R/ModelFunctions.R

In SebastianBehrens/SolowVariants: Provide framework to simulate macroeconomic Solow growth models

# Model Functions of Basic Solow Model ---------------------------------
BS_MF_KN <- function(s, Y, delta, K){s * Y + (1-delta)*K}
BS_MF_LN <- function(n, L){(1+n) * L}
BS_MF_RR <- function(B, K, L, alpha){alpha * B * (K/L)^(alpha - 1)}
BS_MF_WR <- function(B, K, L, alpha){(1-alpha) * B * (K/L)^alpha}
BS_MF_Y <- function(B, K, L, alpha){B * K^alpha * L^(1-alpha)}

BS_SS_KpW <- function(B, alpha, s, n, delta){
    #' @export
    B^(1/(1-alpha))*(s/(n+ delta))^(1/(1-alpha))
    }
BS_SS_YpW <- function(B, alpha, s, n, delta){
    #' @export
    B^(1/(1-alpha))*(s/(n+ delta))^(alpha/(1-alpha))
    }
BS_SS_CpW <- function(B, alpha, s, n, delta){
    #' @export
    B^(1/(1-alpha))*(1-s)*(s/(n+delta))^(alpha/(1-alpha))
    }
BS_SS_RR <- function(alpha, s, n, delta){
    #' @export
    alpha * (s/(n + delta))^(-1)
    }
BS_SS_WR <- function(B, alpha, s, n, delta){
    #' @export
    (1-alpha)*B^(1/(1-alpha))*(s/(n+delta))^(alpha/(1-alpha))
    }

BS_TE <- function(s, n, B, x, alpha, delta){
    #' @export  
    (1/(1+n)) * (s * B * x^alpha + (1- delta) * x)
}
BS_SE_pt1 <- function(s, B, x, alpha){
    #' @export  
    s * B * x^alpha
}
BS_SE_pt2 <- function(n, delta, x){
    #' @export  
    (n + delta) * x
}
BS_SE <- function(s, B, x, alpha, n, delta){
    #' @export  
    (1/(1+n)) * (BS_SE_pt1(s, B, x, alpha) - BS_SE_pt2(n, delta, x))
}
# Model Functions of General Solow Growth Model ---------------------------------
GS_MF_KN <- function(s, Y, delta, K){s * Y + (1-delta)*K}
GS_MF_LN <- function(n, L){(1+n) * L}
GS_MF_AN <- function(g, A){(1+g) * A}
GS_MF_RR <- function(A, K, L, alpha){alpha * (K/(A*L))^(alpha - 1)}
GS_MF_WR <- function(A, K, L, alpha){A* (1-alpha) * (K/(A*L))^alpha }
GS_MF_Y <- function(A, K, L, alpha){K^alpha * (A*L)^(1-alpha)}

# from book
GS_SS_KpW <- function(s, n, g, delta, alpha, A){
    #' @export
    A * (s/(n + g + delta + n*g))^(1/(1-alpha))
    }
GS_SS_YpW <- function(s, n, g, delta, alpha, A){
    #' @export
    A * (s/(n + g + delta + n*g))^(alpha/(1-alpha))
    }
GS_SS_CpW <- function(s, n, g, delta, alpha, A){
    #' @export
    A * (1-s) * (s/(n + g + delta + n * g))^(alpha/(1-alpha))
    }
GS_SS_RR <- function(alpha, s, n, g, delta){
    #' @export
    alpha * (s/(n + g + delta + n * g))^(-1)
    }
GS_SS_WR <- function(alpha, s, n, g, delta, A){
    #' @export
    A * (1- alpha) * (s/(n + g + delta + n * g))^(alpha/(1-alpha))
    }

# from slides
GS_SS_KpEW <- function(s, n, g, delta, alpha){
    #' @export
    (s/(n + g + delta + n*g))^(1/(1-alpha))
    }
GS_SS_YpEW <- function(s, n, g, delta, alpha, A){
    #' @export
    (s/(n + g + delta + n*g))^(alpha/(1-alpha))
    }

