15. The production q (in units) of a certain good in terms of the amount q of labour invested and the amount qc of capital invested is given by the production function q = 2qL0.5.q.5. At this moment, 6 units of labour and 6 units of capital are invested daily. So, by the production function, 12 units are produced daily. The graphs below both show the contour line of level 12 of the production function q. 20 18 16 14 12 10 8 6 4 2 qc 1 2 3 4 5 6 7 (a) Contour line of level 12 8 20 18 16 14 12 10 8 6 4 2 SH qL 9 10 qc 1 2 3 45 6 7 (b) Contour line of level 12 Figure 7: Contour plots to draw on qL 8 9 10 (a) Give an implicit and explicit equation of this contour line. (b) Assume for this and the later parts of the question that a unit of labour costs €84 and a unit of capital costs €35. Moreover, there is a fixed cost of €486 per day. Give the equation of the cost function c (qL, qc). (c) Graph, in Figure 7a, a line that allows you to decide that there exists another combination (qL, qc) to generate 12 units a day with exactly the same cost as the combination (qL, 9c) = (6,6). (d) Explain how you can approximately read this second combination from the graph above. (e) Calculate the second combination mentioned in part d exactly.
15. The production q (in units) of a certain good in terms of the amount q of labour invested and the amount qc of capital invested is given by the production function q = 2qL0.5.q.5. At this moment, 6 units of labour and 6 units of capital are invested daily. So, by the production function, 12 units are produced daily. The graphs below both show the contour line of level 12 of the production function q. 20 18 16 14 12 10 8 6 4 2 qc 1 2 3 4 5 6 7 (a) Contour line of level 12 8 20 18 16 14 12 10 8 6 4 2 SH qL 9 10 qc 1 2 3 45 6 7 (b) Contour line of level 12 Figure 7: Contour plots to draw on qL 8 9 10 (a) Give an implicit and explicit equation of this contour line. (b) Assume for this and the later parts of the question that a unit of labour costs €84 and a unit of capital costs €35. Moreover, there is a fixed cost of €486 per day. Give the equation of the cost function c (qL, qc). (c) Graph, in Figure 7a, a line that allows you to decide that there exists another combination (qL, qc) to generate 12 units a day with exactly the same cost as the combination (qL, 9c) = (6,6). (d) Explain how you can approximately read this second combination from the graph above. (e) Calculate the second combination mentioned in part d exactly.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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could you please answer with pen and paper ?
Transcribed Image Text:15. The production q (in units) of a certain good in terms of the amount q of labour invested and
the amount qc of capital invested is given by the production function q = 2.qL0.5.qc.5. At this
moment, 6 units of labour and 6 units of capital are invested daily. So, by the production function, 12
units are produced daily. The graphs below both show the contour line of level 12 of the production
function q.
20
18
16
14
12
10
8
6
4
2
098
1
2 3 4 5 6 7
(a) Contour line of level 12
8
20
18
16
14
12
10
8
6
4
2
qL
9 10
qc
1
2 3 4 5 6 7
(b) Contour line of level 12
Figure 7: Contour plots to draw on
qL
8 9 10
(a) Give an implicit and explicit equation of this contour line.
(b) Assume for this and the later parts of the question that a unit of labour costs €84 and a unit
of capital costs €35. Moreover, there is a fixed cost of €486 per day.
Give the equation of the cost function c (qL, 9c).
(c) Graph, in Figure 7a, a line that allows you to decide that there exists another combination (qL, qc)
to generate 12 units a day with exactly the same cost as the combination (qL, 9c) = (6,6).
(d) Explain how you can approximately read this second combination from the graph above.
(e) Calculate the second combination mentioned in part d exactly.
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