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ECO 352 – Spring 2010 – PreceptsWeeks 1, 2 – Feb. 1, 8
REVIEW OF MICROECONOMICS
Concepts to be reviewed
Budget constraint: graphical and algebraic representationPreferences, indifference curves. Utility functionMarginal rate of substitution (MRS), diminishing MRS
algebraic formulation of MRS in terms of the utility functionUtility maximization: Tangency, corner, and kink optima
Demand functions, their homogeneity propertyHomothetic preferences. Form of demand functions for theseAggregation of demand over consumersRelative demand, elasticity of substitutionSpecial cases: Linear and Leontief preferences; Cobb-DouglasRelated concepts for production: Production function. Isoquants.
Marginal products. Marginal rate of technical substitution (MRTS)
Output transformation frontier. Marginal rate of transformation (MRT)
Achieving the optimum as a market equilibrium
BUDGET CONSTRAINT
Equation : P
X
X + P
Y
Y = I
Moving along line, P
X
X + P
Y
Y = 0
So slope
Y/
X =
- P
X
/ P
Y
Or solve for Y in terms of X:Y = (I /
P
Y
) - (P
X
/ P
Y
) X
Derivative
dY/dX =
- P
X
/ P
Y
Economic interpretation:
price of X “relative to Y” or“measured in units of Y”: How much Y you have to give up to get 1 more of X
(Opportunity cost of consuming more X)
Intercepts: (I /
P
Y
) on Y-axis, (I /
P
X
) on X-axis
Economic interpretation: how much of each good could you buy
if you bought nothing of the other
X
Y
I / P
I / P
X
Y
)
)
X
Y
MARGINAL RATE OF SUBSTITUTION (MRS)MRS along an indifference curveHow much Y is the consumer willing to
give up in order to get 1 more of X Usually shown positive (numerical value)Arc: Slope of chord
MRS = ( -
Y)/(
X) = -
Y /
X
Point: slope of tangent
MRS = - dY/dX along indifference curve
If U(X,Y) represents preferences,then for a small move along an indifference curve,
U/
X)
X + (
U/
Y)
Y = 0, therefore MRS =
U/
X) / (
U/
Y)
Diminishing MRS: As X increases and Y decreases along an indifference curve,
the curve becomes flatter, and MRS decreases. Indiff. curves are convex
Intuition – As consumer has more X and less Y (retaining indifference)
values X
relatively
less: willing to give up less of Y to get even more X
This may not always be true, but failure of this assumption is not so important
in international trade context so we assume it holds.
UTILITY MAXIMIZATIONRegular case: TangencyAt optimum choice B, slopes ofbudget
line & indiff. curve equal
U/
X) / (
U/
Y) =
P
X
/ P
Y
Other possibilities:Corner solution:
Indifference curve has kink:
MRS at X = 0
smaller than P
X
/ P
Y
Most important: Leontief preferences
so don’t want to buy any X
X
X
Y
Y
COBB-DOUGLAS UTILITY FUNCTION
U(X,Y) = X
α
Y
β
, or equally valid representation ln U(X,Y) =
α
ln(X) +
β
ln(Y)
MRS = (
α
/ X ) / (
β
/ Y ) =
α
Y ) / (
β
X ) =
P
X
/ P
Y
P
X
X /
α
= P
Y
Y /
β
, so each = ( P
X
X + P
Y
Y ) / (
α
β
) = I /
α
β
Demand functions:
X = [
α
α
β
)] I / P
X
Y = [
β
α
β
)] I / P
Y
Expenditure shares:
P
X
X / I = [
α
α
β
)],
P
Y
Y / I = [
β
α
β
)]
Example on previous page:
α
β
= 1. Therefore
When
P
X
= P
Y
= 1, I = 20, X = 10, Y = 10, U = 100
When
P
X
= 2, P
Y
= 1, I = 20, X = 5, Y = 10, U = 50
When
P
X
= 2, P
Y
= 1, for general I, X = I/4, Y = I/2, U = I
2
so to achieve old U = 100 needs I
2
= 800, or I =
HOMOTHETIC PREFERENCESIndifference is preserved by equiproportionate scale changes for all goods:
indifference curves are radial magnifications or reductions of each other.
In figure, MRS at A same as at C
MRS at B same as at D
So MRS depends only on ratio Y/X,
not on absolute scale
Cobb-Douglas is an example:
MRS = (
α
β
) (Y / X)
Useful properties:[1] Tangency condition solves to
Relative demand function
Y / X = f(P
X
/ P
Y
The elasticity of this function is the elasticity of substitution in consumption.For Cobb-Douglas it = 1. For Leontief, it = 0.Straight line preferences (perfect substitutes) is the limiting case, el. of sub. =
20
15
10
5
0 20 15 10 5 0
X
Y
A
C B
D
PRODUCTIONConcepts are analogous, with simple reinterpretations of those for consumption:Preference map Isoquant mapUtility function Production function (PLUS output quantities are cardinal)MRS Marginal rate of technical (input) substitution (MRTS)Additional useful concept:
Marginal product. If output Q = F(K,L), marginal products are
Q/
K,
Q/
L
Cobb-Douglas production function Q = K
α
L
β
Exercise: calculate its marginal products
Returns to scale: If both inputs are doubled, output becomes
(2K)
α
(2L)
β
α+
β
K
α
L
β
, = or < 2 K
α
L
β
as
α
β
, = or < 1
Perfect competition requires constant or diminishing returns to scale.
OUTPUT TRANSFORMATION FRONTIERTT' = Output transformation orproduction possibility frontierMRT = its slope at a point.Assume exact aggregation,social indifference curvesgenerate aggregate demands.Then social optimum maximizessocial preference over TT'Optimum can be achieved as acompetitive market equilibrium,Relative price = slope of PP'the common tangent
x
y
T
T
´
x
y
T
T
´
I
I´
P
P
´
