# Molly W. Dahl Georgetown University Econ 101 – Spring 2009

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Molly W. Dahl Georgetown University Econ 101 – Spring 2009
The Market Molly W. Dahl Georgetown University Econ 101 – Spring 2009

Economic Modeling Construct a model Solve the model Evaluate the model
Choose simplifications Solve the model Come up with a prediction Set S = D, etc. Evaluate the model Was it too simple? Do we learn anything about the real world?

Modeling the Apartment Market
How are apartment rents determined? Suppose apartments are close or distant, but otherwise identical distant apartments rents are exogenous (determined outside the model) and known many potential renters and landlords

Modeling the Apartment Market
Who will rent close apartments? At what price? Will the allocation of apartments be desirable in any sense? How can we construct an insightful model to answer these questions?

Economic Modeling Assumptions
Two basic assumptions: Rational Choice: Each person tries to choose the best alternative available to him or her. Equilibrium: Market price adjusts until quantity demanded equals quantity supplied.

Modeling Apartment Demand
Demand: Suppose the most any one person is willing to pay to rent a close apartment is \$500/month. Then p = \$500  QD = 1. Suppose the price has to drop to \$490 before a 2nd person would rent. Then p = \$490  QD = 2.

Modeling Apartment Demand
The lower is the rental rate p, the larger is the quantity of close apartments demanded p   QD . The quantity demanded vs. price graph is the market demand curve for close apartments.

Market Demand Curve for Apartments

Market Demand Curve for Apartments
QD

Modeling Apartment Supply
Supply: It takes time to build more close apartments so in this short-run the quantity available is fixed (at say 100).

Market Supply Curve for Apartments
100 QS

Competitive Market Equilibrium
“low” rental price  quantity demanded of close apartments exceeds quantity available  price will rise. “high” rental price  quantity demanded less than quantity available  price will fall.

Competitive Market Equilibrium
Quantity demanded = quantity available  price will neither rise nor fall so the market is at a competitive equilibrium.

Competitive Market Equilibrium
100 QD,QS

Competitive Market Equilibrium
Q: Who rents the close apartments? A: Those most willing to pay. Q: Who rents the distant apartments? A: Those least willing to pay. So the competitive market allocation is by “willingness-to-pay”.

Comparative Statics What is exogenous in the model?
price of distant apartments quantity of close apartments incomes of potential renters. What happens if these exogenous variables change?

Comparative Statics Suppose the price of distant apartment rises.
Demand for close apartments increases (rightward shift), causing a higher price for close apartments.

Market Equilibrium p pe 100 QD,QS

Market Equilibrium p Higher demand pe 100 QD,QS

Market Equilibrium p Higher demand causes higher market price; same quantity traded. pe 100 QD,QS

Comparative Statics Suppose there were more close apartments.
Supply is greater, so the price for close apartments falls.

Market Equilibrium p pe 100 QD,QS

Market Equilibrium p Higher supply pe 100 QD,QS

Market Equilibrium p Higher supply causes a lower market price and a larger quantity traded. pe 100 QD,QS

Comparative Statics Suppose potential renters’ incomes rise, increasing their willingness-to-pay for close apartments. Demand rises (upward shift), causing higher price for close apartments.

Market Equilibrium p pe 100 QD,QS

Market Equilibrium p Higher incomes cause higher willingness-to-pay pe
100 QD,QS

Market Equilibrium p Higher incomes cause higher willingness-to-pay, higher market price, and the same quantity traded. pe 100 QD,QS

Market Equilibrium p pe 100 QD,QS

Molly W. Dahl Georgetown University Econ 101 – Spring 2009
Budget Constraints Molly W. Dahl Georgetown University Econ 101 – Spring 2009

Consumption Choice Sets
A consumption choice set is the collection of all consumption choices available to the consumer. What constrains consumption choice? Budgetary, time and other resource limitations.

Budget Constraints A consumption bundle containing x1 units of commodity 1, x2 units of commodity 2 and so on up to xn units of commodity n is denoted by the vector (x1, x2, … , xn). Commodity prices are p1, p2, … , pn.

Budget Constraints Q: When is a bundle (x1, … , xn) affordable at prices p1, … , pn? A: When p1x1 + … + pnxn £ m where m is the consumer’s (disposable) income.

Budget Constraints The bundles that are only just affordable form the consumer’s budget constraint. This is the set { (x1,…,xn) | x1 ³ 0, …, xn ³ 0 and p1x1 + … + pnxn = m }.

Budget Constraints The consumer’s budget set is the set of all affordable bundles; B(p1, … , pn, m) = { (x1, … , xn) | x1 ³ 0, … , xn ³ 0 and p1x1 + … + pnxn £ m } The budget constraint is the upper boundary of the budget set.

