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Lagrange Method. Lagrange Method. Why do we want the axioms 1 – 7 of consumer theory? Answer: We like an easy life!. By that we mean that we want well behaved demand curves. Let’s look at a Utility Function: U = U( ,y) Take the total derivative:. - PowerPoint PPT Presentation
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Lagrange Method
Lagrange Method
• Why do we want the axioms 1 – 7 of consumer theory?
• Answer: We like an easy life!
By that we mean that we want well behaved demand curves.
Let’s look at a Utility Function: U = U(,y)
Take the total derivative:
dy
y
yxUdx
x
yxUdU
,,
1131.2 dU
0233.2 dU
For example if MUx = 2 MUy = 3
dyMUdxMUdU yx
Look at the special case of the total derivative along a given indifference curve:
10U
dy
dx
0
dyy
Udx
x
UdU
dyy
Udx
x
U
dx
dy
yUxU
xMRSydx
dy
MUy
MUx,
• Taking the total derivative of a B.C. yields
Px dx + Py dy = dM
• Along a given B.C. dM = 0
Px dx + Py dy = 0
MyPxP yx
Py
Px
dx
dy
y
x
=> Slope of the Indifference Curve
= Slope of the Budget Constraint
Py
Px
yUxU
MU
MU
y
x
Equilibrium
x
y
We have a general method for finding a point of tangency between an Indifference Curve and the
Budget Constraint:
The Lagrange Method
Widely used in Commerce, MBA’s
and Economics.
y
xy
x
u0
u1u2
Idea: Maximising U(x,y) is like climbing happiness mountain.
But we are restricted by how high we can go
since must stay on BC - (path on mountain).
y
x
u0
u1
u2
So to move up happiness Mountain is subject to being on a
budget constraint path.
Maximize U (x,y) subject to Pxx+ Pyy=M
yPxPMyxULMax yxyx
),(,
Known: Px, Py & M Unknowns: x,y,l
3 Equations: 3 Unknowns: Solve
32.. yxUge
yPxPMyxLMax yx 32
xx PxyL 32.1
PyyxLy 223.2
yPxPML yx .3
= 0
= 0
= 0
Trick:
But:
32xy
xx
yx 22 32
x
yx 322 Note:
U
Known: Px, Py & M Unknowns: x,y,
3 Equations: 3 Unknowns: Solve
32 yxU
yPxPMyxLMax yx 32
xx PxyL 32.1
PyyxLy 223.2
yPxPML yx .3
= 0
= 0
= 0
xPx
U 2
Pyy
U 3
32 yxU
yPxPMyxLMax yx 32
02
.1 xx Px
UL
03
.2 Pyy
ULy
0.3 yPxPML yx
xPx
U 2
Pyy
U 3
yx yP
U
xP
U 32
yx yPxP
32
y
x
P
P
x
y
3
2
32 yxU
yPxPMyxLMax yx 32
02
.1 xx Px
UL
03
.2 Pyy
ULy
xPx
U 2
Pyy
U 3
Notice:
U = x2 y3
xMUx
U
x
yxxy
x
U
22
232
3yMU
y
U
y
U
3
MUy
MUx
yUxU
yUxU
3
2
U
y
x
U
3
2
x
y
3
2
Recall Slope of Budget Constraint = y
x
P
P
y
x
y
x
P
P
x
y
MU
MUSo
3
2
<=> Slope of the Indifference Curve
Slope of IC = slope of BC
Back to the Problem:
Py
Px
x
y
3
2
yPxPM yx
xP
Py
y
x2
3
xP
PPxPM
y
xyx
2
3
y
x
P
P
x
y
3
2
+
But
But +
Back to the Problem:
xP
Py
y
x2
3
xPxPM xx 2
3
y
x
P
P
x
y
3
2
xPM x
2
31
+
But
But +
Py
Px
x
y
3
2
yPxPM yx
xPM x
2
5
xP
Mx
5
2
xx
D PMxP
Mx ,
5
2
So the Demand Curve for x when U=x2y3
If M=100:
x
D
Px
5
200
Demand Curve for X
0
2
4
6
8
10
12
Quantity
Pri
ce
of
x
Price of x 10 8 5 2
4 5 8 20
PxxD
10 4
8 5
5 8
2 20
Recall that: U = x2 y3
Let: U = xa yb
For Cobb - Douglas Utility Function
xPM
x
D
P
M
ba
ax
32
2
x
D
P
Mx
5
2
Note that: Cobb-Douglas is a special result
In general: MPPxx yx ,,
MPPyy yx ,,
MPxx x ,
MPyy y ,
For Cobb - Douglas:
Why does the demand for x not depend on py?
Share of x in income =
M
Pxs x
x
.
In this example: M
Ps x
x M
P
P
Ms x
xx
5
2
5
2
5
2
M
P
P
Ms x
xx Constant
Similarly share of y in income is constant:
5
3
5
3
M
P
P
Ms y
yy
So if the share of x and y in income is constant => change in Px only effects demand for x in C.D.
yPxPMyxL yx 32
2U
1U
M
U
ML So l tells us the change in U as M rises
Increase M
Increase from U1 to U2
in constraint
ConstraintObjective fn
in objective fn
