} \right.\\ y_{C}&=&\left\{ \begin{array}{cc} \frac{q^{2}H^{2}}{18\beta} & \textit{if }q\leq 2/3 \\ \frac{q^{4}H^{2}}{8\beta} & \textit{if }q>2/3, \end{array} \right.\\ y_{E}&=&\left\{ \begin{array}{cc} \frac{qH^{2}}{27\beta} & \textit{if } q\leq 2/3\\ \frac{q^{3}(1-q)H^{2}}{4\beta} & \textit{if } q>2/3. \end{array} \right. \end{array} $$
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Proof of the optimal socialization effort choice of parents (subsection 4.2)
Parents of type i choose the socialization effort \({d_{t}^{i}}\) at generation t that maximizes the following:
$$\underset{{d_{t}^{i}}}{\max}\{P_{t}^{\text{ii}}V^{\text{ii}}+P_{t}^{\text{ij}}V^{\text{ij}}-c({d_{t}^{i}})\}. $$
The first order condition is as follows :
$$\frac{\partial P_{t}^{\text{ii}}}{\partial {d_{t}^{i}}}V^{\text{ii}}+\frac{\partial P_{t}^{\text{ij}}}{\partial {d_{t}^{i}}}V^{ij}=k{d_{t}^{i}}. $$
In order to guarantee \({d_{t}^{i}}\in [0,1)\), a sufficient condition is k > max{ΔVE,ΔVN}, i.e., the marginal cost of effort 1 is greater than the value of cultural assimilation. In that case, the marginal cost for parents to ensure that their child acquires the same preferences as their own is too high.
Differentiating the transition probabilities in Eq. 8 and substituting in the first order condition, we obtain the parents’ socialization efforts as follows:
$$\begin{array}{@{}rcl@{}} d_{t}^{\ast E} &=&\frac{1}{k}\left( V^{\text{EE}}-V^{\text{EN}}\right) (1-q_{t}) \\ d_{t}^{\ast N} &=&\frac{1}{k}\left( V^{\text{NN}}-V^{\text{NE}}\right) q_{t}. \end{array} $$
Hereafter, to simplify the notation, we do not include this auxiliary parameter k, although it should be remembered that it is implicit in the formulas. When \({d_{t}^{E}}-{d_{t}^{N}}\) is written, one should actually read \( k\left ({d_{t}^{E}}-{d_{t}^{N}}\right )\).
Proof of Proposition 2:
Equilibrium in the long run when non-entrepreneurs are majority.
In a non-entrepreneurial economy (q < 1/2), where the majority of non-entrepreneurs vote for confiscatory taxes τ = 1, the values of cultural assimilation are follows:
$$\begin{array}{@{}rcl@{}} {\Delta} V^{E} &=&\gamma -y^{N}\!\rightarrow\! {\Delta} V^{E}=\gamma -\left[ \alpha y^{C}+(1-\alpha )R\right]\\ {\Delta} V^{N} &=&\gamma +y^{N}\!\rightarrow\! {\Delta} V^{N}=\gamma +\left[ \alpha y^{C}+(1-\alpha )R\right] . \end{array} $$
The difference between the socialization efforts of the two types of parents is as follows:
$$\begin{array}{@{}rcl@{}} d^{E}-d^{N} &=&(1-q){\Delta} V^{E}-q{\Delta} V^{N}\\ &=&{\Delta} V^{E}-q\left( {\Delta} V^{E}+{\Delta} V^{N}\right) = \\ &=&\gamma (1-2q)-y_{N}. \end{array} $$
If q < 1/2, the payoffs are \(y_{C}=\frac {2q^{2}H^{2}}{9\alpha \beta }\), yE = 0 (see proof of Proposition 1), therefore
$$d^{E}-d^{N}=\gamma (1-2q)-\left[ \frac{2q^{2}H^{2}}{27(1-q)\beta }+\frac{2-3q }{3(1-q)}R\right] . $$
We have to prove that qc effectively exists and that is unique. If q < 1/2, we have the following:
$$d^{E}-d^{N}=\underset{<0}{\underbrace{-\frac{q^{2}H^{2}}{2(1-q)^{2}\alpha \beta}- \frac{2-3q}{3(1-q)}R}} + \underset{>0}{\underbrace{\gamma (1-2q)}} \\ $$
and substituting for q = 1/2 and q = 0, the difference between socialization efforts results:
$$\begin{array}{@{}rcl@{}} \left. d^{E}-d^{N}\right\vert_{q = 1/2} &=&-\frac{2H^{2}}{54\beta }-\frac{5}{ 6}R<0 \\ \left. d^{E}-d^{N}\right\vert_{q = 0} &=&\gamma -\frac{2}{3}R>0 \text{under Assumption 1} \end{array} $$
As dE − dN is continuous in [0, 1/2], ∃qc ∈ (0, 1/2) : dE − dN = 0, and therefore at this point Δqt = 0.
