3 Consider the production function of candles, with capital (K) and labour (L) as inputs: Qe = Kº.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 Consider the production function of candles, with capital (K) and labour (L) as inputs: Qe = Kº.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.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![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
125,2
r and wages paid to workers w, as follows: K =
_8w and L =
10r3
Use the Jacobian
derivative matrix of Qe and Qeg to calculate how production changes in response to an
aQ.
ar
increase in interest rates:
when r = 2 and w = 1000.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87b53351-2067-4a4d-a81c-a5cf0ffaa5a8%2Fb1ffb71d-7410-403b-b4d1-87aeb586cd27%2Flhkn13c_processed.jpeg&w=3840&q=75)
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
125,2
r and wages paid to workers w, as follows: K =
_8w and L =
10r3
Use the Jacobian
derivative matrix of Qe and Qeg to calculate how production changes in response to an
aQ.
ar
increase in interest rates:
when r = 2 and w = 1000.
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