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\author{131 Undergraduate Public Economics \\ Emmanuel Saez \\ UC Berkeley}
\date{}
\title{Theoretical Tools of Public Finance \\ (Chapter 2 in Gruber's textbook)} \onlyslides{1-300}
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\begin{document}
\begin{slide}
\maketitle
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%\begin{slide}
%\begin{center}
%{\bf OUTLINE}
%\end{center}
%
%2.1 Constrained Utility Maximization
%
%2.2 Putting the Tools to Work: TANF and Labor Supply Among Single Mothers
%
%2.3 Equilibrium and Social Welfare
%
%2.4 Welfare Implications of Benefit Reductions: The TANF Example Continued
%
%2.5 Conclusion
%\end{slide}
\begin{slide}
\begin{center}
{\bf THEORETICAL AND EMPIRICAL TOOLS}
\end{center}
{\bf Theoretical tools}:
The set of
tools designed to understand
the mechanics behind economic decision making.
Economists model individuals' choices using the concepts of utility function maximization subject
to budget constraint
{\bf Empirical tools}:
The set of
tools designed to analyze data
and answer questions raised by theoretical analysis.
\end{slide}
%2.1 Constrained Utility Maximization
%\begin{slide}
%\begin{center}
%{\bf CONSTRAINED UTILITY MAXIMIZATION}
%\end{center}
%
%Economists model individuals' choices using the concepts of utility function maximization subject
%to budget constraint.
%
%{\bf Utility function}:
%A mathematical function representing an individual's set of preferences, which translates her well-being from different consumption bundles into units that can be compared in order to determine choice.
%
%{\bf Constrained utility maximization}:
%The process of maximizing the well-being (utility) of an individual, subject to her resources (budget constraint).
%
%\end{slide}
\begin{slide}
\begin{center}
{\bf UTILITY MAPPING OF PREFERENCES}
\end{center}
\textbf{Utility function:} A utility function is some mathematical function translating consumption into utility:
$$U = u(X_1, X_2, X_3,...) $$
where $X_1, X_2, X_3,$ and so on are the quantity of goods 1,2,3,... consumed by the individual
Example with two goods: $u(X_1,X_2) = \sqrt{X_1 \cdot X_2}$ with $X_1$ number of movies, $X_2$ number of music songs
Individual utility increases with the level of consumption of each good
\end{slide}
\begin{slide}
\begin{center}
{\bf PREFERENCES AND INDIFFERENCE CURVES}
\end{center}
{\bf Indifference curve:}
A graphical
representation of all bundles of goods that make an individual
equally well off
Mathematically, indifference curve giving utility level $\bar{U}$ is given by the set of bundles $(X_1,X_2)$
such that $u(X_1,X_2)=\bar{U}$
Indifference curves have two essential properties, both of which follow naturally from the more-is-better assumption:
1. Consumers prefer higher indifference curves.
2. Indifference curves are always downward sloping.
\end{slide}
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\includepdf[pages={1}]{tools_ch02_new_attach.pdf}
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\begin{center}
{\bf MARGINAL UTILITY}
\end{center}
{\bf Marginal utility}:
The additional
increment to utility obtained by
consuming an additional unit of
a good:
Marginal utility of good $1$ is defined as:
\[MU_1=\frac{ \partial u}{\partial X_1} \simeq \frac{u(X_1+dX_1,X_2)- u(X_1,X_2)}{dX_1} \]
It is the derivative of utility with respect to $X_1$ keeping $X_2$ constant (called the partial derivative)
Example: \[ u(X_1,X_2) = \sqrt{X_1 \cdot X_2} \Rightarrow \frac{ \partial u}{\partial X_1}= \frac{ \sqrt {X_2}}
{ 2\sqrt{X_1}} \]
This utility function described exhibits the important principle of \textbf{diminishing marginal utility}: $\partial u/ \partial X_1$ decreases with $X_1$: the consumption of each additional unit of a good gives less extra utility than the consumption of the previous unit
\end{slide}
%\begin{slide}
%\includepdf[pages={3}]{Gruber2e_ch02_attach.pdf}
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\begin{center}
{\bf MARGINAL RATE OF SUBSTITUTION}
\end{center}
{\bf Marginal rate of substitution (MRS)}:
The $MRS$ is equal to (minus) the slope of the indifference curve, the rate at which the consumer will trade the good on the vertical axis for the good on the horizontal axis.
Marginal rate of substitution between good 1 and good 2 is:
\[ MRS_{1,2} = \frac{MU_1}{MU_2} \]
Individual is indifferent between 1 unit of good 1 and $MRS_{1,2}$ units of good 2.
Example:
\[ u(X_1,X_2) = \sqrt{X_1 \cdot X_2} \Rightarrow MRS_{1,2}= \frac{X_2}{X_1} \]
\end{slide}
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\includepdf[pages={3}]{tools_ch02_new_attach.pdf}
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\begin{slide}
\begin{center}
{\bf BUDGET CONSTRAINT}
\end{center}
{\bf Budget constraint}:
A mathematical representation of all the combinations of goods an individual can afford to buy if she spends her entire income.
\[ p_1 X_1 + p_2 X_2 = Y\]
with $p_i$ price of good $i$, and $Y$ disposable income.
