Properties of Production Sets
edit
The production vector
Y
=
(
y
1
,
y
2
,
…
y
n
)
{\displaystyle Y=(y_{1},y_{2},\ldots y_{n})}
y
i
>
0
{\displaystyle y_{i}>0}
y
i
<
0
{\displaystyle y_{i}<0}
Y is non empty
Y is closed (includes its boundary)
No free lunch -
y
≥
0
→
Y
=
0
{\displaystyle y\geq 0\rightarrow Y=0}
possibility of inaction
(
0
∈
Y
)
{\displaystyle (0\in Y)}
Free disposal
Irreversability - can't make output into inputs
Returns to scale:
Non-increasing:
∀
y
∈
Y
,
α
y
∈
Y
∀
α
∈
[
0
,
1
]
{\displaystyle \forall y\in Y,\,\alpha y\in Y\forall \alpha \in [0,1]}
Non-decreasing:
∀
y
∈
Y
,
α
y
∈
Y
∀
α
>
1
{\displaystyle \forall y\in Y,\,\alpha y\in Y\forall \alpha >1}
Constant:
∀
y
∈
Y
,
α
y
∈
Y
∀
α
≥
0
{\displaystyle \forall y\in Y,\,\alpha y\in Y\forall \alpha \geq 0}
Additivity:
y
∈
Y
and
y
′
∈
Y
→
y
+
y
′
∈
Y
{\displaystyle y\in Y{\mbox{ and }}{y}^{\prime }\in Y\rightarrow y+{y}^{\prime }\in Y}
Convexity:
y
,
y
′
∈
Y
and
a
∈
[
0
,
1
]
→
a
y
+
(
1
−
a
)
y
′
∈
Y
{\displaystyle y,{y}^{\prime }\in Y{\mbox{ and }}a\in [0,1]\rightarrow ay+(1-a){y}^{\prime }\in Y}
max
p
2
y
2
−
p
1
y
1
s.t.
[
y
1
,
y
2
]
∈
Y
f
(
y
1
,
y
2
)
≤
k
L
(
y
1
,
y
2
,
λ
)
=
p
2
y
2
−
p
1
y
1
+
λ
[
k
−
f
(
y
1
,
y
2
)
]
L
1
=
−
p
1
−
λ
f
1
=
0
L
2
=
p
1
−
λ
f
2
=
0
L
λ
=
k
−
f
(
y
1
,
y
2
)
=
0
{\displaystyle {\begin{aligned}\max &\;p_{2}y_{2}-p_{1}y_{1}\\{\mbox{s.t. }}&[y_{1},y_{2}]\in Y\\&f(y_{1},y_{2})\leq k\\{\mathcal {L}}(y_{1},y_{2},\lambda )&=p_{2}y_{2}-p_{1}y_{1}+\lambda [k-f(y_{1},y_{2})]\\{\mathcal {L}}_{1}&=-p_{1}-\lambda f_{1}=0\\{\mathcal {L}}_{2}&=p_{1}-\lambda f_{2}=0\\{\mathcal {L}}_{\lambda }&=k-f(y_{1},y_{2})=0\\\end{aligned}}}
y
=
f
(
Z
)
{\displaystyle y=f(Z)}
Z
=
(
z
1
,
z
2
,
…
,
z
n
)
{\displaystyle Z=(z_{1},z_{2},\ldots ,z_{n})}
max
y
,
Z
p
y
−
w
subject to
y
=
f
(
z
)
-or-
max
Z
p
f
(
Z
)
−
w
Z
∂
L
∂
z
i
=
p
f
i
−
w
i
≤
0
{\displaystyle {\begin{aligned}&\max _{y,Z}py-w\\{\mbox{subject to }}&y=f(z)\\&{\mbox{ -or- }}\\&\max _{Z}pf(Z)-wZ\\{\frac {\partial {\mathcal {L}}}{\partial z_{i}}}&=pf_{i}-w_{i}\leq 0\\\end{aligned}}}
marginal revenue product
edit
Marginal revenue product is the price of output times the marginal product of input
M
R
P
=
p
⋅
f
i
{\displaystyle MRP=p\cdot f_{i}}
p
f
i
=
w
i
∀
z
i
>
0
{\displaystyle pf_{i}=w_{i}\;\forall z_{i}>0}
marginal rate of techical substitution
edit
p
f
1
=
w
1
p
f
2
=
w
2
→
f
1
f
2
=
w
1
w
2
f
(
z
1
,
z
2
)
=
y
¯
f
1
d
z
1
+
f
2
d
z
2
=
0
d
z
2
d
z
1
=
−
f
1
f
2
{\displaystyle {\begin{aligned}pf_{1}&=w_{1}\\pf_{2}&=w_{2}\\&\rightarrow {\frac {f_{1}}{f_{2}}}={\frac {w_{1}}{w_{2}}}f(z_{1},z_{2})&={\bar {y}}\\f_{1}dz_{1}&+f_{2}dz_{2}=0\\{\frac {dz_{2}}{dz_{1}}}&=-{\frac {f_{1}}{f_{2}}}\end{aligned}}}
Properties of profit functions and supply
edit
Profit functions exhibit homogeneity of degree 1
π
=
p
f
(
z
)
−
w
z
{\displaystyle \pi =pf(z)-w{z}}
supply functions exhibit homogeneity of degree 0
The optimal CMP gives cost function <align>\funcd{c}{w,q}</align>
min
z
1
,
z
2
w
1
z
1
+
w
2
z
2
s.t.