GS_TE <- function(s, n, B, x, alpha, delta){
    #' @export  
    (1/((1+n)* (1 + g))) * (s  * x^alpha + (1- delta) * x)
}
GS_SE_pt1 <- function(s, B, x, alpha){
    #' @export  
    s * B * x^alpha
}
GS_SE_pt2 <- function(n, delta, x){
    #' @export  
    (n + delta) * x
}
GS_SE <- function(s, B, x, alpha, n, delta){
    #' @export  
    (1/((1+n)* (1 + g))) * (GS_SE_pt1(s, B, x, alpha) - GS_SE_pt2(n, delta, x))
}
# Model Functions of Extended Solow Growth Model for the Small and Open Economy ---------------------------------
ESSOE_MF_LN <- function(n, L){(1+n) * L}
ESSOE_MF_RR <- function(B, K, L, alpha){alpha * B * (K/(L))^(alpha - 1)}
ESSOE_MF_WR <- function(B, K, L, alpha){B* (1-alpha) * (K/(L))^alpha}
ESSOE_MF_K <- function(r, alpha, B, L){L/((r/(alpha * B))^(1/(1-alpha)))}
ESSOE_MF_Y <- function(B, K, L, alpha){B* K^alpha * (L)^(1-alpha)}
ESSOE_MF_Yn <- function(Y, r, F_var){Y + r * F_var} # F_var for variable F since F in the R programming language stands for FALSE
ESSOE_MF_VN <- function(Y_nat, s, V_previous){s*Y_nat + V_previous}
ESSOE_MF_F <- function(V, K){V - K}
ESSOE_MF_VpWN <- function(s, r, n, vL, w){((1 + (s * r))/(1 + n))*vL + (s * w)/(1 + n)}

ESSOE_SS_KpW <- function(B, alpha, r){
    #' @export
    B^(1/(1-alpha))*(alpha/r)^(1/(1-alpha))
    }
ESSOE_SS_YpW <- function(B, alpha, r){
    #' @export
    B^(1/(1-alpha))*(alpha/r)^(alpha/(1-alpha))
    }
ESSOE_SS_YnpW <- function(alpha, B, r, s, n){
    #' @export
    B^(1/(1-alpha))*(alpha/r)^(alpha/(1-alpha)) * (1-alpha) * (1/(1- (s/n)*r))
    }
ESSOE_SS_CtO <- function(alpha, r){
    #' @export
    alpha/r
}
ESSOE_SS_WR <- function(B, alpha, r){
    #' @export
    (1-alpha) * B^(1/(1-alpha))*(alpha/r)^(alpha/(1-alpha))
    }
ESSOE_SS_VpW_alt <- function(s, n, r, w){
    #' @export
    ((s/n)/(1 - (s/n)*r))*w
    } # old
ESSOE_SS_VpW <- function(s, n, r, w){
    #' @export
    (s/(n - (s * r))) * w
    }
ESSOE_SS_FpW <- function(alpha, s, n, r, w){
    #' @export
    (1/(1- alpha))* (s/n) * (1/r)*((r - ((alpha* n)/s))/(1 - (s/n)*r))*w
    }
ESSOE_TE <- function(s, n, r, x, w){
    ((1 + s * r)/(1 + n)) * x + ((s * w)/(1 + n))
}

# Model Functions of Extended Solow Growth Model with Human Capital ---------------------------------
ESHC_MF_KN <- function(sK, Y, delta, K){sK * Y + (1-delta)*K}
ESHC_MF_HN <- function(sH, Y, delta, H){sH * Y + (1-delta)*H}
ESHC_MF_LN <- function(n, L){(1+n) * L}
ESHC_MF_AN <- function(g, A){(1+g) * A}
ESHC_MF_RR <- function(A, H, K, L, alpha, phi){alpha * (K/(A*L))^(alpha - 1)*(H/(A*L))^phi}
ESHC_MF_WR <- function(A, H, K, L, alpha, phi){(1-alpha) * (K/(A*L))^(alpha)*(H/(A*L))^phi * A}
ESHC_MF_Y <- function(A, H, K, L, alpha, phi){K^alpha * H^phi * (A*L)^(1-alpha - phi)}
ESHC_MF_KpEWN <- function(n, g, sK, YpEW, delta, KpEW){(1/((1+n)*(1+g))) * (sK * YpEW + (1-delta)*KpEW)}
ESHC_MF_HpEWN <- function(n, g, sH, YpEW, delta, KpEW){(1/((1+n)*(1+g))) * (sH * YpEW + (1-delta)*KpEW)}
# from book
ESHC_SS_KpEW <- function(sK, sH, alpha, phi, n, g, delta){
    #' @export
    ((sK^(1-phi)*sH^phi)/(n + g + delta + n * g))^(1/(1-alpha - phi))
    }
ESHC_SS_HpEW <- function(sK, sH, alpha, phi, n, g, delta){
    #' @export
    ((sK^alpha * sH^(1-alpha))/(n + g + delta + n * g))^(1/(1-alpha- phi))
    } 

