Question

An economy is characterized by the following production function: \[ Y = A K^{0.25} L^{0.75} \] where \(K\) denotes capital, \(L\) denotes labour, \(A\) denotes the total factor productivity, and \(Y\) denotes output produced. All capital and labour are fully employed. Suppose that the growth rate of labour is 1%, the growth rate of capital is 4%, and the growth rate of output is 4%. The growth rate of total factor productivity \(A\) is:

• A. 1.5%
• B. 1.75%
• C. 2.25%
• D. 2.5%

Hint:

In growth rate calculations for production functions, apply the weighted growth rates of capital and labour to find the growth rate of total factor productivity.

Answer 1

The Correct Option is C

Step 1: Use the growth rate formula

The growth rate of output can be expressed as:

\[ \text{Growth rate of } Y = \text{Growth rate of } A + 0.25 \times \text{Growth rate of } K + 0.75 \times \text{Growth rate of } L \]

Step 2: Substitute the given values

We are given that:

• Growth rate of output = 4%
• Growth rate of labour = 1%
• Growth rate of capital = 4%

Substitute these values into the equation:

\[ 4 = \text{Growth rate of } A + 0.25 \times 4 + 0.75 \times 1 \]

Now, simplifying the equation:

\[ 4 = \text{Growth rate of } A + 1 + 0.75 \]

4 = \text{Growth rate of } A + 1.75

Step 3: Solve for the growth rate of A

To solve for the growth rate of A, subtract 1.75 from both sides:

\[ \text{Growth rate of } A = 4 - 1.75 = 2.25\% \]

Thus, the growth rate of total factor productivity (A) is 2.25%.