In addition, \(\frac {\partial \left (d^{E}-d^{N}\right )}{\partial q}=-\frac {2H^{2}}{27\beta }\frac {2(1-q)}{{~}^{(1-q)2}}-2\gamma -\frac {R}{3^{(1-q)2}} <0,\forall q^{c}\in [0,1/2],\) therefore dE − dN is decreasing in [0, 1/2]. Therefore, this qc ∈ (0, 1/2) is unique. □
Proof of Proposition 2:
Equilibrium in the long run when entrepreneurs are majority, q ≥ 1/2.
In an entrepreneurial economy (q ≥ 1/2) where the majority of entrepreneurs vote for non-confiscastory taxes τ = 1/2. The values of cultural assimilation if qt < 2/3 are now given by the following:
$$\begin{array}{@{}rcl@{}} {\Delta} V^{E} &=&\gamma +\left[y^{E}-y^{N}\right] \rightarrow {\Delta} V^{E}\\&=&\gamma -\left( 1-\frac{1}{3(1-q)}\right) R+\left[ y^{E}-\frac{1}{3(1-q) }y^{C}\right] ,\\ {\Delta} V^{N} &=&\gamma -\left[ y^{E}-y^{N}\right] \rightarrow {\Delta} V^{N}\\&=&\gamma +\left( 1-\frac{1}{3(1-q)}\right) R-\left[ y^{E}-\frac{1}{3(1-q) }y^{C}\right]. \end{array} $$
The differences between socialization effort, considering (14) and (15) are follows:
$$\begin{array}{@{}rcl@{}} d^{E}-d^{N} &=&(1-q){\Delta} V^{E}-q{\Delta} V^{N}\\ &=&{\Delta} V^{E}-q\left( {\Delta} V^{E}+{\Delta} V^{N}\right) \\ &=&y_{E}-\frac{1}{3(1-q)}y_{C}-\left( 1-\frac{1}{3(1-q)}\right) R\\ &&+\gamma (1-2q). \end{array} $$
If q > 2/3 then (1 − q) < α∗ = 1/3 and all non-entrepreneurs become civil servants and yN = yC. Therefore,
$$d^{E}-d^{N}=\gamma (1-2q)+\frac{q^{3}H^{2}}{8\beta }\left( 2-3q\right), $$
which is negative for all q > 2/3.
However, if q < 2/3 then
$$\begin{array}{@{}rcl@{}} d^{E}-d^{N} &=&\frac{qH^{2}}{27\beta }-\frac{1}{3(1-q)}\frac{q^{2}H^{2}}{ 18\beta }\\&-&\left( \frac{2-3q}{3(1-q)}\right) R+\gamma (1-2q)\\ &=&\frac{2-3q}{54(1-q)\beta }qH^{2}-\frac{2-3q}{3(1-q)}R+\gamma (1-2q) \end{array} $$
We have to prove that qe effectively exists and that is unique. We have that the first term is positive and the second is negative if q < 2/3.
$$d^{E}-d^{N}= \underset{>0}{\underbrace{\frac{2-3q}{54(1-q)\beta }qH^{2}}}\quad \underset{<0}{\underbrace{-\frac{2-3q}{3(1-q)}R+\gamma (1-2q)}} $$
Substituting for q = 1/2 and q = 2/3 in this expression, we have the following:
$$\begin{array}{@{}rcl@{}} \left. d^{E}-d^{N}\right\vert_{q = 1/2} &=&-\frac{1}{3}\left( R-\frac{H^{2}}{ 36\beta }\right) \\&>&0\text{if}H\geq H^{\prime},\text{where}H^{\prime }= 6 \sqrt{\beta R}\\ \left. d^{E}-d^{N}\right\vert_{q = 2/3} &=&-\frac{1}{3}\gamma <0. \end{array} $$
As dE − dN is continuous in [1/2, 2/3], ∃qe ∈ (1/2, 2/3): dE − dN = 0, and therefore at this point Δqt = 0.
