3 Consider the production function of candles, with capital (K) and labour (L) as inputs: Qe=K0.1L0.9 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 = K0.6L0.4. Show, using the Jacobian derivative matrix of Qe 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=1252. Use the Jacobian derivative matrix of Qe and Qeg to calculate how production changes in response to an 10-3 aQc Ər increase in interest rates: when r 2 and w = 1000. aQeg Ər
3 Consider the production function of candles, with capital (K) and labour (L) as inputs: Qe=K0.1L0.9 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 = K0.6L0.4. Show, using the Jacobian derivative matrix of Qe 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=1252. Use the Jacobian derivative matrix of Qe and Qeg to calculate how production changes in response to an 10-3 aQc Ər increase in interest rates: when r 2 and w = 1000. aQeg Ər
Chapter9: Production Functions
Section: Chapter Questions
Problem 9.2P
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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 Qe 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
125. Use the Jacobian
r and wages paid to workers w, as follows: K =
derivative matrix of Qe and Qeg to calculate how production changes in response to an
and L =
10r3
ar
increase in interest rates:
when r = 2 and w = 1000.
Or
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