WebR — FPP, Imposto e Perda de Eficiencia

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# ============================================
# FPP, imposto e perda de eficiencia
# Exercicio Resolvido 14.5
# ============================================

# --- FPP: X1^2 + X2^2 = 200 ---
# Consumidores identicos: U = x1^(1/2) * x2^(1/2)
# Imposto ad valorem t sobre bem 1

cat("====== FPP E DISTORCAO TRIBUTARIA ======\n\n")

# --- Equilibrio sem imposto ---
cat("--- Equilibrio sem imposto ---\n")
X1_star <- sqrt(100)  # X1 = X2 = 10
X2_star <- sqrt(100)
U_star <- (X1_star/2)^0.5 * (X2_star/2)^0.5
TMT_star <- X1_star / X2_star
TMS_star <- X2_star / X1_star  # = 1 (simetria)

cat("X1* =", X1_star, "  X2* =", X2_star, "\n")
cat("TMT = X1/X2 =", TMT_star, "\n")
cat("TMS = X2/X1 =", TMS_star, "\n")
cat("TMT = TMS?", abs(TMT_star - TMS_star) < 1e-10, "\n")
cat("Utilidade por consumidor:", U_star, "\n\n")

# --- Equilibrio com imposto ---
taxas <- c(0.1, 0.25, 0.5, 1.0, 2.0)
cat("--- Efeito do imposto t sobre bem 1 ---\n")
cat(sprintf("%-6s %-8s %-8s %-8s %-8s %-10s %-8s\n",
    "t", "X1", "X2", "TMT", "TMS", "U/cons", "Perda%"))
cat(strrep("-", 60), "\n")

resultados <- data.frame()
for (t in taxas) {
  # X2^2 = (1+t)*X1^2, na FPP: X1^2*(2+t) = 200
  X1_t <- sqrt(200 / (2 + t))
  X2_t <- sqrt(200 - X1_t^2)
  TMT_t <- X1_t / X2_t
  TMS_t <- (1 + t) * TMT_t  # consumidor ve preco (1+t)*p1
  U_t <- (X1_t/2)^0.5 * (X2_t/2)^0.5
  perda <- (U_star - U_t) / U_star * 100
  cat(sprintf("%-6.2f %-8.2f %-8.2f %-8.3f %-8.3f %-10.4f %-8.2f\n",
      t, X1_t, X2_t, TMT_t, TMS_t, U_t, perda))
  resultados <- rbind(resultados,
    data.frame(t = t, X1 = X1_t, X2 = X2_t, U = U_t, perda = perda))
}

cat("\nCONCLUSAO: O imposto cria uma cunha TMS != TMT.\n")
cat("Quanto maior t, maior a distorcao e a perda de bem-estar.\n")

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

# Painel 1: FPP com e sem imposto
theta <- seq(0, pi/2, length = 300)
X1_fpp <- sqrt(200) * cos(theta)
X2_fpp <- sqrt(200) * sin(theta)

plot(X1_fpp, X2_fpp, type = "l", lwd = 3, col = "#343a40",
     xlab = expression(X[1]), ylab = expression(X[2]),
     main = "FPP e distorcao tributaria",
     xlim = c(0, 16), ylim = c(0, 16), asp = 1)

# Ponto sem imposto
points(X1_star, X2_star, pch = 19, col = "#198754", cex = 2)
text(X1_star + 0.3, X2_star + 0.5, "Sem imposto\n(TMS = TMT)",
     col = "#198754", cex = 0.7, font = 2)

# Ponto com t = 0.5
t05 <- 0.5
X1_05 <- sqrt(200/(2+t05)); X2_05 <- sqrt(200 - X1_05^2)
points(X1_05, X2_05, pch = 17, col = "#dc3545", cex = 2)
text(X1_05 - 0.3, X2_05 + 0.6, "t = 0.5\n(TMS != TMT)",
     col = "#dc3545", cex = 0.7, font = 2, pos = 2)

# Ponto com t = 1.0
t10 <- 1.0
X1_10 <- sqrt(200/(2+t10)); X2_10 <- sqrt(200 - X1_10^2)
points(X1_10, X2_10, pch = 15, col = "#fd7e14", cex = 1.8)
text(X1_10 - 0.3, X2_10 + 0.6, "t = 1.0",
     col = "#fd7e14", cex = 0.7, font = 2, pos = 2)

# Linha 45 graus (eficiencia)
abline(a = 0, b = 1, col = "#198754", lty = 3, lwd = 1)

legend("topright", legend = c("FPP", "Eficiente", "t=0.5", "t=1.0"),
       col = c("#343a40", "#198754", "#dc3545", "#fd7e14"),
       pch = c(NA, 19, 17, 15), lwd = c(3, NA, NA, NA),
       cex = 0.7, bg = "white")

# Painel 2: Perda de bem-estar vs taxa
t_seq <- seq(0, 3, length = 200)
perda_seq <- sapply(t_seq, function(t) {
  X1 <- sqrt(200/(2+t)); X2 <- sqrt(200 - X1^2)
  U <- (X1/2)^0.5 * (X2/2)^0.5
  (U_star - U) / U_star * 100
})

plot(t_seq, perda_seq, type = "l", lwd = 3, col = "#dc3545",
     xlab = "Taxa de imposto (t)",
     ylab = "Perda de bem-estar (%)",
     main = "Peso morto em equilibrio geral")
abline(h = 0, col = "#adb5bd", lty = 2)
points(0.5, resultados$perda[resultados$t == 0.5],
       pch = 19, col = "#dc3545", cex = 1.5)
text(0.5, resultados$perda[resultados$t == 0.5] + 1,
     "t = 0.5", col = "#dc3545", cex = 0.8, font = 2)

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