Budget constraint defines a linear set of bundles the consumer can purchase with its disposable income $Y$
\[ X_2 = \frac{Y}{p_2} - \frac{p_1}{p_2} X_1 \]
The slope of the budget constraint is $-p_1/p_2$
%If the consumer gives up 1 unit of good 1, it has
%$p_1$ more to spend, and can buy $p_1/p_2$ units of good 2.
%{\bf Opportunity cost}:
%The cost of
%any purchase is the next best
%alternative use of that money,
%or the forgone opportunity.
%When a person's budget is fixed, if he buys one thing he is, by definition, reducing the money he has to spend on other things. Indirectly, this purchase has the same effect as a direct good-for-good trade.
\end{slide}
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\includepdf[pages={4}]{tools_ch02_new_attach.pdf}
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\begin{slide}
\begin{center}
{\bf UTILITY MAXIMIZATION}
\end{center}
Individual maximizes utility subject to budget constraint:
\[ \max_{X_1,X_2} u(X_1,X_2) \quad \text{subject to} \quad p_1 X_1 + p_2 X_2 = Y \]
\[ \textbf{Solution:} \quad MRS_{1,2} = \frac{p_1}{p_2} \]
\small
Proof: Budget implies that $X_2=(Y-p_1X_1)/p_2$
Individual chooses $X_1$ to
maximize $u(X_1,(Y-p_1X_1)/p_2)$
The first order condition (FOC) is:
\[ \frac{ \partial u}{\partial X_1} - \frac{p_1}{p_2} \cdot \frac{ \partial u}{\partial X_2} =0. \]
\normalsize
At the optimal choice, the individual is indifferent between buying 1 extra unit of good 1 for \$ $p_1$ and
buying $p_1/p_2$ extra units of good 2 (also for \$ $p_1$).
\end{slide}
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\includepdf[pages={5}]{tools_ch02_new_attach.pdf}
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\begin{center}
{\bf INCOME AND SUBSTITUTION EFFECTS}
\end{center}
Let us denote by $p=(p_1,p_2)$ the price vector
Individual maximization generates demand functions $X_1(p,Y)$ and $X_2(p,Y)$
How does $X_1(p,Y)$ vary with $p$ and $Y$?
Those are called price and income effects
\small
Example: $u(X_1,X_2)= \sqrt{X_1 \cdot X_2}$ then $MRS_{1,2}=X_2/X_1$.
Utility maximization implies $X_2/X_1=p_1/p_2$ and hence $p_1 X_1 = p_2 X_2$
Budget constraint $p_1 X_1+p_2 X_2 = Y$ implies $p_1 X_1 = p_2 X_2 = Y/2$
Demand functions: $X_1(p,Y)=Y/(2 p_1)$ and $X_2(p,Y)=Y/(2 p_2)$
\end{slide}
\begin{slide}
\begin{center}
{\bf INCOME EFFECTS}
\end{center}
Income effect is the effect of giving extra income $Y$ on the demand for goods:
How does $X_1(p,Y)$ vary with $Y$?
{\bf Normal goods}:
Goods for which demand increases as income $Y$ rises: $X_1(p,Y)$ increases with $Y$
(most goods are normal)
{\bf Inferior goods}:
Goods for which demand falls as income $Y$ rises: $X_1(p,Y)$ decreases with $Y$
(example: you use public transportation less when you are
rich enough to buy a car)
Example: if leisure is a normal good, you work less (i.e. get more leisure) if you are given
a transfer
\end{slide}
\begin{slide}
\begin{center}
{\bf PRICE EFFECTS}
\end{center}
How does $X_1(p_1,p_2,Y)$ vary with $p_1$?
Changing $p_1$ affects the slope of the budget constraint and can be decomposed into
2 effects:
{\bf 1) Substitution effect}:
Holding
utility constant, a relative rise in
the price of a good will always
cause an individual to choose
less of that good
{\bf 2) Income effect}:
A rise in the
price of a good will typically
cause an individual to choose
less of all goods because her
income can purchase less than
before
For normal goods, an increase in $p_1$ reduces $X_1(p_1,p_2,Y)$ through both substitution
and income effects
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%\includepdf[pages={11}]{Gruber2e_ch02_attach.pdf}
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%2.3 Equilibrium and Social Welfare
%\begin{slide}
%\begin{center}
%{\bf EQUILIBRIUM AND SOCIAL WELFARE}
%\end{center}
%
%{\bf Welfare economics:}
%The study
%of the determinants of wellbeing,
%or welfare, in society.
%
%{\bf Demand curves:}
%A curve showing the quantity of a good
%demanded by individuals at
%each price.