f
(
z
1
,
z
2
)
≤
q
L
(
z
1
,
z
2
,
λ
)
=
w
1
z
1
+
w
2
z
2
−
λ
[
f
(
z
1
,
z
2
)
−
q
]
L
1
=
w
1
−
λ
f
1
=
0
L
2
=
w
2
−
λ
f
2
=
0
L
λ
=
f
(
z
1
,
z
2
)
−
q
=
0
{\displaystyle {\begin{aligned}\min _{z_{1},z_{2}}&\,w_{1}z_{1}+w_{2}z_{2}{\mbox{ s.t. }}f(z_{1},z_{2})\leq q\\{\mathcal {L}}(z_{1},z_{2},\lambda )&=w_{1}z_{1}+w_{2}z_{2}-\lambda [f(z_{1},z_{2})-q]\\{\mathcal {L}}_{1}&=w_{1}-\lambda f_{1}=0\\{\mathcal {L}}_{2}&=w_{2}-\lambda f_{2}=0\\{\mathcal {L}}_{\lambda }&=f(z_{1},z_{2})-q=0\\\end{aligned}}}
w
1
w
2
=
f
1
f
2
m
p
1
m
p
2
{\displaystyle {\frac {w_{1}}{w_{2}}}={\frac {f_{1}}{f_{2}}}\;{\frac {mp_{1}}{mp_{2}}}}
w
1
f
1
=
w
2
f
2
{\displaystyle {\frac {w_{1}}{f_{1}}}={\frac {w_{2}}{f_{2}}}}
z
2
{\displaystyle z_{2}}
C
(
w
1
,
w
2
,
q
)
=
w
1
z
1
∗
+
w
2
z
2
∗
=
w
1
z
1
∗
+
w
2
z
2
∗
−
λ
[
f
(
z
1
∗
,
z
2
∗
)
−
q
]
∂
C
∂
w
1
=
z
1
∗
∂
C
∂
w
2
=
z
2
∗
∂
C
∂
q
=
λ
- marginal cost
{\displaystyle {\begin{aligned}{C}(w_{1},w_{2},q)&=w_{1}z_{1}^{*}+w_{2}z_{2}^{*}\\&=w_{1}z_{1}^{*}+w_{2}z_{2}^{*}-\lambda [f(z_{1}^{*},z_{2}^{*})-q]\\{\frac {\partial C}{\partial w_{1}}}&=z_{1}^{*}\\{\frac {\partial C}{\partial w_{2}}}&=z_{2}^{*}\\{\frac {\partial C}{\partial q}}&=\lambda {\mbox{- marginal cost}}\\\end{aligned}}}
The solution to the CMP gives factor demands,
z
i
∗
=
z
i
(
w
,
q
)
{\displaystyle z_{i}^{*}=z_{i}({w},q)}
∑
w
i
z
i
=
c
(
w
,
q
)
{\displaystyle \sum w_{i}z_{i}=c(w,q)}
![{\displaystyle \forall y\in Y,\,\alpha y\in Y\forall \alpha \in [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bd2a2b637ae0004a05dcf1c70863e4577cb6b90)
![{\displaystyle y,{y}^{\prime }\in Y{\mbox{ and }}a\in [0,1]\rightarrow ay+(1-a){y}^{\prime }\in Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13772dd431db09aadecfb1d91935cdeb25419396)
![{\displaystyle {\begin{aligned}\max &\;p_{2}y_{2}-p_{1}y_{1}\\{\mbox{s.t. }}&[y_{1},y_{2}]\in Y\\&f(y_{1},y_{2})\leq k\\{\mathcal {L}}(y_{1},y_{2},\lambda )&=p_{2}y_{2}-p_{1}y_{1}+\lambda [k-f(y_{1},y_{2})]\\{\mathcal {L}}_{1}&=-p_{1}-\lambda f_{1}=0\\{\mathcal {L}}_{2}&=p_{1}-\lambda f_{2}=0\\{\mathcal {L}}_{\lambda }&=k-f(y_{1},y_{2})=0\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba3fde8c9cc0fb48a85cc4027131d6d30d6b6480)
![{\displaystyle {\begin{aligned}\min _{z_{1},z_{2}}&\,w_{1}z_{1}+w_{2}z_{2}{\mbox{ s.t. }}f(z_{1},z_{2})\leq q\\{\mathcal {L}}(z_{1},z_{2},\lambda )&=w_{1}z_{1}+w_{2}z_{2}-\lambda [f(z_{1},z_{2})-q]\\{\mathcal {L}}_{1}&=w_{1}-\lambda f_{1}=0\\{\mathcal {L}}_{2}&=w_{2}-\lambda f_{2}=0\\{\mathcal {L}}_{\lambda }&=f(z_{1},z_{2})-q=0\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4de4110341e16e71eeb0111affe082fe383de99c)
![{\displaystyle {\begin{aligned}{C}(w_{1},w_{2},q)&=w_{1}z_{1}^{*}+w_{2}z_{2}^{*}\\&=w_{1}z_{1}^{*}+w_{2}z_{2}^{*}-\lambda [f(z_{1}^{*},z_{2}^{*})-q]\\{\frac {\partial C}{\partial w_{1}}}&=z_{1}^{*}\\{\frac {\partial C}{\partial w_{2}}}&=z_{2}^{*}\\{\frac {\partial C}{\partial q}}&=\lambda {\mbox{- marginal cost}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/558c7476c10d26a35f511e036fc9fbeb67cbb60e)
