Question

5. The Cobb–Douglas production function is

y " za

1 zb

2 a ' 0, b ' 0

Show that MP1 " ay/z1, MP2 " by/z2 .

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

Draw the isoquant m

Answer 1

$Y=Z_1^{\alpha}Z_2^{\beta}\\ MP_1= \alpha Z_1^{\alpha -1}Z_2^{\beta}\\ MP_1= \frac{\alpha Z_1^{\alpha }Z_2^{\beta}}{Z_1}\\ MP_1= \frac{\alpha Y}{Z_1}\\$

$MP_2= \beta Z_1^{\alpha }Z_2^{\beta-1}\\ MP_2= \frac{\beta Z_1^{\alpha }Z_2^{\beta}}{Z_2}\\ MP_2= \frac{\beta Y}{Z_2}\\$

$MRTS= \frac{MP_1}{MP_2}\\ MRTS= \frac{\frac{\alpha Y}{Z_1}}{\frac{\beta Y}{Z_2}}\\ MRTS= \frac{\alpha}{\beta}*\frac{Z_2}{Z_1}\\$

a. MRTS would not be affected by the change in Y. As MRTS is not the function of Y.

b. As $\frac{Z_2}{Z_1}$ changes MRTS changes.

An increase in $\frac{Z_2}{Z_1}$ MRTS increases and vice versa.