Question

the production function for T-shirts can be represented as q = L0.25 K0.75.
Draw on a graph a set of isoquants for q=100, q=200 and q=400. What is the value of K
and L used on each isoquant when the firm uses the same amount of K and L? Calculate the
MRTS as function of L and K, and show that the isoquant is convex
Show if this production function has increasing, decreasing or constant return to scale.
Then consider a general Cobb-Douglas production function of the form q = La Kb where a and
b are any positive value and find the conditions for those two parameters, under which the
production function has increasing return to scale.

Answer 1

let K=L=x

than let's make an equation

$100=x^{0.25}\times x^{0.75}$

x=100

$200=x^{0.25}\times x^{0.75}$

x=200

$400=x^{0.25}\times x^{0.75}$

x=400

Calculate the MRTS by formula:

$MRTS=-\frac{\Delta K}{\Delta L}$

K=90, L=50

$MRTS=-\frac{60-90}{100-50}=0.6$

similarly, we will present calculations in Excele:

$a+b>1$ a=0.25 b=1.25

this ratio characterizes an increasing return on scale, for example, with an increase in labor and capital spent by 100%, the volume of production will increase, for example, by 120%, that is, more than twice