清华大学中级微观经济学讲义
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This gives us: pMP1=w1
Important: We have assumed that MP1 is decreasing in x1
.
10
Comparative Statics of Short-Run Profit-Maximization
What happens to the short-run profit-maximizing production plan as the output price p changes?
function and the maximal iso-profit line are
equal.
y*
M P1
w1 p
Slopesw1 p
a
t
(
x
* 1
,
x~
2
,
y*
)
x
* 1
x1
.
8
Short-Run Profit-Maximization
M P 1w p 1 pM P 1w 1
pMP1 is the marginal revenue product of input 1, the rate at which revenue increases with the amount used of input 1. If p M P 1w 1then profit increases with x1. If p M P 1w 1then profit decreases with x1.
.
11
Comparative Statics of Short-Run Profit-Maximization
The equation of a short-run iso-profit line
is
yw p 1x1p w 2x ~2
so an increase in p causes a reduction in the slope of the family of iso-profit lines
.Байду номын сангаас
9
A Mathematical Approach to Short-Run Profit-Maximization
Mathematically, the firm’s short run
problem is: Maximize p y w 1 x 1 w 2 x ~ 2 .
Subject to: y f(x 1 ,x ~ 2 ).
y
y*
y f(x 1 ,x ~ 2 )
Slopesw1 p
x
* 1
.
x1
14
A Useful Math: The Envelope
Theorem
Suppose x* maximizes g (x; t), where t is a parameter;
Then x* varies with t, i.e, x*=x*(t);
y
y f(x 1 ,x ~ 2 )
Slopesw1 p
x1
.
7
Short-Run Profit-Maximization
At the short-run profit-maximizing plan, y the slopes of the short-run production
We have
gx [x*(t), t]=0; Let g*(t) = Max g(x,t) = g[x*(t), t]
.
2
Economic Profit
The economic profit generated by (x1,…,xm,y1,…,yn) is
p 1 y 1 p n y n w 1 x 1 w m x m .
Notes: For the time being, w e restrict to the case of
y f(x 1 ,x ~ 2 ).
The firm’s fixed cost is F C w 2 x ~ 2 and its profit function is
p y w 1 x 1 w 2 x ~ 2 .
.
4
Short-Run Iso-Profit Lines
An iso-profit line contains all the production plans that yield the same profit level.
The equation of an iso-profit line is
p y w 1 x 1 w 2 x ~ 2 . I.e. yw p 1x1p w 2x ~2.
.
5
Short-Run Iso-Profit Lines
y
Slopesw1 p
x1
.
6
Short-Run Profit-Maximization
Profit-Maximization
.
1
What Do We Do in this Chapter?
After working our “producer’s budget sets” (production sets), We are working on “producer’s choices” Please pay attention to the similarity and differences between “producer’s choices” and “consumer’s choices”
a competitive firm, which is a tiny relative to the market size and takes prices p1,…,pn w1,…,wm as given constants;
.
3
Short-run Economic Profit
Suppose the firm is in a short-run circumstance in which x2x ~2. Its short-run production function is
.
12
Comparative Statics of Short-Run
Profit-Maximization
y
y f(x 1 ,x ~ 2 )
y* Slopesw1 p
x
* 1
.
x1
13
Comparative Statics of Short-Run
Profit-Maximization
Important: We have assumed that MP1 is decreasing in x1
.
10
Comparative Statics of Short-Run Profit-Maximization
What happens to the short-run profit-maximizing production plan as the output price p changes?
function and the maximal iso-profit line are
equal.
y*
M P1
w1 p
Slopesw1 p
a
t
(
x
* 1
,
x~
2
,
y*
)
x
* 1
x1
.
8
Short-Run Profit-Maximization
M P 1w p 1 pM P 1w 1
pMP1 is the marginal revenue product of input 1, the rate at which revenue increases with the amount used of input 1. If p M P 1w 1then profit increases with x1. If p M P 1w 1then profit decreases with x1.
.
11
Comparative Statics of Short-Run Profit-Maximization
The equation of a short-run iso-profit line
is
yw p 1x1p w 2x ~2
so an increase in p causes a reduction in the slope of the family of iso-profit lines
.Байду номын сангаас
9
A Mathematical Approach to Short-Run Profit-Maximization
Mathematically, the firm’s short run
problem is: Maximize p y w 1 x 1 w 2 x ~ 2 .
Subject to: y f(x 1 ,x ~ 2 ).
y
y*
y f(x 1 ,x ~ 2 )
Slopesw1 p
x
* 1
.
x1
14
A Useful Math: The Envelope
Theorem
Suppose x* maximizes g (x; t), where t is a parameter;
Then x* varies with t, i.e, x*=x*(t);
y
y f(x 1 ,x ~ 2 )
Slopesw1 p
x1
.
7
Short-Run Profit-Maximization
At the short-run profit-maximizing plan, y the slopes of the short-run production
We have
gx [x*(t), t]=0; Let g*(t) = Max g(x,t) = g[x*(t), t]
.
2
Economic Profit
The economic profit generated by (x1,…,xm,y1,…,yn) is
p 1 y 1 p n y n w 1 x 1 w m x m .
Notes: For the time being, w e restrict to the case of
y f(x 1 ,x ~ 2 ).
The firm’s fixed cost is F C w 2 x ~ 2 and its profit function is
p y w 1 x 1 w 2 x ~ 2 .
.
4
Short-Run Iso-Profit Lines
An iso-profit line contains all the production plans that yield the same profit level.
The equation of an iso-profit line is
p y w 1 x 1 w 2 x ~ 2 . I.e. yw p 1x1p w 2x ~2.
.
5
Short-Run Iso-Profit Lines
y
Slopesw1 p
x1
.
6
Short-Run Profit-Maximization
Profit-Maximization
.
1
What Do We Do in this Chapter?
After working our “producer’s budget sets” (production sets), We are working on “producer’s choices” Please pay attention to the similarity and differences between “producer’s choices” and “consumer’s choices”
a competitive firm, which is a tiny relative to the market size and takes prices p1,…,pn w1,…,wm as given constants;
.
3
Short-run Economic Profit
Suppose the firm is in a short-run circumstance in which x2x ~2. Its short-run production function is
.
12
Comparative Statics of Short-Run
Profit-Maximization
y
y f(x 1 ,x ~ 2 )
y* Slopesw1 p
x
* 1
.
x1
13
Comparative Statics of Short-Run
Profit-Maximization