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3 Consider the production function of candles, with capital (K) and labour (L) as inputs: Qc = K0.1L0.9 .1 Show that both additional labour and additional capital increase production at a decreasing rate by computing and referring to some of the second derivatives of Qc. .2 Now consider the production function of an additional product, electrical generators, de- fined by the following function: Qeg K0.6L0.4 Show, using the Jacobian derivative matrix of Qc and Qeg, how increasing capital by 10 units and decreasing labour by 5 units, would approximately change production of both goods. Start from an initial combination of K 100 and L = 50. = .3 Suppose that demand for capital and labour are in turn determined by the interest rate r and wages paid to workers w, as follows: K Use the Jacobian derivative matrix of Qc and Qeg to calculate how production changes in response to an 8w 10r3 and L= 125r² vw. increase in interest rates: ƏQc Ər ƏQeg Ər when r = 2 and w = 1000. =

3 Consider the production function of candles, with capital (K) and labour (L) as inputs: Qc = K0.1L0.9 .1 Show that both additional labour and additional capital increase production at a decreasing rate by computing and referring to some of the second derivatives of Qc. .2 Now consider the production function of an additional product, electrical generators, de- fined by the following function: Qeg K0.6L0.4 Show, using the Jacobian derivative matrix of Qc and Qeg, how increasing capital by 10 units and decreasing labour by 5 units, would approximately change production of both goods. Start from an initial combination of K 100 and L = 50. = .3 Suppose that demand for capital and labour are in turn determined by the interest rate r and wages paid to workers w, as follows: K Use the Jacobian derivative matrix of Qc and Qeg to calculate how production changes in response to an 8w 10r3 and L= 125r² vw. increase in interest rates: ƏQc Ər ƏQeg Ər when r = 2 and w = 1000. =

Microeconomic Theory
Microeconomic Theory
12th Edition
ISBN:9781337517942
Author:NICHOLSON
Publisher:NICHOLSON
Chapter9: Production Functions
Section: Chapter Questions
Problem 9.2P
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Could you please help with 3.2 and 3.3.

3 Consider the production function of candles, with capital (K) and labour (L) as inputs: Qc = K0.1 L0.9. 3.1 Show that both additional labour and additional capital increase production at a decreasing rate by computing and referring to some of the second derivatives of Qe. 3.2 Now consider the production function of an additional product, electrical generators, de- fined by the following function: Qeg K0.6 L0.4. Show, using the Jacobian derivative matrix of Q. and Qeg, how increasing capital by 10 units and decreasing labour by 5 units, would approximately change production of both goods. Start from an initial combination of K = 100 and L = 50. 3.3 Suppose that demand for capital and labour are in turn determined by the interest rate r and wages paid to workers w, as follows: K 8w and L 125r2 Use the Jacobian 10r3 derivative matrix of Qc and Q to calculate how production changes in response to an eg increase in interest rates: when r = 2 and w = 1000. aQeg ar
Transcribed Image Text:3 Consider the production function of candles, with capital (K) and labour (L) as inputs: Qc = K0.1 L0.9. 3.1 Show that both additional labour and additional capital increase production at a decreasing rate by computing and referring to some of the second derivatives of Qe. 3.2 Now consider the production function of an additional product, electrical generators, de- fined by the following function: Qeg K0.6 L0.4. Show, using the Jacobian derivative matrix of Q. and Qeg, how increasing capital by 10 units and decreasing labour by 5 units, would approximately change production of both goods. Start from an initial combination of K = 100 and L = 50. 3.3 Suppose that demand for capital and labour are in turn determined by the interest rate r and wages paid to workers w, as follows: K 8w and L 125r2 Use the Jacobian 10r3 derivative matrix of Qc and Q to calculate how production changes in response to an eg increase in interest rates: when r = 2 and w = 1000. aQeg ar
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