Question

A closed economy has the following production function: Y=5 K¹/⁵ L⁴/⁵ where Y denotes output, K denotes capital, and L denotes labor.

If both capital and labour increase by 5% what would be the percentage Change in the output?

Answer 1

The production function of a closed economy is given, and when both capital and labour increase by
`5%,` the output also increases by
`5%` . this is determined by applying the properties of exponents to the given Cobb-Douglas production function.

Step-by-step explanation:

A student inquires about the percentage change in output of a closed economy with a specific production function when both capital (K) and labor (L) increase by
`5%.` The production function given is:

`Y=5 K⁹/₅ L⁴/₅`

To determine the percentage change in output, we can use the properties of exponents in this Cobb-Douglas production function.

Since both K and L are increasing by 5%, we calculate the new output as:

`Y' = 5 (K * 1.05)⁹/₅ (L * 1.05)⁴/₅`

We can express the increase in terms of exponents:

`Y' = 5 K⁹/₅ L⁴/₅ * 1.05⁹/₅ * 1.05⁴/₅`

`1.05⁹/₅ * 1.05⁴/₅ = 1.05`

The overall increase of output:
`Y' is Y * 1.05,`

Which indicates a
`5%` increase in output. Hence, if both capital and labour increase by
`5%` the output would also increase by
`5%.`