QuestionTwo companies have production functions:
1) $y_{1}=2 K+3 L$ 2) $y_{2}=2 K^{0.5} L^{0.5}$
a) Find returns to scale for $y_{1}$ . b) Calculate and analyze the marginal rate of technical substitution for $y_{1}$ . c) Find returns to scale for $y_{2}$ . d) Calculate and analyze the marginal rate of technical substitution for $y_{2}$ .
Check if company 1 with 5L and 10K minimizes costs ( $w=2$ , $r=3$ ) and do the same for company 2 with 3L and 2K.

Studdy Solution

STEP 1

Question: What does it mean if a production function exhibits constant returns to scale?
A) Output increases by more than the factor by which inputs are increased. B) Output increases by less than the factor by which inputs are increased. C) Output increases by the same factor by which inputs are increased. D) Output does not change no matter how much inputs are increased.
Answer: C) Output increases by the same factor by which inputs are increased.

STEP 2

Question: Given the production function $y_{1}=2 K+3 L$ , what are its returns to scale?
A) Increasing returns to scale. B) Decreasing returns to scale. C) Constant returns to scale. D) The production function does not exhibit any returns to scale.
Answer: C) Constant returns to scale.

STEP 3

Question: What is the marginal rate of technical substitution (MRTS) of the production function $y_{1}=2 K+3 L$ ?
A) -1 B) -1.5 C) 1 D) 1.5
Answer: B) -1.5

STEP 4

Question: Given the production function $y_{2}=2 K^{0.5} L^{0.5}$ , what are its returns to scale?
A) Increasing returns to scale. B) Decreasing returns to scale. C) Constant returns to scale. D) The production function does not exhibit any returns to scale.
Answer: C) Constant returns to scale.

STEP 5

Question: The first company with the production function $y_{1}=2 K+3 L$ uses 5 units of labor and 10 units of capital. The cost of labor is $w=2$ and the cost of capital is $r=3$ . Is the company behaving rationally, i.e., minimizing production costs?
A) Yes B) No C) It cannot be determined from the given information. D) The company is behaving rationally only if it is not making a profit.
Answer: B) No

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