Functions of two variables and partial derivatives, question 14

Topic: Functions of two variables and partial derivatives
Difficulty: Capstone
Suggested time: 10 min
Marks: 5
Theory: UBC Math 100 §15.1Key results

A firm’s weekly output is $Q(K, L) = 15K^{1/3}L^{2/3}$ units, where $K$ and $L$ are its weekly spending on capital and on labour, in thousands of dollars. It now spends $K = 64$ and $L = 8$ . Find the two marginal products, decide which way one thousand dollars of spending should be moved between the inputs to raise output, and estimate the resulting increase in output.

Give the estimated increase in weekly output, in units.

Your answer

units

A fraction or a decimal.

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Theory UBC Math 100 §15.1Key results

Worked solution

Strategy

K and L are both measured in thousands of dollars, so each marginal product is the extra output per extra thousand dollars on that input. Compare them to see where money does more, then estimate the move as the output lost on one input plus the output gained on the other.

Working

$Q(64, 8) = 15\cdot 64^{1/3}\cdot 8^{2/3} = 15\cdot 4\cdot 4 = 240$
Current output, for reference.
$\frac{\partial Q}{\partial K} = 5K^{-2/3}L^{2/3} = 5\cdot\frac{4}{16} = \frac{5}{4}$
Hold L fixed. 64^2/3 = 16 and 8^2/3 = 4.
$\frac{\partial Q}{\partial L} = 10K^{1/3}L^{-1/3} = 10\cdot\frac{4}{2} = 20$
Hold K fixed. 64^1/3 = 4 and 8^1/3 = 2.
$\frac{\partial Q}{\partial L} = 20 > \frac{5}{4} = \frac{\partial Q}{\partial K}$
A thousand dollars more on labour adds about 20 units; on capital, about 1.25. Move money from capital to labour.
$\Delta Q \approx \frac{\partial Q}{\partial K}(-1) + \frac{\partial Q}{\partial L}(+1) = -\frac{5}{4} + 20 = \frac{75}{4} = 18.75$
Cutting capital by one thousand dollars loses about 1.25 units; adding it to labour gains about 20.
$Q(63, 9) - Q(64, 8) \approx 18.2$
The exact change confirms the estimate.

Answer

\Delta Q \approx 20 - \frac{5}{4} = \frac{75}{4} = 18.75 \text{ units}

Check

Actually moving the thousand gives Q(63, 9) ≈ 258.2, about 18.2 more than the present 240, close to the estimate.

Where this goes wrong

Counting only the gain from labour, 20 units, and forgetting that the money leaves capital. Moving spending is a trade: the estimate is the marginal product gained minus the marginal product given up.