WebR — Modelo Trabalho-Lazer com Cobb-Douglas

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# ============================================
# Modelo Trabalho-Lazer: Cobb-Douglas
# Exercicio Resolvido 17.1
# ============================================

alpha <- 0.5   # peso do consumo
TT <- 16       # dotacao de tempo (horas)
V <- 32        # renda nao salarial

cat("====== MODELO TRABALHO-LAZER ======\n")
cat("U(C,L) = C^alpha * L^(1-alpha)\n")
cat("alpha =", alpha, "  T =", TT, "  V =", V, "\n\n")

# Demandas marshallianas (Cobb-Douglas):
# L*(w) = (1-alpha)*(wT + V) / w
# h*(w) = T - L*(w) = alpha*T - (1-alpha)*V/w
# C*(w) = alpha*(wT + V)

# --- Salario de reserva ---
# h* > 0 => alpha*T - (1-alpha)*V/w > 0
# => w > (1-alpha)*V / (alpha*T)
w_R <- (1 - alpha) * V / (alpha * TT)
cat("Salario de reserva: w_R =", w_R, "\n\n")

# --- Tabela para varios salarios ---
w_vals <- c(1, 2, 3, 4, 6, 8, 10, 15, 20)
cat(sprintf("%-6s %-8s %-8s %-8s %-8s\n",
    "w", "L*", "h*", "C*", "Renda"))
cat(strrep("-", 42), "\n")
for (w in w_vals) {
  L_star <- (1-alpha) * (w*TT + V) / w
  h_star <- max(TT - L_star, 0)
  C_star <- alpha * (w*TT + V)
  renda <- w * h_star + V
  if (h_star > 0) {
    cat(sprintf("%-6.0f %-8.2f %-8.2f %-8.1f %-8.1f\n",
        w, L_star, h_star, C_star, renda))
  } else {
    cat(sprintf("%-6.0f %-8s %-8s %-8.1f %-8.1f\n",
        w, "16.00", "0.00", V, V))
  }
}

cat(sprintf("\ndh*/dw = (1-alpha)*V / w^2 = %.4f (em w=4)\n",
    (1-alpha)*V / 4^2))
cat("Oferta sempre positivamente inclinada (sem backward-bending)\n")
cat("Com V=0: h* = alpha*T =", alpha*TT, "(nao depende de w)\n")

# --- Grafico ---
par(mfrow = c(1, 2), mar = c(4.5, 4.5, 3, 1), bg = "#f8f9fa")

# Painel 1: Escolha otima no espaco (L, C)
w_ex <- 4
L_opt <- (1-alpha)*(w_ex*TT + V)/w_ex
C_opt <- alpha*(w_ex*TT + V)
M_star <- w_ex*TT + V

# Restricao orcamentaria
L_seq <- seq(0, TT, length=200)
C_reta <- M_star - w_ex * L_seq

plot(L_seq, C_reta, type="l", lwd=3, col="#0d6efd",
     xlim=c(0, TT+1), ylim=c(0, M_star+10),
     xlab="Lazer (L)", ylab="Consumo (C)",
     main=paste0("Escolha otima (w=", w_ex, ")"))

# Curva de indiferenca pelo ponto otimo
U_opt <- C_opt^alpha * L_opt^(1-alpha)
L_ci <- seq(0.5, TT, length=300)
C_ci <- (U_opt / L_ci^(1-alpha))^(1/alpha)
valid <- C_ci <= M_star * 1.2
lines(L_ci[valid], C_ci[valid], lwd=2, col="#6f42c1", lty=2)

points(L_opt, C_opt, pch=19, col="#dc3545", cex=2)
text(L_opt-0.8, C_opt+4,
     paste0("(", round(L_opt,1), ", ", round(C_opt,1), ")"),
     col="#dc3545", cex=0.8, font=2)

# Dotacao
points(TT, V, pch=17, col="#198754", cex=1.5)
text(TT-1, V+4, paste0("Dotacao (", TT, ",", V, ")"),
     col="#198754", cex=0.7, font=2)

legend("topright",
       legend=c("Restricao orc.", "Curva de indiferenca", "Otimo"),
       col=c("#0d6efd", "#6f42c1", "#dc3545"),
       lwd=c(3,2,NA), lty=c(1,2,NA), pch=c(NA,NA,19),
       cex=0.65, bg="white")

# Painel 2: Oferta de trabalho h*(w)
w_seq <- seq(0.5, 25, length=300)
h_seq <- pmax(alpha*TT - (1-alpha)*V/w_seq, 0)

plot(h_seq, w_seq, type="l", lwd=3, col="#dc3545",
     xlab="Horas trabalhadas (h)", ylab="Salario (w)",
     main="Oferta de trabalho h*(w)",
     xlim=c(0, alpha*TT + 1))
abline(h=w_R, col="#adb5bd", lty=3, lwd=1.5)
text(2, w_R + 0.5, paste0("w_R = ", w_R), col="#adb5bd",
     cex=0.75, font=2)

# Caso V=0
h_V0 <- rep(alpha*TT, length(w_seq))
lines(h_V0, w_seq, lwd=2, col="#0d6efd", lty=2)
text(alpha*TT + 0.3, 20, "V=0\n(inelastica)",
     col="#0d6efd", cex=0.65, font=2)

legend("bottomright",
       legend=c(paste0("V=", V), "V=0"),
       col=c("#dc3545", "#0d6efd"),
       lwd=c(3,2), lty=c(1,2), cex=0.65, bg="white")

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