The Cobb$-$Douglas production function $is$ commonly used $to$ model the production $ofan$ output, based $on$ two $or$

more inputs. $In$ this project, $we$ will consider the inputs $of$ capital and labour:

Capital, denoted $byK,is$ the business's capital expenditures over a given time period. This could include

money spent $on$ space $or$ equipment rental, maintenance, supplies, etc.

Labour, denoted $byL,is$ the number $of$ worker$-$hours used over the same time period.

The output $is$ the quantity $of$ items produced, $Q$ over the same time period. The key assumptions leading $to$

the Cobb$-$Douglas production function are $as$ follows. Assume $A1$ and $A2$ are true throughout this individual

submission.

$A1.If$ the labour $is$ held fixed, a very small relative change $in$ labour $K,\frac{K}{K}$ should always lead $to$ a corresponding

relative change $inQ$ that $is$ proportional $to$ the relative change $in$ labour. $In$ other words,

$\frac{Q}{Q}=\frac{K}{K}$

for some constant $>0.$

$A2.If$ the capital expenditures are held fixed, a very small relative change $in$ labour $L,\frac{L}{L},$ should always lead $to$

a corresponding relative change $inQ$ that $is$ proportional $to$ the relative change $in$ labour. $In$ other words,

$\frac{Q}{Q}=\frac{L}{L}$

for some constant $>0.$