Lily Wong (CUTIE)
Econ 1011
6d. The Calculus of Optimization
As we know from the previous section, the goal of consumer is to Maximize Utility. There are two conditions
to Maximize Utility:
1.
Slope of Indifference Curve = Slope of Budget Line
In other words,
Brief Definitions
:
*Marginal Rate of Substitution (MRS)
MRS is the maximum amount of y that a consumer is willing to give up to get one more x.
Thus,
MRS =
¶
Y /
¶
X = dY/dX = Slope of IC
Or
MRS = MUx/MUy
where MUx =
∆
U/
∆
X =
∂
U/
∂
X; MUy =
∆
U/
∆
Y =
∂
U/
∂
Y
Proof
:
Given: U(X,Y)
Take total differential
dU =
∂
U/
∂
X • dX +
∂
U/
∂
Y • dY
dU = MUx • dX + MUy • dY
Along the IC, dU = 0
Thus, 0 = MUx • dX + MUy • dY
dY/dX = MUx/MUy
*Slope of Budget Line (also called Budget Constraint)
In Figure 6.d.1, the blue line is the Budget Constraint. In order to find the intercept of X
which represents the quantity of X, we have to divide the income (I) by the price of X (Px).
Similarly, we have to divide the income (I) by the price of Y (Py) to find the intercept of Y
which represents the quantity of Y. Thus,
The intercept of X = I/ Px
The
intercept
of
Y = I/ Py
Then,
Slope of B.C.= Rise/Run
= (I/ Py) / (I/ Px)
= (I/ Py) • (Px /I)
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 Spring '07
 Cheng
 Optimization, Utility, Px, blue line, Lily Wong, py, maximize utility

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