Problem Set V Maxim Pechyonkin D201260019 Problem 7.3 We are given the following production function for the stools: q = 0.1 k 0.2 l 0.8 , where k represents the number of hours of bar stool lahtes used, and l represents the number of worker hours. Sam wants to produce 10 new stools. He is also facing the following budget constraint: 50 k + 50 l = 10 000, since the prices are equal, and Sam only has $10,000 to spend on renovation. (a) Sam wants to hire both inputs in the same amount, since the prices are the same. This means that k = l, inserting this as well as desirable amount of stools produced yields: 10 = 0.1 k 0.2 k 0.8 k = 100 k = l = 100, i.e. he should hire 100 hours of ether input. Sam will spend: 50 · 100 + 50 · 100 = 10 000 dollars to produce 10 new stools. (b) Norm argues that Sam should choose such inputs, so that marginal productivities are equal: MP k = ¶q ¶k = 0.02 I l k M 0.8 ; MP l = ¶q ¶l = 0.08 I k l M 0.2 MP k = MP l 0.02 I l k M 0.8 = 0.08 I k l M 0.2 I l k M 0.8 = 4 I l k M -0.2 l k = 4
Transcript
Problem Set V
Maxim Pechyonkin
D201260019
Problem 7.3
We are given the following production function for the stools:
q = 0.1 k0.2
l0.8, where k represents the number of hours of bar stool lahtes used, and l
represents the number of worker hours.
Sam wants to produce 10 new stools. He is also facing the following budget constraint:
50 k + 50 l = 10 000, since the prices are equal, and Sam only has $10,000 to spend on
renovation.
(a) Sam wants to hire both inputs in the same amount, since the prices are the same.
This means that k = l, inserting this as well as desirable amount of stools
produced yields:
10 = 0.1 k0.2
k0.8 Þ k = 100
k = l = 100, i.e. he should hire 100 hours of ether input. Sam will spend:
50 ´ 100 + 50 ´ 100 = 10 000 dollars to produce 10 new stools.
(b) Norm argues that Sam should choose such inputs, so that marginal productivities
are equal:
MPk =¶q
¶k= 0.02 I l
kM0.8
; MPl =¶q
¶l= 0.08 I k
lM0.2
MPk = MPl
0.02 I l
kM0.8
= 0.08 I k
lM0.2
I l
kM0.8
= 4 I l
kM
-0.2
l
k= 4
:l = 4 k
q = 10 = 0.1 k0.2
l0.8
Þ 10 = 0.1 k0.2H4 kL0.8
Þ :k =
10
0.1´40.8= 32.9877
l = 131.951
If Sam sticks to this plan, he will produce 10 new stools and he will spend:
50 ´ 32.9877 + 50 ´ 131.951 = 8246.94 dollars.
(c) If Sam follows Norm’s plan, but he wishes to spend all of his money to produce
more than 10 stools, he should do the following. First, he has to calculate the inputs
according to his budget constraint:
:l = 4 k from HbL
50 k + 50 l = 10 000 Sam' s budged constraint
After some simple algebra we get:
:k = 40
l = 160, these are desired inputs to spend all the Sam’s money on new stools,
which will yield:
q = 0.1 ´ 400.2 ´ 1600.8 =12.1257, which is 2.1257 stools more than the original
plan.
In fact, you can’t really produce 12.1257 stools, so the better solution is to aim at
12 stools:
:l = 4 k
q = 12 = 0.1 k0.2
l0.8
Þ 12 = 0.1 k0.2H4 kL0.8
Þ :k =
12
0.1´40.8= 39.5852
l = 158.341
E = 9896.31 total spendings
This means that Sam will produce 12 new stools, and still he will save himself
103.69 dollars, which he can probably go and spend at a good restaurant to celebrate his
successful renovation project.
(d) Carla could suggest producing only 10 stools and putting the saved money in
decorating the restaurant, which would probably bring more customers. Or she
could suggest spending this money on advertisement. Everything depends on
whether the alternative spending of savings would bring more benefits than simply
producing 12 stools insted of 10 in long run.
Problem 7.3
We are given the following production function for the stools:
q = 0.1 k0.2
l0.8, where k represents the number of hours of bar stool lahtes used, and l
represents the number of worker hours.
Sam wants to produce 10 new stools. He is also facing the following budget constraint:
50 k + 50 l = 10 000, since the prices are equal, and Sam only has $10,000 to spend on
renovation.
