Question

 1) The Cobb-Douglas production function is 

y=z1αz2β

, with 

α>0

 and 

β>0

. Show that 

MP1=αy/z1

, 

MP2=βy/z2

. What is the 

MRTS21

? How does it vary with: (a) y; (b) 

z2/z1

.

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. A change in Y would not affect MRTS since MRTS is not a 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