%\end{slide}
%\begin{slide}
%\includepdf[pages={7}]{tools_ch02_new_attach.pdf}
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\begin{center}
{\bf AGGREGATE DEMAND}
\end{center}
Each individual has a demand for each good that depends on the price $p$ of the good.
Aggregating across all individuals, we get
aggregate demand $D(p)$ for the good
%Demand graph: quantity on X-axis, price on Y-axis
Basic rationalization: consumers maximize $v(Q)- p \cdot Q$ where $v(Q)$ is utility of consuming $Q$ units (increasing and concave): First order condition $v'(Q)=p$ defines $Q=D(p)$.
At price $p$, demand is $D(p)$ and $p$ is the \$ value for consumers of the marginal
(last) unit consumed
First unit consumed generates utility $v'(0)=D^{-1}(0)$ and hence surplus $D^{-1}(0) - p$, last (marginal) unit consumed generates surplus $v'(Q)-p=0$
$\Rightarrow$ Consumer surplus can be measured as area below the demand curve
and above the price horizontal line
\end{slide}
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\begin{center}
{\bf ELASTICITY OF DEMAND}
\end{center}
The \% change in demand caused by a 1\% change in the price of that good:
\[ \varepsilon^D=\frac{\mbox{\% change in quantity demanded}}{\mbox{\% change in price}}=\frac{\Delta D/D}{\Delta p/p}= \frac{p}{D} \frac{dD}{dp} \]
Elasticities are widely used because they are \textbf{unit free}
$\varepsilon^D=p D'(p)/D(p)$ is a function of $p$ and hence can vary with $p$ along the demand
curve
When $D(p)=D_0 \cdot p^{\varepsilon}$ with $D_0, \varepsilon$ fixed parameters,
then $\varepsilon^D = \varepsilon$ is constant (called iso-elastic demand function)
\end{slide}
\begin{slide}
\begin{center}
{\bf PROPERTIES OF ELASTICITY OF DEMAND}
\end{center}
1) Typically negative, since quantity demanded typically falls as price rises.
2) Typically not constant along a demand curve.
3) With vertical demand curve, demand is \textbf{perfectly inelastic} ($\varepsilon=0$).
4) With horizontal demand curve, demand is \textbf{perfectly elastic} ($\varepsilon=-\infty$).
5) The effect of one good's prices on the demand for another good is the \textbf{cross-price} elasticity. Typically, not zero.
\end{slide}
\begin{slide}
\begin{center}
{\bf PRODUCERS}
\end{center}
Producers (typically firms) use technology to transform inputs (labor and capital) into outputs (consumption goods)
Goal of producers is to maximize profits = sales of outputs minus costs of inputs
Production decisions (for given prices) define supply functions
Simple case: Profits $ \Pi = p \cdot Q - c(Q)$ where $c(Q)$ is cost of producing quantity $Q$.
$c(Q)$ is increasing and convex (means that $c'(Q)$ increases with $Q$).
Profit maximization: $\max_Q [ p \cdot Q - c(Q)]$
$\Rightarrow$ $c'(Q) = p$:
marginal cost of production equals price
Defines the supply curve $Q=S(p)$.
\end{slide}
\begin{slide}
\begin{center}
{\bf SUPPLY CURVES}
\end{center}
%{\bf Supply curve}:
%A curve showing the quantity of a good that firms in aggregate are willing to supply at each price:
%{\bf Marginal productivity}:
%The impact of a one unit change in any input, holding other inputs constant, on the firm's output.
\textbf{Supply curve $S(p)$} is the quantity that firms in aggregate are willing to supply at each price:
typically upward sloping with price due to decreasing returns to scale
At price $p$, producers produce $S(p)$, and the \$ cost of producing the marginal (last) unit is $p$
Elasticity of supply $\varepsilon_S$ is defined as
\[ \varepsilon_S=\frac{\mbox{\% change in quantity supplied}}{\mbox{\% change in price}}=\frac{\Delta S/S}{\Delta p/p}= \frac{p}{S} \frac{dS}{dp} \]
$\varepsilon^S=p S'(p)/S(p)$ is a function of $p$ and hence can vary with $p$ along the supply
curve
When $S(p)=S_0 \cdot p^{\varepsilon}$ with $S_0, \varepsilon$ fixed parameters,
then $\varepsilon^S = \varepsilon$ is constant (called iso-elastic supply function)
\end{slide}
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\includepdf[pages={17}]{tools_ch02_new_attach.pdf}
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\begin{center}
{\bf MARKET EQUILIBRIUM}
\end{center}
Demanders and suppliers interact on markets
{\bf Market equilibrium}: The equilibrium is the price $p^*$ such that $D(p^*)=S(p^*)$
In the simple diagram, $p^*$ is unique if $D(p)$ decreases with $p$ and $S(p)$ increases with $p$
If $p>p^*$, then supply exceeds demand, and price needs to fall to equilibrate supply and demand
If $p