Suppose an economy has an increase in labor input of 60 percent, while output increased by 65 percent. Assuming no change in total factor productivity, calculate the percentage increase in the capital input. (Use the Cobb-Douglas production function Y=AK0.40L0.60). Round your answer to 2 decimal places. Type in your answer without the percent sign. For example, if your answer is 55%, simply type 55.

Question

BlackTom AI · Accepted Answer

We start by restating the problem: an economy uses Y = A K^0.40 L^0.60, with no change in TFP (A constant). Labor input L increases by 60%, output Y increases by 65%. We need the percentage change in capital input K.

Because A is constant, we can use a log-differential (or percentage change) approach for a Cobb-Douglas production function. For Y = A K^0.40 L^0.60, the approximate relationship is:
d ln Y ≈ 0.40 d ln K + 0.60 d ln L.
Here, d ln Y corresponds to the percentage change in Y (approximately 0.65 for a 65% increase), and d ln L corresponds to the percentage change in L (0.60 for a 60% increase).

Plugging in the values, 0.65 ≈ 0.40 d ln K + 0.60 × 0.60.
Compute 0.60 × 0.60 = 0.36. So, 0.65 ≈ 0.40 d ln K + 0.36.
Subtract 0.36 from both sides: 0.65 − 0.36 = 0.29 ≈ 0.40 d ln K.
Solve for d ln K: d ln K ≈ 0.29 / 0.40 = 0.725.
Interpretation: the approximate percentage change in K is about 72.5%. Using a more exact calculation (without the small-change approximation) involves solving 1.65 = K^0.40 × L^0.60 with L = 1.60, yielding K^0.40 = 1.65 / 1.60^0.60, so K = [1.65 / 1.60^0.60]^(1/0.40) ≈ [1.245]^(2.5) ≈ 1.727 ≈ 72.79%.

Step-by-step numerical check:
- Calculate 1.60^0.60: use exponentiation to get about 1.326.
- Then 1.65 / 1.326 ≈ 1.245.
- Raise to the power 1/0.40 = 2.5: 1.245^2.5 ≈ 1.727, which corresponds to a 72.7%–72.9% increase depending on rounding.

With the requested rounding to two decimal places, the capital input increase is approximately 72.79%.

Summary: Using the Cobb-Douglas form Y = A K^0.40 L^0.60 and the given changes in Y and L, solving for the required K change yields about a 72.79% increase in K.

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