# YpEW complicated variant (sH and sK occuring twice per factor)
# ESHC_SS_YpEW <- function(sK, sH, n, g, delta, alpha, phi){(((sK^(1-phi)*sH^phi)/(n + g + delta + n * g))^(1/(1-alpha - phi)))^alpha *(((sK^(1- alpha) * sH^alpha)/(n + g + delta + n * g))^(1/(1-alpha- phi)))^phi}

# YpEW simplified sK and sH occuring just once in separate factors
ESHC_SS_YpEW <- function(sK, sH, n, g, delta, alpha, phi){
    #' @export
    (((sK)/(n + g + delta + n * g))^(1/(1-alpha - phi)))^alpha *(((sH)/(n + g + delta + n * g))^(1/(1-alpha- phi)))^phi
    }

ESHC_SS_YpW <- function(A, sK, sH, n, g, delta, alpha, phi){
    #' @export
    A * (((sK)/(n + g + delta + n * g))^(1/(1-alpha - phi)))^alpha *(((sH)/(n + g + delta + n * g))^(1/(1-alpha- phi)))^phi
    }
# ESHC_SS_YpW <- function(HpEW, KpEW, A, alpha, phi){A * KpEW^alpha * HpEW^phi}

ESHC_SS_CpW <- function(A, sK, sH, n, g, delta, alpha, phi){
    #' @export
    A * (1- sK - sH) * (((sK)/(n + g + delta + n * g))^(1/(1-alpha - phi)))^alpha *(((sH)/(n + g + delta + n * g))^(1/(1-alpha- phi)))^phi
    }

# SS_WR and SS_RR missing. formula nowhere to be found — must be computed myself
# ESHC_SS_RR <- function(alpha, s, n, g, delta){alpha * (s/(n + g + delta + n * g))^(-1)}
# ESHC_SS_WR <- function(alpha, s, n, g, delta, A){A * (1- alpha) * (s/(n + g + delta + n * g))^(alpha/(1-alpha))}


# Model Functions of Extended Solow Growth Model with the Scarce Resource Oil ---------------------------------
ESSRO_MF_Y <- function(A, K, L, E, alpha, beta){K^alpha * (A*L)^(beta) * E^(1- alpha - beta)}
ESSRO_MF_KN <- function(s, Y, delta, K){s * Y + (1-delta)*K}
ESSRO_MF_LN <- function(n, L){(1+n) * L}
ESSRO_MF_RN <- function(E, R){R - E}
ESSRO_MF_E <- function(sE, R){sE * R}
ESSRO_MF_AN <- function(g, A){(1+g) * A}

# ESSRO_MF_RR <- function(){}
# ESSRO_MF_WR <- function(){}

ESSRO_SS_YpW <- function(KpW, YpW, A, R, L, alpha, beta){
    #' @export
    epsilon <- 1- alpha - beta
    (KpW/YpW)^(alpha/(beta + epsilon)) * A^(beta/(beta + epsilon)) * (R/L)^(epsilon/(beta + epsilon))
    }
ESSRO_SS_KpW <- function(n, delta, s, E, L, alpha, beta){
    #' @export
    epsilon <- 1- alpha - beta
    (n+delta)/(s * (E/L)^epsilon * A^beta ) ^(1/(alpha-1) )
    }


# Remark regarding dynamics of E and R
# R_0 => E_1 = s_ER_0 => R_1=R_0 - E_1 => E_2=sER_1 => ...

# Model Functions of Extended Solow Growth Model with the Scarce Resource Land ---------------------------------
ESSRL_MF_Y <- function(A, K, L, X, alpha, beta){K^alpha * (A * L)^(beta) * X^(1-alpha - beta)}

ESSRL_MF_KN <- function(s, Y, delta, K){s * Y + (1-delta)*K}
ESSRL_MF_LN <- function(n, L){(1+n) * L}
ESSRL_MF_AN <- function(g, A){(1+g) * A}