In addition, \(\frac {\partial \left (d^{E}-d^{N}\right )}{\partial q} =-~2\gamma +\frac {R}{3(1-q)^{2}}+\frac {H^{2}}{54\beta }\frac {3q^{2}-6q + 2}{ (1-q)^{2}}<0\) ∀q ∈ [1/2, 2/3], taking into account that \(\frac {2}{3}R<\gamma \) and that \(H\geq H^{\prime }= 6\sqrt {\beta R}\); therefore, dE − dN is decreasing in [1/2, 2/3]. Hence, this qe ∈ (1/2, 2/3) is unique.
Nonetheless, if H < H′, the differences between socialization efforts (dE − dN) is negative ∀qt ∈ [1/2, 1] and non-entrepreneurial traits spreads among the population. Eventually, the proportion qt will arise to qt < 1/2 and the economy converges to qc as proved in Proposition 2. □
Proof of Proposition 4
Comparison of the traditional and entrepreneurial equilibria
Taking into account the steady state points qc and qe from Propositions 2 and 3, we calculate the administration efficiency, civil servants’ payoff, and entrepreneurs’ payoff in each equilibrium. Table 1 provides the results to facilitate the comparison
- 1.
Comparing administration efficiency, we have that ee > ec if \(\frac {q^{e}H}{3\beta }>\frac {2q^{c}H}{3\beta }\rightarrow q^{e}>2q^{c}.\) We know that 1/2 < qe < 2/3 and qc < 1/2. Therefore, it must hold that 1/2 > 2qc. A sufficient condition is qc < 1/4. If dE − dN| q= 1/4 < 0, then qc < 1/4. The difference between socialization efforts for q < 1/2 is given by the following:
$$d^{E}-d^{N}=-\frac{2q^{2}H^{2}}{27\beta (1-q)}-\frac{2-3q}{3(1-q)}R+\gamma \left( 1-2q\right) $$
and evaluated in q = 1/4 is
$$\left. d^{E}-d^{N}\right\vert_{q = 1/4}=-\frac{H^{2}}{162\beta } -5/9R + 1/2\gamma . $$
Recall that for H < H′, there is only a steady state qc. Therefore, we analyze the case where H ≥ H′. Substituting for \( H^{\prime }= 6\sqrt {\beta R},\) we obtain that under Assumption 2, for any H ≥ H′→ dE − dN| q= 1/4 < 0, which implies that ee > ec.
- 2.
Civil servants’ payoff is higher in the entrepreneurial economy than in the traditional economy: \({y_{C}^{e}}>{y_{C}^{c}}\) if \(\frac {\left (q^{e}\right )^{2}H^{2}}{18\beta }>\frac {\left (q^{c}\right )^{2}H^{2}}{9\beta } \rightarrow q^{e}>2q^{c}\), which holds if ee > ec that has been proved above.
- 3.
It is trivial because \(y_{E}^{c}= 0\)
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Proof of Proposition 5
First, we compare the utilities \({U_{E}^{e}}\) and \({U_{N}^{e}}\):
$$\begin{array}{@{}rcl@{}} {U_{E}^{e}} &=&\frac{q^{e}H^{2}}{27\beta }+\gamma \\ {U_{N}^{e}} &=&\frac{\left( q^{e}\right)^{2}H^{2}}{54(1-q^{e})\beta }+\frac{ 2-3q^{e}}{3(1-q^{e})}R. \end{array} $$
Comparing the first terms, we obtain that \(\frac {q^{e}H^{2}}{27\beta }>\frac { \left (q^{e}\right )^{2}H^{2}}{54(1-q^{e})^{2}\beta }\) if \(q^{e}<\frac {2}{3}\) which holds because qe ∈ (1/2, 2/3). Moreover, \(\gamma > \frac {2-3q^{e}}{3(1-q^{e})}R\) ∀qe ∈ (1/2, 2/3) as \( \gamma >\frac {2}{3}R\) because of Assumption 1, then \({U_{E}^{e}}>{U_{N}^{e}}\).
Table 1 Efficiency and payoffs at the steady state equilibria
Second, we compare \({y_{E}^{e}}\) with \({y_{C}^{e}}\). The payoff of entrepreneurs is higher than the payoff of a civil servant if \(\frac {q^{e}H^{2}}{27\beta }> \frac {\left (q^{e}\right ) ^{2}H^{2}}{18\beta }\rightarrow \)qe < 2/3 which holds because qe ∈ (1/2, 2/3). □