(a) Sam wants to hire both inputs in the same amount, since the prices are the same.
This means that k = l, inserting this as well as desirable amount of stools
produced yields:
10 = 0.1 k0.2
k0.8 Þ k = 100
k = l = 100, i.e. he should hire 100 hours of ether input. Sam will spend:
50 ´ 100 + 50 ´ 100 = 10 000 dollars to produce 10 new stools.
(b) Norm argues that Sam should choose such inputs, so that marginal productivities
are equal:
MPk =¶q
¶k= 0.02 I l
kM0.8
; MPl =¶q
¶l= 0.08 I k
lM0.2
MPk = MPl
0.02 I l
kM0.8
= 0.08 I k
lM0.2
I l
kM0.8
= 4 I l
kM
-0.2
l
k= 4
:l = 4 k
q = 10 = 0.1 k0.2
l0.8
Þ 10 = 0.1 k0.2H4 kL0.8
Þ :k =
10
0.1´40.8= 32.9877
l = 131.951
If Sam sticks to this plan, he will produce 10 new stools and he will spend:
50 ´ 32.9877 + 50 ´ 131.951 = 8246.94 dollars.
(c) If Sam follows Norm’s plan, but he wishes to spend all of his money to produce
more than 10 stools, he should do the following. First, he has to calculate the inputs
according to his budget constraint:
:l = 4 k from HbL
50 k + 50 l = 10 000 Sam' s budged constraint
After some simple algebra we get:
:k = 40
l = 160, these are desired inputs to spend all the Sam’s money on new stools,
which will yield:
q = 0.1 ´ 400.2 ´ 1600.8 =12.1257, which is 2.1257 stools more than the original
plan.
In fact, you can’t really produce 12.1257 stools, so the better solution is to aim at
12 stools:
:l = 4 k
q = 12 = 0.1 k0.2
l0.8
Þ 12 = 0.1 k0.2H4 kL0.8
Þ :k =
12
0.1´40.8= 39.5852
l = 158.341
E = 9896.31 total spendings
This means that Sam will produce 12 new stools, and still he will save himself
103.69 dollars, which he can probably go and spend at a good restaurant to celebrate his
successful renovation project.
(d) Carla could suggest producing only 10 stools and putting the saved money in
decorating the restaurant, which would probably bring more customers. Or she
could suggest spending this money on advertisement. Everything depends on
whether the alternative spending of savings would bring more benefits than simply
producing 12 stools insted of 10 in long run.
2 Problem Set 5.nb
Problem 7.3
We are given the following production function for the stools:
q = 0.1 k0.2
l0.8, where k represents the number of hours of bar stool lahtes used, and l
represents the number of worker hours.
Sam wants to produce 10 new stools. He is also facing the following budget constraint:
50 k + 50 l = 10 000, since the prices are equal, and Sam only has $10,000 to spend on
renovation.
(a) Sam wants to hire both inputs in the same amount, since the prices are the same.
This means that k = l, inserting this as well as desirable amount of stools
produced yields:
10 = 0.1 k0.2
k0.8 Þ k = 100
k = l = 100, i.e. he should hire 100 hours of ether input. Sam will spend:
50 ´ 100 + 50 ´ 100 = 10 000 dollars to produce 10 new stools.
(b) Norm argues that Sam should choose such inputs, so that marginal productivities
are equal:
MPk =¶q
¶k= 0.02 I l
kM0.8
; MPl =¶q
¶l= 0.08 I k
lM0.2
MPk = MPl
0.02 I l
kM0.8
= 0.08 I k
lM0.2
I l
kM0.8
= 4 I l
kM
-0.2
l
k= 4
:l = 4 k
q = 10 = 0.1 k0.2
l0.8
Þ 10 = 0.1 k0.2H4 kL0.8
Þ :k =
10
0.1´40.8= 32.9877
l = 131.951
If Sam sticks to this plan, he will produce 10 new stools and he will spend:
50 ´ 32.9877 + 50 ´ 131.951 = 8246.94 dollars.
(c) If Sam follows Norm’s plan, but he wishes to spend all of his money to produce
more than 10 stools, he should do the following. First, he has to calculate the inputs
according to his budget constraint:
:l = 4 k from HbL
50 k + 50 l = 10 000 Sam' s budged constraint
After some simple algebra we get:
:k = 40
l = 160, these are desired inputs to spend all the Sam’s money on new stools,
which will yield:
q = 0.1 ´ 400.2 ´ 1600.8 =12.1257, which is 2.1257 stools more than the original
plan.