ESSRL_MF_WR <- function(A, K, L, X, alpha, beta){beta * (K/(A*L))^alpha * (X/(A*L))^(1-alpha - beta) * A}
ESSRL_MF_RR <- function(A, K, L, X, alpha, beta){alpha * (K/(A*L))^(alpha - 1) * (X/(A*L))^(1-alpha - beta)}
ESSRL_MF_LR <- function(A, K, L, X, alpha, beta){(1-alpha - beta) * (K/(A*L))^(alpha) * (X/(A*L))^((1-alpha - beta) - 1)}

# from book (but also on slides)
ESSRL_SS_YpW <- function(KpW, YpW, A, X, L, alpha, beta){
    #' @export
    (KpW/YpW)^(alpha/(beta + (1-alpha - beta))) * A^(beta/(beta + (1-alpha - beta))) * (X/L)^((1-alpha - beta)/(beta + (1-alpha - beta)))
}
# Capital to Output Ratio
ESSRL_SS_CtO <- function(s, n, g, delta, alpha, beta){
    #' @export
    s/(((1 + n) * (1 + g))^(beta/(beta + (1-alpha - beta))) - (1- delta))
}


# Remember: (1-alpha - beta) => this is kappa.

# Model Functions of Extended Solow Growth Model with the Scarce Resources Oil and Land ---------------------------------
ESSROL_MF_Y <- function(A, K, L, E, X, alpha, beta, kappa){K^alpha * (A*L)^(beta) * X^kappa * E^(1-alpha-beta-kappa)}
ESSROL_MF_KN <- function(s, Y, delta, K){s * Y + (1-delta)*K}
ESSROL_MF_LN <- function(n, L){(1+n) * L}
ESSROL_MF_RN <- function(E, R){R - E}
ESSROL_MF_E <- function(sE, R){sE * R}
ESSROL_MF_AN <- function(g, A){(1+g) * A}
# ESSRO_MF_RR <- function(){}
# ESSRO_MF_WR <- function(){}

ESSROL_SS_gY <- function(alpha, beta, kappa, n, g, sE){
    #' @export
    (beta/(beta + kappa + (1- alpha - beta - kappa)))* g - 
        ((kappa + (1- alpha - beta- kappa))/(beta + kappa + (1- alpha - beta - kappa))) * n - 
        ((1- alpha - beta- kappa)/(beta + kappa  + (1- alpha - beta - kappa))) * sE
}

ESSROL_SS_YpW <- function(alpha, beta, kappa, A, K, L, X, R, sE){
    #' @export
    epsilon <- 1- alpha - beta - kappa
    (K/L)^(alpha/(beta + kappa + epsilon)) * A^(beta/(beta + kappa + epsilon)) * (X/L)^(kappa/(beta + kappa + epsilon)) * sE^(epsilon/(beta + kappa + epsilon)) * (R/L)^(epsilon/(beta + kappa + epsilon))
}
# Remark regarding dynamics of E and R
# R_0 => E_1 = s_ER_0 => R_1=R_0 - E_1 => E_2=sER_1 => ...

# Model Functions of Extended Solow Growth Model with Endogeneous Growth ---------------------------------
ESEG_MF_KN <- function(s, Y, delta, K){s * Y + (1-delta)*K}
ESEG_MF_LN <- function(n, L){(1+n) * L}
# ESEG_MF_RR <- function(K, L, alpha){alpha * B * (K/L)^(alpha - 1)}
# ESEG_MF_WR <- function(K, L, alpha){(1-alpha) * B * (K/L)^alpha}
ESEG_MF_Y <- function(K, L, alpha, phi, A){
    if(phi < 0.95){
        K^alpha * (K^phi*L)^(1-alpha)
    }else if(phi >= 0.95 && phi < 1){
        A * K
    }

}

ESEG_MF_WR <- function(A, K, L, alpha, phi){(1-alpha) * ( K/ (A * L) )^alpha * A}
ESEG_MF_RR <- function(A, K, L, alpha, phi){alpha * (K/(A*L))^(alpha - 1)}
# ESEG_SS_gY <- function(endogenous_type, input_list){
#     if(endogenous_type == "AK")
#     (1 + n)^((phi)/(1- phi)) - 1
# }

