Question

In: Economics

2. Consider a firm with the following production function: Q = K 1/3 L 2/3 (a)...

2. Consider a firm with the following production function: Q = K ^1/3 L ^2/3

(a) Consider an output level of Q = 100. Find the expression of the isoquant for this output level.

(b) Find the marginal product of labor, MPL. Is it increasing, decreasing, or constant in the units of labor, L, that the firm uses?

(c) Find the marginal product of capital, MPK. Is it increasing, decreasing, or constant in the units of capital, K, that the firm uses?

(d) Use your result in parts (b)-(c) to find the marginal rate of technical substitution, MRTS, for this firm.

(e) Is the MRTS increasing or decreasing in the units of labor, L? What does that imply about the shape of the isoquant?

(f) Given your result in part (d), what can you say about the firm’s ability to substitute one input for another? (g) Assume now that the firm were to increase all inputs by a common factor > 0. What happens to the output that the firm produces? [Hint: check whether the firm’s production function exhibits increasing, decreasing, or constant returns to scale.]

Solutions

Expert Solution

2. (a) The isoquant equation for Q=100 would be as .

In terms of two variable equaiton/function, it would be as or .

(b) The MPL would be as or or or .

We have or , meaning that MPL is decreasing in L, ie MPL decreases as L increases.

(c) The MPK would be as or or or .

We have or , meaning that MPK is decreasing in K, ie MPK decreases as K increases.

(d) The MRTS would be as or or .

(e) We have or , meaning that the MRTS increases as L increases or MRTS is increasing in L (while MRTS would be decreasing in K, also since MRTS is negative, MRTS increasing in L means that MRTS tends to 0 as L increases). According to the founded condition, the isoquant is negatively sloped and is convex to the origin (MRTS can be taken as first derivative of isoquant while derivative of MRTS can be considered as second derivate of isoquant, which is positive).

(f) The firm would substitute 2K/L units of capital for a marginal unit of labor. This means that if firm wants to increase a marginal unit of labor, firm must reduce 2K/L units of capital, in order to remain at the same production level.

(g) The firm's production function is , and increasing inputs by a>0 times, we have or or . This means that the firm have constant returns to scale, ie the output increases by 'a' times for an increase in input by 'a' times.

Rahul Sunny answered 2 weeks ago

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