The production function per worker relates how much output a worker can produce to the level of technology and the amount of capital they have to work with. A commonly-used production function per worker is: yt=A(t)∙k(t)1/3, where y is output per worker, A is a measure of technology, k is capital per worker, and all variables are functions of time. If technology is growing at a rate of 1% and capital per worker is growing at a rate of 3% then output per worker will grow at a rate of

1%

2%

3%

4%

The production function per worker relates how much output a worker can produce to the level of technology and the amount of capital they have to work with. In this case, the production function is given by the equation yt=A(t)∙k(t)1/3, where y is output per worker, A is a measure of technology, and k is capital per worker. Given that technology is growing at a rate of 1% and capital per worker is growing at a rate of 3%, we can calculate the rate of growth of output per worker.

To calculate the rate of growth, we differentiate the production function with respect to time:

dy/dt = (dA/dt)(k^(1/3)) + (1/3)(A)(k^(-2/3))(dk/dt)

Substituting the given rates of change, we have:

dy/dt = (0.01)(k^(1/3)) + (1/3)(A)(0.03)(k^(-2/3))

Since we are interested in the percentage growth rate, we can express the derivative as a percentage:

Percentage growth rate = (dy/dt) / y * 100

Substituting the values into the equation, we can calculate the growth rate:

Percentage growth rate = [(0.01)(k^(1/3)) + (1/3)(A)(0.03)(k^(-2/3))] / (A∙k^(1/3)) * 100

Simplifying the expression, we get:

Percentage growth rate = 0.01 + 0.01/3 = 0.01 + 0.0033 = 0.0133 = 1.33%

Therefore, the output per worker will grow at a rate of 1.33%.

PeterKyle PeterKyle · Answer 2 · 2025-05-26T07:16:00-04:00

To determine the growth rate of output per worker, we need to consider the growth rates of technology and capital per worker in the given production function:

yt = A(t) * k(t)^(1/3)

Given that technology is growing at a rate of 1% (0.01) and capital per worker is growing at a rate of 3% (0.03), we can calculate the growth rate of output per worker using the following formula:

Growth Rate of Output per Worker = Growth Rate of Technology + (1/3) * Growth Rate of Capital per Worker

Substituting the values:

Growth Rate of Output per Worker = 0.01 + (1/3) * 0.03

= 0.01 + 0.01

= 0.02

Therefore, output per worker will grow at a rate of 2% (option b) in this scenario.

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