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The key concepts of microeconomics to be reviewed in the ECO 352 course during weeks 1 and 2 of Spring 2010. Topics include budget constraints, preferences and utility, marginal rate of substitution, utility maximization, demand functions, and related production concepts. Students will learn about concepts such as indifference curves, utility functions, and the tangency condition for optimization.
What you will learn
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ECO 352 – Spring 2010 – PreceptsWeeks 1, 2 – Feb. 1, 8
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
Equation : P
X
Y
Moving along line, P
X
Y
So slope
X
Y
Or solve for Y in terms of X:Y = (I /
Y
X
Y
Derivative
dY/dX =
X
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 /
Y
) on Y-axis, (I /
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
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,
Y = 0, therefore MRS =
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
X
Y
Other possibilities:Corner solution:
Indifference curve has kink:
MRS at X = 0
smaller than P
X
Y
Most important: Leontief preferences
so don’t want to buy any X
α
β
, or equally valid representation ln U(X,Y) =
α
ln(X) +
β
ln(Y)
α
β
α
β
X
Y
X
α
Y
β
, so each = ( P
X
Y
α
β
α
β
Demand functions:
α
α
β
X
β
α
β
Y
Expenditure shares:
X
α
α
β
Y
β
α
β
Example on previous page:
α
β
= 1. Therefore
When
X
Y
When
X
Y
When
X
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:
α
β
Useful properties:[1] Tangency condition solves to
Relative demand function
Y / X = f(P
X
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
Cobb-Douglas production function Q = K
α
β
Exercise: calculate its marginal products
Returns to scale: If both inputs are doubled, output becomes
α
β
α+
β
α
β
, = or < 2 K
α
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
y
T
T
´
y
T
T
´
I
I´
P
P
´