1 mark, −0.333MicroeconomicsCosts of productionCost minimisation and long-run input choiceModerateabout 3 min

GATE 2021 Question 3: Costs of production

For the production function $Q=F(K,L)=\sqrt{KL}$ with $P_K=4$ and $P_L=2$ , find the values of $K$ and $L$ that will minimize the cost of producing 2 units of output.

Tap an option to check your answer.

Answer and worked solution

Answer: (C)

Minimize $C=4K+2L$ subject to $\sqrt{KL}=2$ (i.e., $KL=4$ ).

Tangency condition: $\dfrac{MP_K}{MP_L} = \dfrac{P_K}{P_L}$ . With $Q=\sqrt{KL}=(KL)^{1/2}$ :

$MP_K = \frac{1}{2}\sqrt{\frac{L}{K}}, \qquad MP_L = \frac{1}{2}\sqrt{\frac{K}{L}}$

$\frac{MP_K}{MP_L} = \frac{L}{K} = \frac{P_K}{P_L} = \frac{4}{2}=2 \quad\Rightarrow\quad L=2K$

Substituting into the output constraint $KL=4$ : $K(2K)=4 \Rightarrow 2K^2=4 \Rightarrow K^2=2 \Rightarrow K=\sqrt2$ .

Then $L=2K=2\sqrt2$ .

Therefore $K=\sqrt2, L=2\sqrt2$ , matching option (C).

The trap: (B)

K=2\sqrt2\ ;\ L=\sqrt2

Students sometimes set up the tangency ratio inverted ( $K/L$ instead of $L/K$ , or vice versa relative to which input's price is in the numerator), producing a value with the correct magnitude structure but K and L swapped or scaled incorrectly, landing on a plausible-looking but wrong option like (B).

Plain-text version

For the production function Q=F(K,L)=√KL with P_K=4 and P_L=2, find the values of K and L that will minimize the cost of producing 2 units of output. (A) K=2√3; L=3√2 (B) K=2√2; L=√2 (C) K=√2; L=2√2 (D) K=2; L=2

Also written as: For the production function Q=F(K,L)=sqrt(KL) with P_K=4 and P_L=2, find the values of K and L that will minimize the cost of producing 2 units of output.

Costs of production has come up 83 times in 7 exams since 2006.

What this question tests

This question is on Cost minimisation and long-run input choice, part of Costs of production in Microeconomics.

Choosing inputs to produce a given output at least cost, including across several plants.

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