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1 Consider the following production functions: 1. Y = AK 1/3L2/3 2. Y = A(K +L) 3. Y = (AL)1/4K1/4 4. Y = AHL¹/2 For each of the production functions listed above: a. Determine whether the production function exhibits CRS, diminishing returns to physical capital (or human capital, when applicable), and diminishing returns to labor. b. Check whether each production function satisfies the Inada conditions. c. Compute the per capita production function. For the INADA condition, the constant return to scale and the diminishing marginal return are not the prerequisites. In other words, an equation can satisfy the INADA condition without satisfying any other conditions. For the purpose of this question please assume the human capital H behaves like the physical capital K. Treat L as an input similar to physical capital and human capital. 1.1 Inada Conditions 1.1.1 Conceptually: If there is very little capital, then adding a little bit of capital is extremely useful. Conversely, if there is already a lot of capital, then adding a little bit more capital is almost useless. X 1.1.2 Mathematically lim FK (K, H) = lim F₁ (K, H) = ∞0< K-0 H-0 lim FK (K, H) = lim F₁ (K,H) = 0< K→∞ H→∞

1 Consider the following production functions: 1. Y = AK 1/3L2/3 2. Y = A(K +L) 3. Y = (AL)1/4K1/4 4. Y = AHL¹/2 For each of the production functions listed above: a. Determine whether the production function exhibits CRS, diminishing returns to physical capital (or human capital, when applicable), and diminishing returns to labor. b. Check whether each production function satisfies the Inada conditions. c. Compute the per capita production function. For the INADA condition, the constant return to scale and the diminishing marginal return are not the prerequisites. In other words, an equation can satisfy the INADA condition without satisfying any other conditions. For the purpose of this question please assume the human capital H behaves like the physical capital K. Treat L as an input similar to physical capital and human capital. 1.1 Inada Conditions 1.1.1 Conceptually: If there is very little capital, then adding a little bit of capital is extremely useful. Conversely, if there is already a lot of capital, then adding a little bit more capital is almost useless. X 1.1.2 Mathematically lim FK (K, H) = lim F₁ (K, H) = ∞0< K-0 H-0 lim FK (K, H) = lim F₁ (K,H) = 0< K→∞ H→∞

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
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I J Consider the following production functions: 1. Y = AK ¹/³L2/3 2. Y = A(K +L) 3. Y = (AL) ¹/4K 1/4 4. Y = AHL ¹/2 For each of the production functions listed above: a. Determine whether the production function exhibits CRS, diminishing returns to physical capital (or human capital, when applicable), and diminishing returns to labor. b. Check whether each production function satisfies the Inada conditions. c. Compute the per capita production function. For the INADA condition, the constant return to scale and the diminishing marginal return are not the prerequisites. In other words, an equation can satisfy the INADA condition without satisfying any other conditions. For the purpose of this question please assume the human capital H behaves like the physical capital K. Treat L as an input similar to physical capital and human capital. 1.1 Inada Conditions 1.1.1 Conceptually: If there is very little capital, then adding a little bit of capital is extremely useful. Conversely, if there is already a lot of capital, then adding a little bit more capital is almost useless.< X H 1.1.2 Mathematically lim FK (K,H) = lim F₁ (K, H) = ∞0< K→0 H→0 lim FK (K, H) = lim F(K,H) = 0< K→∞ H→∞
Transcribed Image Text:I J Consider the following production functions: 1. Y = AK ¹/³L2/3 2. Y = A(K +L) 3. Y = (AL) ¹/4K 1/4 4. Y = AHL ¹/2 For each of the production functions listed above: a. Determine whether the production function exhibits CRS, diminishing returns to physical capital (or human capital, when applicable), and diminishing returns to labor. b. Check whether each production function satisfies the Inada conditions. c. Compute the per capita production function. For the INADA condition, the constant return to scale and the diminishing marginal return are not the prerequisites. In other words, an equation can satisfy the INADA condition without satisfying any other conditions. For the purpose of this question please assume the human capital H behaves like the physical capital K. Treat L as an input similar to physical capital and human capital. 1.1 Inada Conditions 1.1.1 Conceptually: If there is very little capital, then adding a little bit of capital is extremely useful. Conversely, if there is already a lot of capital, then adding a little bit more capital is almost useless.< X H 1.1.2 Mathematically lim FK (K,H) = lim F₁ (K, H) = ∞0< K→0 H→0 lim FK (K, H) = lim F(K,H) = 0< K→∞ H→∞
1.1.3 Examples This production function satisfies the Inada conditions:< 3 F(K, H) A.K ³. H = FK ===// AK³²₂H ² lim — A K³² H = ₂ lim — A K-DO K-DO K4O F₁ = =—=—= A·K ²³²³. H ² ² ² & 6 K→∞ lim I lim + AK ³² H ²₂ lim / AH = = 0 — ㅎ K+∞ K-D00 K²/ HO 6 lim H∞ 6 -5 A.K ²³²³. N = {/ = 6 =—=— A. K = . H = {/ FK = 2K H lim (2K⋅H ²) + ∞0 2 lim (2K.H²) +0 lim H∞ 6 lim K² This production function doesn't satisfy the Inada conditions:< 2 2 F(K₁ H) = K² - H² -110 =1 — - A.K ²³ · H/5/6 = 00 =H+α = A·K ³ 6 HT -15/02 = 0 N FM₂ = 2K²³. H lim (2K² H) + ∞ E lim (2K² H) +0 HDO
Transcribed Image Text:1.1.3 Examples This production function satisfies the Inada conditions:< 3 F(K, H) A.K ³. H = FK ===// AK³²₂H ² lim — A K³² H = ₂ lim — A K-DO K-DO K4O F₁ = =—=—= A·K ²³²³. H ² ² ² & 6 K→∞ lim I lim + AK ³² H ²₂ lim / AH = = 0 — ㅎ K+∞ K-D00 K²/ HO 6 lim H∞ 6 -5 A.K ²³²³. N = {/ = 6 =—=— A. K = . H = {/ FK = 2K H lim (2K⋅H ²) + ∞0 2 lim (2K.H²) +0 lim H∞ 6 lim K² This production function doesn't satisfy the Inada conditions:< 2 2 F(K₁ H) = K² - H² -110 =1 — - A.K ²³ · H/5/6 = 00 =H+α = A·K ³ 6 HT -15/02 = 0 N FM₂ = 2K²³. H lim (2K² H) + ∞ E lim (2K² H) +0 HDO
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