Nt1330 Unit 1 Math Term Paper

3. There is a partition of the groups $\{ \mathcal{Q}, \mathcal{N} \setminus \mathcal{Q}\}${Q,N∖Q} of $ \mathcal{N}$N with $Q=\#\mathcal{Q}$Q=#Q such that \[\begin{align} P^i_{ss}=\left\{ \begin{array}{cc} \frac{1}{Q} \frac{\varphi -\sqrt{\varphi^2-4 \gamma Q (1-\alpha +\gamma) \left(1-\sum _{j\notin \mathcal{Q}} n^j\right)}}{2 (1-\alpha +\gamma) } & \text{for} \ i \in \mathcal{Q} \\ \frac{n^i}{\sum _{j\notin \mathcal{Q}} n^j} \frac{\varphi +\sqrt{\varphi^2-4 \gamma Q (1-\alpha +\gamma) \left(1-\sum _{j\notin \mathcal{Q}} n^j\right)}}{2 (1-\alpha +\gamma)}& \text{for} \ i \in \mathcal{N} \setminus \mathcal{Q} \end{array} \right. \end{align}\] Piss= 1Q ϕ− ϕ2−4γQ(1−α+γ) 1− j∉Q nj2(1−α+γ)for i∈Q ni j∉Q nj ϕ+ ϕ2−4γQ(1−α+γ)
\label{eqEequil} \end{align}\] Eiss= ψ− ψ2−4γ2Q(Q−1)( j∉Q nj)( j∉Q nj−1)2(Q−1)Qγfor i∈Qnifor i∈N∖Q. where $\psi=2-\alpha+\gamma ((2 Q-1) (\sum_{j \notin \mathcal{Q}} n^j) -Q) $ψ=2−α+γ((2Q−1)( j∉Q nj)−Q)
The proof of this proposition is in Appendix C. Similarly to the previous existence result, Proposition ?.(i) follows from the continuity of the influence dynamics correspondence from a simplex to itself and Brouwer’s fixed-point theorem. Uniqueness here is somewhat less simple to demonstrate and follows essentially from the first order conditions. Proposition ?.(iii) is the solution of the system of first order conditions at the steady state.
Part (iii) of the proposition expresses the steady state efforts, and influence shares, again distinguishing between constrained ($i \in \mathcal{N} \setminus \mathcal{Q}$i∈N∖Q), and unconstrained groups ($ i \in \mathcal{Q}$i∈Q). An important element is that all groups have strictly positive efforts at the steady state. The groups in $\mathcal{N} \setminus \mathcal{Q}$N∖Q make a political effort equal to $n^i$ni and their steady state influence is proportional to that. The groups in $\mathcal{Q}$Q make an interior political effort and their steady state influence share is equal to the rapport of their political effort to the total effort. Steady states efforts and influence shares are the same across all these unconstrained
Consequently, the capital accumulation dynamics defined by (?) leads to a unique steady state characterized by a level of output $Y_{ss}$Yss. This is formalized in Proposition ?.
Proposition 3 There exists a unique steady state of the capital accumulation defined by equation (?) characterized by a level of output \[\begin{align} Y_{ss}&=&\left(A (\iota_{ss})^{\alpha} (L_{ss})^{1-\alpha} \right)^{\frac{1}{1-\alpha}} \label{YssSol} \end{align}\] Yss = A(ιss)α(Lss)1−α 11−α where $\iota_{ss}$ιss is the investment share at the steady state of the power dynamics and $L_{ss}$Lss is the labour supply at the steady state.
The proof is in Appendix D. The mechanism of this paper appears more clearly in equation (?). The discussions above have highlighted that an increase in appropriative competition intensity $\gamma$γ tend to have a negative effect of on $L_{ss}$Lss and a positive effect on $\iota_{ss}$ιss. How these two effects balance in total is not a priori determined. In the next section, I show by the way of a counter-example that it is not necessarily true that $\gamma$γ affects $Y_{ss}$Yss negatively, even if this is the most likely