theoretical tools of public...

Theoretical Tools of Public Economics

Part-2

Previous Lecture

• Definitions and Properties

– Utility functions

• Marginal utility: positive (negative) if x is a ‘good’(‘bad’)

• Diminishing marginal utility

– Indifferences curves

• Downward sloping

• Consumers prefer higher indifference curves

• Do not intersect

Previous Lecture

• Definitions and Properties

– Optimization with constraints

• Utility maximization subject to a budget constraint

Application in Public Economics(TANF Example-Chapter 2.2)

– Temporary Assistance to Needy Families: provides a monthly support check to families with incomes below a threshold level that varies by state.

– Largely targeted to single-female-headed households with children.


– Suppose Joelle is a single mother who spends all of her earnings and TANF benefits on food for her and her children (consumption denoted by C)

– The cost of working more is spending less time with her family (time worked denoted by W or leisure-time denoted by L)


– Can use

1. Consumption (C) and time-worked (W): one good and one bad

2. Consumption (C) and leisure-time (L): two goods

– We will use the second approach and then calculate the time-worked by subtracting the leisure-time from the total work hours.


– Assume that Joelle can work a maximum of 2000 hours per year at $10/hour.

– Assume further that a unit of food costs $1.


• Budget constraint without TANF

– With the first approach (using C and W)

WC 10≤


• Budget constraint without TANF

– With the second approach (using C and L)

– In other words, the price of leisure is $10/hour.

( )LC −≤ 200010

2000010 ≤+ LC


• The budget constraint without TANF


• The budget constraint under TANF

– Scenario-1: Guarantee of $5000 and 50% reduction rate


• Budget constraints under TANF (Scenario-1)

If or 1000≤W 20001000 ≤< L

( )WWC 105.0105000 −+≤

( )LC −+≤ 200055000

150005 ≤+ LC



If or 1000>W 10000 ≤< L

WC 10≤

2000010 ≤+ LC


• The budget constraint under TANF

– Scenario-2: Guarantee of $3000 and 50% reduction rate



If or 1400≤W 20001400 ≤< L

( )WWC 105.0103000 −+≤

( )LC −+≤ 200053000

130005 ≤+ LC



If or 600>W 6000 ≤< L

WC 10≤

2000010 ≤+ LC


• What if we switch from Scenario-1 to Scenario-2?

– Substitution effects

– Income effects


• How do we calculate the magnitude of the impact of the switch from Scenario-1 to Scenario-2?

• Assume that Joelle’s utility function takes the following forms

– Case 1:

– Case 2:

– Case 3:

( ) ( ) ( )LCLCU ln175ln100, +=

( ) ( ) ( )LCLCU ln300ln75, +=

( ) ( ) ( )LCLCU ln75ln300, +=


• Case-1


• Case-2

Equilibrium and Social Welfare (Chapter 2.3)

• Now that we examined the individual level, what happens at the aggregate level?

– How does the aggregate labor supply change with TANF among low income families?


• Welfare economics: the study of determinants of well-being, or welfare, in society.


• How do we determine social welfare?

– First we define the determinants of social efficiency (the size of the economic pie),

• Demand curves

• Supply curves

• Equilibrium

– Then we discuss how to introduce equality (how to distribute the pie)

• Social welfare function

• How to choose the social welfare function?

Equilibrium and Social WelfareDemand Curves

• Demand curve: A curve showing the aggregate quantity of a good demanded by individuals at each price.


• Elasticity of demand: The percentage change in the quantity demanded of a good caused by each 1% change in the price of that good.


• Elasticity of demand:– They are typically negative, since quantity demanded typically

falls as price rises.

– A vertical demand curve is one for which the quantity demanded does not change when price rises; in this case, demand is perfectly inelastic.

– A horizontal demand curve is one where quantity demanded changes infinitely for even a very small change in price; in this case, demand is perfectly elastic.

– The effect of one good’s prices on the demand for another good is the cross-price elasticity.

Equilibrium and Social WelfareSupply Curves

• Supply curve: A curve showing the quantity of a good that firms are willing to supply at each price.

– Analogous to the utility maximization problem


• Analogous to utility maximization

– Utility functions � Profit functions

• Profit function:

– Total revenue (p*q(K,L)) – Total cost (C(q(K,L))

– Price*Production function – Cost function

– Maximize with respect to goods � Maximize with respect to inputs and outputs

• q: output, K (capital) and L (labor) input


• Analogous to utility maximization

– Marginal utility � Marginal productivity

• Marginal productivity of K: Keeping all other inputs, the change in output as a result of a unit change in capital

– Diminishing marginal utility � Diminishing marginal productivity

( )LKqK

MPK ,∂

∂=

0<∂

∂KMP

K


• The optimum quantity of output is achieved at the point where marginal revenue equals marginal cost.

– Marginal cost: The incremental cost to a firm of producing one more unit of a good.

• (In a competitive market), All firms with initial marginal costs lower than the market price will produce until their marginal costs equal the market price.

Equilibrium and Social WelfareMarket Equilibrium

• Market: the arena in which demanders and suppliers interact.

• Market equilibrium: The combination of price and quantity that satisfies both demand and supply determined by the interaction between the supply and demand curves.

Equilibrium and Social WelfareMarket Equilibrium

Equilibrium and Social WelfareSocial Efficiency

• Consumer surplus: the gain to consumers from trades in a market for consumer goods.


• Produces surplus: the benefit that producers derive from selling a good, above and beyond the cost of producing that good.


• Social surplus: the sum of consumer surplus and producer surplus


• First Fundamental Theorem of Welfare Economics: the competitive equilibrium, where supply equals demand, maximizes social surplus (social efficiency).

Equilibrium and Social WelfareSocial Welfare

• Social welfare: the level of social well-being in society; determined both by social efficiency (the size of the economic pie) and the equitable distribution of society’s resources (the distribution of the pie)

• Second Fundamental Theorem of Welfare Economics: Society can attain any socially efficient outcome by suitably redistributing resources among individuals and then allowing them to trade freely.


• Equity-efficiency trade-off: The choice society must make between the total size of the economic pie and its distribution among individuals.

• How to measure this trade-off?

– Social welfare functions

Equilibrium and Social WelfareSocial Welfare Functions

• SWF maps the set of individual utilities in society into an overall society utility function.

• Examples:

– Utilitarian SWF:

– Rawlsian SWF:

NUUUSWF +++= ...21

( )NUUUSWF ,...,,min 21=


• How to choose a SWF?

– Two different approaches

• Commodity egalitarianism: The principle that society

should ensure that individuals meet a set of basic needs,

but that beyond that point income distribution is irrelevant.

• Equality of opportunity: The principle that society should

ensure that all individuals have equal opportunities for

success, but not focus on the outcomes of choices made.

Equilibrium and Social WelfareTANF Example

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