WebR — Minimização de Custo: Isoquanta e Isocusto

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# ============================================
# Minimizacao de Custo: Isoquanta + Isocusto
# Cobb-Douglas: q = K^alpha * L^beta
# ============================================

# --- Parametros (altere e re-execute!) ---
alpha <- 0.4     # expoente do capital
beta  <- 0.6     # expoente do trabalho
w <- 8           # salario
v <- 2           # custo do capital
q0 <- 10         # nivel de produto alvo

# --- Demandas condicionais ---
s <- alpha + beta
Lc <- q0^(1/s) * (beta * v / (alpha * w))^(alpha/s)
Kc <- q0^(1/s) * (alpha * w / (beta * v))^(beta/s)
C_min <- w * Lc + v * Kc

# TMST no otimo
TMST <- (beta/alpha) * (Kc/Lc)
razao_precos <- w/v

cat("====== MINIMIZACAO DE CUSTO ======\n")
cat("Tecnologia: q = K^", alpha, " * L^", beta, "\n")
cat("Precos: w =", w, " v =", v, " | q0 =", q0, "\n\n")

cat("--- Solucao otima ---\n")
cat("L^c =", round(Lc, 4), "\n")
cat("K^c =", round(Kc, 4), "\n")
cat("C_min = w*Lc + v*Kc =", round(C_min, 4), "\n\n")

cat("--- Verificacao da tangencia ---\n")
cat("TMST = (beta/alpha)*(K/L) =", round(TMST, 4), "\n")
cat("w/v =", round(razao_precos, 4), "\n")
cat("TMST = w/v?", ifelse(abs(TMST - razao_precos) < 1e-6, "SIM!", "NAO"), "\n\n")

cat("--- Interpretacao ---\n")
cat("PMgL/w =", round((beta * q0 / Lc) / w, 4), "\n")
cat("PMgK/v =", round((alpha * q0 / Kc) / v, 4), "\n")
cat("Produto marginal por $ gasto e igual para ambos insumos!\n")

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

L_max <- Lc * 3
K_max <- Kc * 3

# Isoquanta
L_seq <- seq(0.1, L_max, length = 500)
K_isoquanta <- (q0^(1/alpha) / L_seq^(beta/alpha))
K_isoquanta[K_isoquanta > K_max] <- NA

# Isocusto otima
K_isocusto <- (C_min - w * L_seq) / v
K_isocusto[K_isocusto < 0] <- NA

# Isocustos subotimas (mais caras)
C_high <- C_min * 1.3
K_high <- (C_high - w * L_seq) / v
K_high[K_high < 0] <- NA

C_low <- C_min * 0.75
K_low <- (C_low - w * L_seq) / v
K_low[K_low < 0] <- NA

plot(L_seq, K_isoquanta, type = "l", lwd = 3, col = "#0d6efd",
     xlab = "L (trabalho)", ylab = "K (capital)",
     main = paste0("Minimizacao de custo: q = K^", alpha, " L^", beta),
     xlim = c(0, L_max), ylim = c(0, K_max), cex.lab = 1.1)

lines(L_seq, K_isocusto, lwd = 2.5, col = "#dc3545")
lines(L_seq, K_high, lwd = 1.5, col = "#dc3545", lty = 3)
lines(L_seq, K_low, lwd = 1.5, col = "#dc3545", lty = 3)

# Ponto otimo
points(Lc, Kc, pch = 19, col = "#198754", cex = 2)
text(Lc, Kc,
     paste0("  (L*=", round(Lc, 1), ", K*=", round(Kc, 1), ")\n",
            "  C_min = ", round(C_min, 1)),
     pos = 4, cex = 0.85, col = "#198754", font = 2)

# Caminho de expansao (K/L constante)
L_exp <- seq(0, L_max, length = 100)
K_exp <- (Kc/Lc) * L_exp
lines(L_exp, K_exp, lwd = 1.5, col = "#6f42c1", lty = 2)

# Anotacoes
text(L_max * 0.7, 0.5, paste0("Incl. isocusto = -w/v = -", w/v),
     col = "#dc3545", cex = 0.8)

legend("topright",
       legend = c(paste0("Isoquanta (q=", q0, ")"),
                  paste0("Isocusto otima (C=", round(C_min, 1), ")"),
                  "Isocustos alternativas",
                  "Caminho de expansao"),
       col = c("#0d6efd", "#dc3545", "#dc3545", "#6f42c1"),
       lwd = c(3, 2.5, 1.5, 1.5), lty = c(1, 1, 3, 2),
       cex = 0.75, bg = "white")

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