Suppose the economy's production function is Y = AK^0.3.N^0.7 . If K = 2000, N = 100, and A = 1,
then Y = 246. If K and N both rise by 20%, and A is unchanged, by how much does Y increase?

The economy's production function is given as:

\[
Y = A \cdot K^{0.3} \cdot N^{0.7}
\]

where:
- \( A \) is the total factor productivity,
- \( K \) is the capital,
- \( N \) is the labor,
- \( Y \) is the output.

### Step 1: Initial Values
You are given the following initial values:
- \( K = 2000 \),
- \( N = 100 \),
- \( A = 1 \),
- \( Y = 246 \) (as calculated using these values).

### Step 2: New Values of \( K \) and \( N \)
Both \( K \) and \( N \) increase by 20%. Therefore, the new values of \( K \) and \( N \) are:
\[
K' = 1.2 \times K = 1.2 \times 2000 = 2400
\]
\[
N' = 1.2 \times N = 1.2 \times 100 = 120
\]

### Step 3: Calculate the New Output \( Y' \)
Using the new values of \( K' = 2400 \) and \( N' = 120 \), and keeping \( A = 1 \), we can calculate the new output \( Y' \):

\[
Y' = A \cdot (K')^{0.3} \cdot (N')^{0.7}
\]
\[
Y' = 1 \cdot (2400)^{0.3} \cdot (120)^{0.7}
\]

Let me calculate this for you.The new output \( Y' \) is approximately 294.77.

### Step 4: Calculate the Percentage Increase in \( Y \)
To find the percentage increase in output, we compare the new output \( Y' \) with the original output \( Y = 246 \):

\[
\text{Percentage increase} = \left( \frac{Y' - Y}{Y} \right) \times 100
\]
\[
\text{Percentage increase} = \left( \frac{294.77 - 246}{246} \right) \times 100
\]

Let me calculate the exact percentage increase.The output \( Y \) increases by approximately 19.83% when both capital \( K \) and labor \( N \) rise by 20%, with \( A \) remaining unchanged.

Would you like further details or explanations?

Here are 5 related questions you might find interesting:
1. How does the elasticity of output to capital affect this result?
2. What happens if only capital increases by 20%, keeping labor constant?
3. How would a change in total factor productivity \( A \) influence the outcome?
4. Can we interpret this result in terms of returns to scale?
5. What is the role of exponents in the production function?

**Tip**: The exponents in production functions reflect the elasticity of output with respect to each input.

Suppose the economy's production function is Y = AK^0.3.N^0.7. If K = 2000, N = 100, and A = 1, then Y = 246. If K and N both rise by 20%, and A is unchanged, by how much does Y increase?

Learn how to calculate the percentage increase in economic output using the Cobb-Douglas production function when both capital (K) and labor (N) increase by 20%. The original values are K = 2000, N = 100, and A = 1, leading to an output increase of approximately 19.83%.

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