ESEG_SS_gYpW <- function(n, phi, s, A, delta){
    #' @export
    if(phi < 0.95){
    (1 + n)^((phi)/(1- phi)) - 1
    }else if(phi >= 0.95 && phi < 1){
            s * A - delta
    }else if(phi > 1){
            NaN
    }
}
ESEG_SS_KpEW <- function(alpha, phi, n, s, delta){
    #' @export
    (s/((1 + n)^(1/(1 - phi)) - (1- delta)))^(1/(1-alpha))
    }
ESEG_SS_YpEW <- function(alpha, phi, n, s, delta){
    #' @export
    (s/((1 + n)^(1/(1 - phi)) - (1- delta)))^(alpha/(1-alpha))
    }
# BS_SS_KpW <- function(B, alpha, s, n, delta){B^(1/(1-alpha))*(s/(n+ delta))^(1/(1-alpha))}
# BS_SS_YpW <- function(B, alpha, s, n, delta){B^(1/(1-alpha))*(s/(n+ delta))^(alpha/(1-alpha))}
# BS_SS_CpW <- function(B, alpha, s, n, delta){B^(1/(1-alpha))*(1-s)*(s/(n+delta))^(alpha/(1-alpha))}
# BS_SS_RR <- function(alpha, s, n, delta){alpha * (s/(n + delta))^(-1)}
# BS_SS_WR <- function(B, alpha, s, n, delta){(1-alpha)*B^(1/(1-alpha))*(s/(n+delta))^(alpha/(1-alpha))}


# Model Functions of Extended Solow Growth Model with Endogeneous Growth as developed by Romer ---------------------------------
ESEGRomer_MF_KN <- function(s, Y, delta, K){s * Y + (1-delta)*K}
ESEGRomer_MF_LN <- function(n, L){(1+n) * L}
ESEGRomer_MF_L_At <- function(sR, L){sR * L}
ESEGRomer_MF_AN <- function(rho, phi, lambda, A, L_At){rho * A^phi * L_At^lambda + A}
# ESEG_MF_RR <- function(K, L, alpha){alpha * B * (K/L)^(alpha - 1)}
# ESEG_MF_WR <- function(K, L, alpha){(1-alpha) * B * (K/L)^alpha}
ESEGRomer_MF_Y <- function(A, K, L, alpha){K^alpha * (A*L)^(1-alpha)} # remember L here referes to L working in the goods sector (1-s_R) * L_t = L_{Y,t}

# ESEG_SS_gY <- function(endogenous_type, input_list){
#     if(endogenous_type == "AK")
#     (1 + n)^((phi)/(1- phi)) - 1
# }

ESEGRomer_SS_gYpW <- function(n, rho, phi, sR, L, lambda){
    #' @export
    if(phi < 0.95){
    (1 + n)^((phi)/(1- phi)) - 1
    }else if(phi >= 0.95 && phi <= 1){
            rho * (sR * L)^lambda
    }else if(phi > 1){
            message("The function ESEGRomer_SS_gYpW is not defined for phi > 1.")
    }
}

ESEGRomer_SS_YpEW <- function(s, n, alpha, rho, phi, sR, L, delta, lambda){
    #' @export
    if(phi < 0.95){
    g_se <- (1 + n)^(lambda/(1-phi)) - 1
    (s/(n + g_se + delta + n*g_se))^(alpha/(1-alpha)) * (1-sR)
    }else if(phi >= 0.95 && phi <= 1){
    g_e <- rho * (sR * L)^lambda
    (s/(g_e + delta))^(alpha/(1-alpha)) * (1-sR)
    }else if(phi > 1){
            message("The function ESEGRomer_SS_gYpW is not defined for phi > 1.")
    }
}

# Model Functions of Extended Solow Growth Model with Endogeneous Growth as developed by Cozzi's Extension to Romer's ESEG ---------------------------------
ESEGCozziOne_MF_KN <- function(s, Y, delta, K){s * Y + (1-delta)*K}
ESEGCozziOne_MF_LN <- function(n, L){(1+n) * L}
ESEGCozziOne_MF_AN <- function(rho, phi, lambda, A, sR){(rho * A * sR^lambda) + A}
ESEGCozziOne_MF_L_At <- function(sR, L){sR * L}
# ESEG_MF_RR <- function(K, L, alpha){alpha * B * (K/L)^(alpha - 1)}
# ESEG_MF_WR <- function(K, L, alpha){(1-alpha) * B * (K/L)^alpha}
ESEGCozziOne_MF_Y <- function(A, K, L, alpha){K^alpha * (A*L)^(1-alpha)} # remember L here referes to L working in the goods sector (1-s_R) * L_t = L_{Y,t}

# ESEG_SS_gY <- function(endogenous_type, input_list){
#     if(endogenous_type == "AK")
#     (1 + n)^((phi)/(1- phi)) - 1
# }

