Question

Y= 4X1^1/2X2^1/2, factor prices are p1=1, p2=36 and the firm can hire as much of either factor it wants at these prices. The firm's marginal cost is

Answer 1

$Y= 4X_1^{\frac{1}{2}}X_2^{\frac{1}{2}}\\MRTS=\frac{Mu_{x1}}{Mu_{x2}}=\frac{\frac{1}{2}(4x_1)^{\frac{1}{2}}x_2^\frac{1}{2}}{\frac{1}{2}(4x_1)^{\frac{1}{2}}x_2^\frac{1}{2}}$

$MRTS=\frac{x_2}{x_1}$

Approximately $\frac{x_2}{x_1}=\frac{p_1}{p_2}=\frac{1}{36}$

$y=4(36x_2)^{\frac{1}{2}}x_2^{\frac{1}{2}}\\x_1=\frac{36}{24}\\TC=p_1x_1+p_2x_2=\frac{36}{24}y+\frac{36}{24}y\\TC=\frac{72}{24}y=3y\\MC=\frac{∆TC}{∆y}=3$

Balance of constant returns to scale,MC is constant.