K(E,)K(E^*,\Theta)K(E,) denotes the elite-cooptation term evaluated at leadership parameters (the particular functional form for KKK was given in Section 2.C).
3. Algebraic reduction --- express TT^*T and eliminate
Equation (4a) is linear in TT^*T. Solve for TT^*T:
TSE(E)BL(1T)=BPP1T=BPTSE(E)BLP.\alpha_T\,S_E(E^*)\,B_L\,(1-T^*) = B_P P^* \quad\Longrightarrow\quad 1 - T^* = \dfrac{B_P}{\alpha_T\,S_E(E^*)\,B_L}\,P^*.TSE(E)BL(1T)=BPP1T=TSE(E)BLBPP.
Hence
T=1BPTBLSE(E)P(5)\boxed{ \; T^* \;=\; 1 \;-\; \dfrac{B_P}{\alpha_T\,B_L\,S_E(E^*)}\, P^* \; } \tag{5}T=1TBLSE(E)BPP(5)
(Important physical constraint: 0T10\le T^*\le10T1 restricts admissible pairs (E,P)(E^*,P^*)(E,P); in particular the denominator must be positive and the right-hand side must be in [0,1][0,1][0,1].)
Substitute (5) into (4b) and (4c) to eliminate TT^*T.
(a) Substitute into economic steady state (4b)
From (4b):
EEE+PPTT=0.\mu_E - \lambda_E E^* + \phi_P P^* - \phi_T T^* = 0.EEE+PPTT=0.
