Welcome To Econ 51

Part 2: Review of Consumer Theory

Econ 51, Lecture 1

Good 1 - Good 2 Space

Two "Goods" : Good 1 and Good 2

\text{Bundle }X\text{ may be written }(x_1,x_2)
x_1 = \text{quantity of good 1 in bundle }X
x_2 = \text{quantity of good 2 in bundle }X
A = (40, 160)
B = (80,80)
\text{Examples:}

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m = \text{money income}
p_1 = \text{price of good 1}
p_2 = \text{price of good 2}
\text{horizontal intercept} = \frac{m}{p_1}
\text{vertical intercept} = \frac{m}{p_2}
\text{slope of budget line} = -\frac{p_1}{p_2}

Budget Constraints

\text{Example: } p_1 = 2, p_2 = 1, m = 240

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Preferences

Definition Review:

Indifference Curves

Preferred/Dispreferred Sets

Marginal Rate of Substitution

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Utility Functions

u(x_1,x_2) = x_1x_2
MU_1(x_1,x_2) = \frac{\partial u(x_1,x_2)}{\partial x_1} =
MU_2(x_1,x_2) = \frac{\partial u(x_1,x_2)}{\partial x_2} =
MRS(x_1,x_2) = \frac{MU_1}{MU_2} =
MRS(40,160)=
MRS(80,80)=
u(40,160) =
u(80,80)=
\text{Example: }

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MRS > \frac{p_1}{p_2}
MRS < \frac{p_1}{p_2}

Indifference curve is
steeper than the budget line

Indifference curve is
flatter than the budget line

Moving to the right
along the budget line
would increase utility

Moving to the left
along the budget line
would increase utility

More willing to give up good 2
than the market requires

Less willing to give up good 2
than the market requires

The “Gravitational Pull" Towards Optimality

POINT A

POINT B

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IF...

THEN...

The consumer's utility function is "well behaved" -- smooth, strictly convex, and strictly monotonic

The indifference curves do not cross the axes

The budget line is a simple straight line

The optimal consumption bundle will be characterized by two equations:

MRS = \frac{p_1}{p_2}
p_1x_1 + p_2x_2 = m

More generally: the optimal bundle may be found using the Lagrange method

Optimal Choice

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Optimal Choice

Otherwise, the optimal bundle may lie at a corner,
a kink in the indifference curve, or a kink in the budget line.
No matter what, you can use the "gravitational pull" argument!

  1. Write an equation for the tangency condition.
  2. Write an equation for the budget line.
  3. Solve for \(x_1^*\) or \(x_2^*\).
  4. Plug value from (3) into either equation (1) or (2).
u(x_1,x_2) = x_1x_2

Solving for Optimality when Calculus Works

p_1 = 2, p_2 = 1, m = 240

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x_1^*(p_1,p_2,m)
x_2^*(p_1,p_2,m)

(Gross) demand functions are mathematical expressions
of endogenous choices as a function of exogenous variables (prices, income).

(Gross) Demand Functions

u(x_1,x_2) = x_1x_2
p_1x_1 + p_2x_2 = m

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x_1^*(p_1,p_2,m) = \frac{a}{a+b}\times \frac{m}{p_1}

For a Cobb-Douglas utility function of the form

Special Case: The “Cobb-Douglas Rule"

u(x_1,x_2) = x_1^ax_2^b

The demand functions will be

x_2^*(p_1,p_2,m) = \frac{b}{a+b}\times \frac{m}{p_2}

That is, the consumer will spend fraction a/(a+b) of their income on good 1, and fraction b/(a+b) of their income on good 2.

This shortcut is very much worth memorizing! We'll use it a lot in the next few weeks in place of going through the whole optimization process.

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Demand and Offer Curves

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\text{Hold }p_2 = 1\text{ and }m = 240\text{ constant, but keep }p_1 \text{ as a variable}.
x_1^*(p_1) = \frac{m}{2p_1} = \frac{240}{2p_1} = \frac{120}{p_1}
x_2^*(p_1) = \frac{m}{2p_2} = \frac{240}{2 \times 1} = 120
x_1^*
x_2^*
p_1
1
2
3
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To Do Before Next Class

Be sure you're signed up for a section.

Do the reading and the quiz -- due at 11:15am on Thursday!

Read the syllabus carefully.

Look over the summary notes for this class.

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