WebR — Lei de Walras e Excesso de Demanda

Carregando WebR...

# ============================================
# Lei de Walras: verificacao numerica
# Exercicio Resolvido 14.6
# ============================================

# --- Economia ---
wA <- c(4, 6); wB <- c(6, 4)
aA <- 0.5   # U_A = x1^(1/2) * x2^(1/2)
aB <- 1/3   # U_B = x1^(1/3) * x2^(2/3)
totais <- wA + wB

cat("====== LEI DE WALRAS ======\n")
cat("Dotacoes: wA =", wA, "  wB =", wB, "\n")
cat("Totais:", totais, "\n\n")

# --- Funcao de excesso de demanda ---
Z <- function(p1, p2 = 1) {
  IA <- p1 * wA[1] + p2 * wA[2]
  IB <- p1 * wB[1] + p2 * wB[2]
  # Demandas Cobb-Douglas
  xA <- c(aA * IA / p1, (1-aA) * IA / p2)
  xB <- c(aB * IB / p1, (1-aB) * IB / p2)
  z <- (xA + xB) - totais
  return(z)
}

# --- Testar a varios precos ---
precos <- c(0.3, 0.5, 0.8, 1.0, 1.5, 2.0, 3.0)
cat("--- Verificacao: p*Z1 + Z2 = 0 para todo p ---\n")
cat(sprintf("%-8s %-10s %-10s %-12s\n", "p1", "Z1", "Z2", "p1*Z1+Z2"))
cat(strrep("-", 42), "\n")

for (p in precos) {
  z <- Z(p)
  walras <- p * z[1] + 1 * z[2]
  cat(sprintf("%-8.2f %-10.4f %-10.4f %-12.2e\n", p, z[1], z[2], walras))
}

# --- Equilibrio ---
# aA*IA/p + aB*IB/p = total_1
# Resolver: p*(aA*wA[1]+aB*wB[1]-T1) + aA*wA[2]+aB*wB[2] = 0
num <- aA * wA[2] + aB * wB[2]
den <- totais[1] - aA * wA[1] - aB * wB[1]
p_star <- num / den

cat("\n--- Equilibrio ---\n")
cat("p* =", round(p_star, 4), "\n")
z_eq <- Z(p_star)
cat("Z1(p*) =", round(z_eq[1], 10), "  Z2(p*) =", round(z_eq[2], 10), "\n\n")

cat("CONCLUSAO: A Lei de Walras (p*Z = 0) vale para TODO\n")
cat("vetor de precos, nao apenas no equilibrio.\n")
cat("E' uma identidade contabil, nao uma condicao de equilibrio.\n")

# --- Grafico ---
par(mar = c(4.5, 4.5, 3, 2), bg = "#f8f9fa")
p_seq <- seq(0.1, 4, length = 400)

Z1_seq <- sapply(p_seq, function(p) Z(p)[1])
Z2_seq <- sapply(p_seq, function(p) Z(p)[2])
walras_seq <- p_seq * Z1_seq + Z2_seq

plot(p_seq, Z1_seq, type = "l", lwd = 3, col = "#0d6efd",
     ylim = range(c(Z1_seq, Z2_seq)),
     xlab = expression(p[1] ~~ (p[2] == 1)),
     ylab = "Excesso de demanda",
     main = "Lei de Walras: Z(p) e identidade contabil")
lines(p_seq, Z2_seq, lwd = 3, col = "#dc3545")
lines(p_seq, walras_seq, lwd = 2, col = "#198754", lty = 3)
abline(h = 0, col = "#adb5bd", lty = 2)
abline(v = p_star, col = "#6f42c1", lty = 2, lwd = 1.5)
points(p_star, 0, pch = 19, col = "#6f42c1", cex = 2)
text(p_star + 0.1, Z1_seq[1] * 0.2,
     paste0("p* = ", round(p_star, 3)),
     col = "#6f42c1", cex = 0.9, font = 2)

legend("topright",
       legend = c(expression(Z^1*(p)), expression(Z^2*(p)),
                  expression(p[1]*Z^1 + Z^2 == 0)),
       col = c("#0d6efd", "#dc3545", "#198754"),
       lwd = c(3, 3, 2), lty = c(1, 1, 3),
       cex = 0.75, bg = "white")

(Aguardando WebR...)