2 marksMicroeconomicsProduction and returns to scaleProducer equilibrium and the expansion pathHardabout 3 min

IIT JAM 2022 Question 59: Production and returns to scale

Let the production function be $Q = \sqrt{L^2 + K^2}$ , with unit prices of labour and capital Rs. 30 and Rs. 40, and total cost Rs. 580. The maximum value of $Q$ subject to the cost constraint is _______ (round off to 2 decimal places).

Answer and worked solution

Answer: 11.55 to 11.65 or 19.30 to 19.40

Maximising $Q = \sqrt{L^2 + K^2}$ is the same as maximising $L^2 + K^2$ on the budget line $30L + 40K = 580$ . Because $L^2 + K^2$ is convex, its largest value on a line segment is at an endpoint.

At $K = 0$ , $L = 580/30 = 19.33$ and $Q = 19.33$ ; at $L = 0$ , $K = 14.5$ and $Q = 14.5$ . So the maximum is $Q = 19.33$ . The tangency (shortest-distance) point gives $Q = 580/\sqrt{30^2+40^2} = 11.60$ , which is actually the minimum; the exam accepts either value.

Plain-text version

Let the production function be Q = √(L² + K²), with unit prices of labour and capital Rs. 30 and Rs. 40, and total cost Rs. 580. The maximum value of Q subject to the cost constraint is _ (round off to 2 decimal places).

Also written as: Let the production function be Q = sqrt(L^2 + K^2), with unit prices of labour and capital Rs. 30 and Rs. 40, and total cost Rs. 580. The maximum value of Q subject to the cost constraint is _ (round off to 2 decimal places).

Production and returns to scale has come up 92 times in 6 exams since 2007.

What this question tests

This question is on Producer equilibrium and the expansion path, part of Production and returns to scale in Microeconomics.

The least-cost mix of inputs at given prices and how it moves as output grows.

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