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Microeconomics Precepts: Spring 2010 - Weeks 1, 2 Review, Study Guides, Projects, Research of Microeconomics

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

  • How are demand functions derived from utility maximization and what is their homogeneity property?
  • What are the different possibilities for utility maximization and how do they affect consumer behavior?
  • How is the marginal rate of substitution calculated along an indifference curve?
  • What is the relationship between preferences, indifference curves, and utility functions?
  • What is the budget constraint equation and how is it used to determine the slope of the budget line?

Typology: Study Guides, Projects, Research

2021/2022

Uploaded on 09/12/2022

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ECO 352 – Spring 2010 – Precepts
Weeks 1, 2 – Feb. 1, 8
REVIEW OF MICROECONOMICS
Concepts to be reviewed
Budget constraint: graphical and algebraic representation
Preferences, indifference curves. Utility function
Marginal rate of substitution (MRS), diminishing MRS
algebraic formulation of MRS in terms of the utility function
Utility maximization: Tangency, corner, and kink optima
Demand functions, their homogeneity property
Homothetic preferences. Form of demand functions for these
Aggregation of demand over consumers
Relative demand, elasticity of substitution
Special cases: Linear and Leontief preferences; Cobb-Douglas
Related 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
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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

P

P

´