Chapter 6 Answer Problems Intermediate Price Theory PDF

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Chapter 6 – Intermediate Price Theory – Perloff (8th) Solved Problems3 If the production function is ��= ��(��,��)= 3��+ 2��, and capital is fixed at ��= 50, what is theshort-run production function? What is the marginal product of labor?Solution: The short-run production function is: �� = 3�� + 2 ∗...

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Chapter 6 – Intermediate Price Theory – Perloff (8th) Solved Problems

3.3 If the production function is 𝑞 = 𝑓(𝐿, 𝐾) = 3𝐿 + 2𝐾 , and capital is fixed at 𝐾 = 50, what is the short-run production function? What is the marginal product of labor? Solution: The short-run production function is:

𝑞 = 3𝐿 + 2 ∗ 50 𝒒 = 𝟑𝑳 + 𝟏𝟎𝟎

The 𝑀𝑃𝐿 is:

𝑀𝑃𝐿 = 1 ∗ 3𝐿1−1 (Ignore the terms that do not have L in them) 𝑀𝑃𝐿 = 3𝐿0 𝑴𝑷𝑳 = 𝟑

Or: 𝑀𝑃𝐿 =

∆𝑞 (3(𝐿 + ∆𝐿) + 100) − (3𝐿 + 100) 3𝐿 + ∆3𝐿 + 100 − 3𝐿 − 100 3∆𝐿 =3 = = = ∆𝐿 ∆𝐿 ∆𝐿 ∆𝐿

3.4 Suppose that the production function is 𝑞 = 𝐿0.75 𝐾 0.25 .

? a. What is the average product of labor, holding capital fixed at 𝐾 b. What is the marginal product of labor? Solution: (a)

0.25  0.25 𝐾 0.25 𝑞 𝐿0.75 𝐾 𝐾 𝐴𝑃𝐿 = = = 0.25 𝑜𝑟 ( ) 𝐿 𝐿 𝐿 𝐿

(b) 𝑀𝑃𝐿 = 0.75 ∗ 𝐿0. 75−1 𝐾 0.25 = 0.75𝐿−0.25 𝐾 0.25 = 0.75 Q3.

0.25 𝐾 0.25 𝐾 ) 𝑜𝑟 0. 75( 𝐿 𝐿0.25

In the short run, we assume that capital is a fixed input and labor is a variable input, so the firm can increase output only by increasing the amount of labor it uses. In the short-run, the firm's ), where 𝑞 is output, 𝐿 is workers, and 𝐾  is the fixed number of production function is 𝑞 = 𝑓(𝐿, 𝐾 units of capital. A specific equation for the production function is given by: Or, when  𝐾 = 20,

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𝑞 = 8𝐾𝐿 + 5𝐿2 − 𝐿3 . 3

1 3 𝑞 = 160𝐿 + 5𝐿2 − 3 𝐿 Round each of your answers up to two decimal places. (a) The level of output 𝑞 for 2 units of labor input is __337.33___. (b) The average productivity of these 2 units of labor is __168.67__. (c) The marginal productivity of using one more unit of labor input is _178.67_. Use the formula ∆𝑞 𝑀𝑃𝐿 = ∆𝐿 (d) Given the relationship between the average productivity and the marginal productivity, the average productivity of labor is __Rising__ (falling, constant, rising). Solution: (a) 𝑞 = 160 ∗ 2 + 5 ∗ 22 − (b) 𝐴𝑃𝐿 = (c)

𝑞 337.33 = 168.67 = 2 𝐿

We know that 𝑞𝐿=2 = 337.33, let us solve for 𝑞𝐿=3 so that we can calculate the change in q, ∆𝑞: 1 1 𝑞𝐿=3 = 160 ∗ 3 + 5 ∗ 32 − 33 = 480 + 45 − 27 = 516 3 3 𝑀𝑃𝐿 =

(d)

1 3 1 2 = 320 + 20 − ∗ 8 = 337.33 3 3

∆𝑞 𝑞𝐿=3 − 𝑞𝐿=2 516 − 337.333 = 178.67 = = 1 3−2 ∆𝐿

Since 𝑀𝑃𝐿 is higher than 𝐴𝑃𝐿 we know that the 𝐴𝑃𝐿 is Rising. -

Average Product of Labor always rises when 𝑀𝑃𝐿 > 𝐴𝑃𝐿 , and Average Product of Labor always decreases when 𝑀𝑃𝐿 < 𝐴𝑃𝐿 for a given amount of Labor.

Q4.

Identify the returns to scale in the functions below. Production Function a. 𝑞 = 4𝐿 + 5𝐾 b. 𝑞 = (𝐿 + 𝐾 )0.5 c. 𝑞 = 2𝐿𝐾 2 1

Returns to Scale Constant Decreasing Increasing Constant

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d. 𝑞 = 𝐿 2 𝐾 2

Solution: a. 𝑞 (2𝐿, 2𝐾) = 4(2𝐿) + 5(2𝐾) = 2(4𝐿 + 5𝐾) → Doubling the inputs doubled the output → Constant Returns to Scale (CRS). 0.5 b. 𝑞 (2𝐿, 2𝐾) = (2𝐿 + 2𝐾)0.5 = (2(𝐿 + 𝐾)) = √2√𝐿 + 𝐾 → Doubling the inputs less than doubled the output → Decreasing Returns to Scale (DRS). c. 𝑞 (2𝐿, 2𝐾) = 2(2𝐿)(2𝐾 )2 = 2 ∗ 2 ∗ 𝐿 ∗ 22 ∗ 𝐾 2 = 8 ∗ 2𝐿𝐾 2 → Doubling the inputs more than doubled the output → Increasing Returns to Scale (IRS). 1

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d. 𝑞 (2𝐿, 2𝐾) = (2𝐿)2 (2𝐾)2 = 2 2 𝐿2 2 2 𝐾 2 = (22 2 2 )𝐿2 𝐾 2 = 2𝐿2 𝐾 2 → Doubling the inputs doubled the output → Constant Returns to Scale (CRS).

Q5.

Under what conditions does a Cobb-Douglas production function, 𝑞 = 𝐴𝐿𝛼 𝐾𝛽 Exhibit decreasing, constant, or increasing returns to scale? Show how output changes if both inputs are doubled. (a) If the inputs double, then, as a function if 𝑞, 𝛼, and 𝛽, output increases by _𝟐𝜶+𝜷 _. (b) Give a rule for determining the returns to scale. Let 𝑔 = 𝛼 + 𝛽 . If 𝑔 = 1 then the production function exhibits __constant__ returns to scale, if 𝑔 < 1, then the production function exhibits __decreasing__ returns to scale, and if 𝑔 > 1, then the production function exhibits __increasing__ returns to scale. (c) For example, if 𝑔 = 1.47, then the production function exhibits ___increasing___ returns to scale, and doubling the inputs will increase output by ___177.02 ___ percent. Solution: (21.47 − 1) ∗ 100 = 𝟏𝟕𝟕. 𝟎𝟐%...