In fact, you can’t really produce 12.1257 stools, so the better solution is to aim at
12 stools:
:l = 4 k
q = 12 = 0.1 k0.2
l0.8
Þ 12 = 0.1 k0.2H4 kL0.8
Þ :k =
12
0.1´40.8= 39.5852
l = 158.341
E = 9896.31 total spendings
This means that Sam will produce 12 new stools, and still he will save himself
103.69 dollars, which he can probably go and spend at a good restaurant to celebrate his
successful renovation project.
(d) Carla could suggest producing only 10 stools and putting the saved money in
decorating the restaurant, which would probably bring more customers. Or she
could suggest spending this money on advertisement. Everything depends on
whether the alternative spending of savings would bring more benefits than simply
producing 12 stools insted of 10 in long run.
Problem 7.7
We are given the following production function for the stools:
q = Β0 + Β1 k l + Β2 k + Β3 l, where 0 £ Βi £ 0
(a) Let us investigate the possibility of getting constant return to scale by increasing
all the inputs by factor of t:
q = Β0 + Β1 t k t l + Β2 t k + Β3 t l = Β0 + tI Β1 k l + Β2 k + Β3 lM,
if Β0 = 0, then qHt k, t lL = t qHk, lL
(b) Calculating marginal productivities and inspecting them:
MPk =¶
¶kI Β1 k l + Β2 k + Β3 lM =
1
2Β1
l
k+ Β2, which is homogeneous of
degree zero
¶
¶kMPk = -
1
4Β1
l
k k
< 0, which means, that marginal productivity of capital is
diminishing
Similarly:
MPl =¶
¶lI Β1 k l + Β2 k + Β3 lM =
1
2Β1
k
l+ Β3, which is homogeneous of
degree zero
¶
¶lMPl = -
1
4Β1
k
l l
< 0, which means, that marginal utility of labor is diminish-
ing
(c) Let us inspect Σ:
Σ ºfk fl
f fk,l
=
1
2Β1
l
k+Β2
1
2Β1
k
l+Β3
J Β1 k l +Β2 k+Β3 lNΒ1
4 l k
=
1
4Β
12+
1
2Β1 Β3
l
k+
1
2Β1 Β2
k
l+Β2 Β3
1
4Β
12+
1
4Β1 Β2
k
l+
1
4Β1 Β3
l
k
=
1
4Β
12+
1
4Β1 Β3
l
k+
1
4Β1 Β2
k
l+
1
4Β1 Β3
l
k+
1
4Β1 Β2
k
l+Β2 Β3
1
4Β
12+
1
4Β1 Β2
k
l+
1
4Β1 Β3
l
k
= 1 +Β1 Β2 k+Β1 Β3 l+Β2 Β3 k l
Β1 Β2 k+Β1 Β3 l+Β12
k l
Σ = 0, if Β1 Β2 k+Β1 Β3 l+Β2 Β3 k l
Β1 Β2 k+Β1 Β3 l+Β12
k l
= -1
Σ = 1, if Β1 Β2 k + Β1 Β3 l + Β2 Β3 k l = 0
Σ = ¥, if Β1 Β2 k + Β1 Β3 l + Β12
k l ® 0+
Problem Set 5.nb 3
Problem 7.7
We are given the following production function for the stools:
q = Β0 + Β1 k l + Β2 k + Β3 l, where 0 £ Βi £ 0
(a) Let us investigate the possibility of getting constant return to scale by increasing
all the inputs by factor of t:
q = Β0 + Β1 t k t l + Β2 t k + Β3 t l = Β0 + tI Β1 k l + Β2 k + Β3 lM,
if Β0 = 0, then qHt k, t lL = t qHk, lL
(b) Calculating marginal productivities and inspecting them:
MPk =¶
¶kI Β1 k l + Β2 k + Β3 lM =
1
2Β1
l
k+ Β2, which is homogeneous of
degree zero
¶
¶kMPk = -
1
4Β1
l
k k
< 0, which means, that marginal productivity of capital is
diminishing
Similarly:
MPl =¶
¶lI Β1 k l + Β2 k + Β3 lM =
1
2Β1
k
l+ Β3, which is homogeneous of
degree zero
¶
¶lMPl = -
1
4Β1
k
l l
< 0, which means, that marginal utility of labor is diminish-