ESEGCozziOne_SS_gYpW <- function(rho, sR, lambda){
    #' @export
    rho * sR^lambda
}
ESEGCozziOne_SS_KpEW <- function(alpha, rho, phi, lambda, n, s, sR, delta){
    #' @export
    g_e <- rho * sR^lambda
    (s/(n + g_e + delta + n * g_e))^(1/(1-alpha)) * (1-sR)
    }

ESEGCozziOne_SS_YpEW <- function(alpha, rho, phi, lambda, n, s, sR, delta){
    #' @export
    g_e <- rho * sR^lambda
    (s/(n + g_e + delta + n * g_e))^(alpha/(1-alpha)) * (1-sR)
    }
# Model Functions of Extended Solow Growth Model with Endogeneous Growth as developed by Cozzi's Extension to Romer's ESEG ---------------------------------
ESEGCozziTwo_MF_KN <- function(s, Y, delta, K){s * Y + (1-delta)*K}
ESEGCozziTwo_MF_LN <- function(n, L){(1+n) * L}
ESEGCozziTwo_MF_AN <- function(k, rho, phi, lambda, A, sR, L_At){k * (rho * A^phi * L_At^lambda) + (1-k)*(A * sR^lambda) + A}
ESEGCozziTwo_MF_L_At <- function(sR, L){sR * L}
# ESEG_MF_RR <- function(K, L, alpha){alpha * B * (K/L)^(alpha - 1)}
# ESEG_MF_WR <- function(K, L, alpha){(1-alpha) * B * (K/L)^alpha}
ESEGCozziTwo_MF_Y <- function(A, K, L, alpha){K^alpha * (A*L)^(1-alpha)} # remember L here referes to L working in the goods sector (1-s_R) * L_t = L_{Y,t}

# ESEG_SS_gY <- function(endogenous_type, input_list){
#     if(endogenous_type == "AK")
#     (1 + n)^((phi)/(1- phi)) - 1
# }

ESEGCozziTwo_SS_gTFP <- function(k, rho, sR, lambda, phi, n){
    #' @export
    conditional_n_bar <- ((1-k) * rho * sR^lambda + 1)^((1-phi)/lambda) - 1 # minimum n for semi edno growth
    if(n > conditional_n_bar){
        (1 + n)^(lambda/(1-phi)) - 1
    }else{
        (1-k)  *rho * sR^lambda
    }
}
ESEGCozziTwo_SS_KpEW <- function(k, rho, sR, lambda, phi, n, s, alpha, delta){
    #' @export
    conditional_n_bar <- ((1-k) * rho * sR^lambda + 1)^((1-phi)/lambda) - 1 # minimum n for semi edno growth
    if(n > conditional_n_bar){
        g_h <- (1 + n)^(lambda/(1-phi)) - 1
    }else{
        g_h <- (1-k)  *rho * sR^lambda
    }
    out <- (s/(n + g_h + delta + n * g_h))^(1/(1-alpha)) * (1-sR)
    return(out)
    }
ESEGCozziTwo_SS_YpEW <- function(k, rho, sR, lambda, phi, n, s, alpha, delta){
    #' @export
    conditional_n_bar <- ((1-k) * rho * sR^lambda + 1)^((1-phi)/lambda) - 1 # minimum n for semi edno growth
    if(n > conditional_n_bar){
        g_h <- (1 + n)^(lambda/(1-phi)) - 1
    }else{
        g_h <- (1-k)  *rho * sR^lambda
    }
    out <- (s/(n + g_h + delta + n * g_h))^(alpha/(1-alpha)) * (1-sR)
    return(out)
    }
# BS_SS_KpW <- function(B, alpha, s, n, delta){B^(1/(1-alpha))*(s/(n+ delta))^(1/(1-alpha))}
# BS_SS_YpW <- function(B, alpha, s, n, delta){B^(1/(1-alpha))*(s/(n+ delta))^(alpha/(1-alpha))}
# BS_SS_CpW <- function(B, alpha, s, n, delta){B^(1/(1-alpha))*(1-s)*(s/(n+delta))^(alpha/(1-alpha))}
# BS_SS_RR <- function(alpha, s, n, delta){alpha * (s/(n + delta))^(-1)}
# BS_SS_WR <- function(B, alpha, s, n, delta){(1-alpha)*B^(1/(1-alpha))*(s/(n+delta))^(alpha/(1-alpha))}