Question

The Cobb–Douglas production function is
y=Z1^aZ2^b a>0, b>0

show that the MP1 = ay/z1, MP2 = by/z2

What is the MRTS21? How does it vary with: (a) y; (b) z2 /z1?

Answer 1

$Y = Z_1^αZ_2^β$

$MP_1 = αZ_1^{α-1}Z_2^β = \frac{αZ_1^αZ_2^β}{Z_1} = \frac{αY}{Z_1}$

$MP_2 = βZ_1^αZ_2^{β -1} = \frac{βZ_1^αZ_2^β}{Z_2} = \frac{βY}{Z_2}$

$MRTS_{21} = \frac{MP_1}{MP_2}$

$MRTS_{21} = \frac{\frac{αY}{Z_1}}{\frac{βY}{Z_2}}$

$MRTS_{21} = \frac{αY}{Z_1} \times \frac{Z_2}{β} = \frac{α}{β} \times \frac{Z_2}{Z_1}$

a) MRTS₂₁ would not be affected by the change in y. MRTS₂₁ is not the function of y.

b) MRTS₂₁ would be affected by the change in $\frac{Z_2}{Z1}$ . An increase in $\frac{Z_2}{Z1}$ would result in increase of MRTS₂₁ and vice versa.