Budget Set and Constraint for Two Commodities
x2 Budget constraint is p1x1 + p2x2 = m. m /p2 m /p1 x1

Budget Set and Constraint for Two Commodities
x2 Budget constraint is p1x1 + p2x2 = m. m /p2 m /p1 x1

Budget Set and Constraint for Two Commodities
x2 Budget constraint is p1x1 + p2x2 = m. m /p2 Just affordable m /p1 x1

Budget Set and Constraint for Two Commodities
x2 Budget constraint is p1x1 + p2x2 = m. m /p2 Not affordable Just affordable m /p1 x1

Budget Set and Constraint for Two Commodities
x2 Budget constraint is p1x1 + p2x2 = m. m /p2 Not affordable Just affordable Affordable m /p1 x1

Budget Set and Constraint for Two Commodities
x2 Budget constraint is p1x1 + p2x2 = m. m /p2 the collection of all affordable bundles. Budget Set m /p1 x1

Budget Set and Constraint for Two Commodities
x2 p1x1 + p2x2 = m is x2 = -(p1/p2)x1 + m/p2 so slope is -p1/p2. m /p2 Budget Set m /p1 x1

Budget Constraints For n = 2 and x1 on the horizontal axis, the constraint’s slope is -p1/p2. What does it mean?

Budget Constraints For n = 2 and x1 on the horizontal axis, the constraint’s slope is -p1/p2. What does it mean? Increasing x1 by 1 must reduce x2 by p1/p2.

Budget Constraints x2 Slope is -p1/p2 -p1/p2 +1 x1

Budget Constraints x2 Opp. cost of an extra unit of commodity 1 is p1/p2 units foregone of commodity 2. -p1/p2 +1 x1

Budget Constraints x2 Opp. cost of an extra unit of commodity 1 is p1/p2 units foregone of commodity 2. And the opp. cost of an extra unit of commodity 2 is p2/p1 units foregone of commodity 1. +1 -p2/p1 x1

Budget Sets & Constraints; Income and Price Changes
The budget constraint and budget set depend upon prices and income. What happens as prices or income change?

How do the budget set and budget constraint change as income m increases?
x2 Original budget set x1

Higher income gives more choice
x2 New affordable consumption choices Original and new budget constraints are parallel (same slope). Original budget set x1

How do the budget set and budget constraint change as income m decreases?
x2 Original budget set x1

How do the budget set and budget constraint change as income m decreases?
x2 Consumption bundles that are no longer affordable. Old and new constraints are parallel. New, smaller budget set x1

Budget Constraints - Income Changes
Increases in income m shift the constraint outward in a parallel manner, thereby enlarging the budget set and improving choice.

Budget Constraints - Income Changes
Increases in income m shift the constraint outward in a parallel manner, thereby enlarging the budget set and improving choice. Decreases in income m shift the constraint inward in a parallel manner, thereby shrinking the budget set and reducing choice.

Budget Constraints - Income Changes
No original choice is lost and new choices are added when income increases, so higher income cannot make a consumer worse off. An income decrease may (typically will) make the consumer worse off.

Budget Constraints - Price Changes
What happens if just one price decreases? Suppose p1 decreases.

How do the budget set and budget constraint change as p1 decreases from p1’ to p1”?
x2 m/p2 -p1’/p2 Original budget set m/p1’ m/p1” x1

How do the budget set and budget constraint change as p1 decreases from p1’ to p1”?
x2 m/p2 New affordable choices -p1’/p2 Original budget set m/p1’ m/p1” x1

How do the budget set and budget constraint change as p1 decreases from p1’ to p1”?
x2 m/p2 New affordable choices Budget constraint pivots; slope flattens from -p1’/p2 to -p1”/p2 -p1’/p2 Original budget set -p1”/p2 m/p1’ m/p1” x1

Budget Constraints - Price Changes
Reducing the price of one commodity pivots the constraint outward. No old choice is lost and new choices are added, so reducing one price cannot make the consumer worse off.

Budget Constraints - Price Changes
Similarly, increasing one price pivots the constraint inwards, reduces choice and may (typically will) make the consumer worse off.

The Food Stamp Program Food stamps are coupons that can be legally exchanged only for food. How does a commodity-specific gift such as a food stamp alter a family’s budget constraint?

The Food Stamp Program Suppose m = \$100, pF = \$1 and the price of “other goods” is pG = \$1. The budget constraint is then F + G =100.

The Food Stamp Program G F + G = 100; before stamps. 100 F 100

The Food Stamp Program G F + G = 100: before stamps. 100 F 100

The Food Stamp Program G F + G = 100: before stamps. 100
Budget set after 40 food stamps issued. F 40 100 140

The Food Stamp Program G F + G = 100: before stamps. 100
Budget set after 40 food stamps issued. The family’s budget set is enlarged. F 40 100 140

The Food Stamp Program What if food stamps can be traded on a black market for \$0.50 each?

The Food Stamp Program G F + G = 100: before stamps. 120
Budget constraint after food stamps issued. 100 Budget constraint with black market trading. F 40 100 140

The Food Stamp Program G F + G = 100: before stamps. 120
Budget constraint after food stamps issued. 100 Black market trading makes the budget set larger again. F 40 100 140

Shapes of Budget Constraints
Q: What makes a budget constraint a straight line? A: A straight line has a constant slope and the constraint is p1x1 + … + pnxn = m so if prices are constants then a constraint is a straight line.

Shapes of Budget Constraints
But what if prices are not constants? E.g. bulk buying discounts, or price penalties for buying “too much”. Then constraints will be curved or kinked.

Shapes of Budget Constraints - Quantity Discounts
Suppose p2 is constant at \$1 but that p1=\$2 for 0 £ x1 £ 20 and p1=\$1 for x1>20.

Shapes of Budget Constraints - Quantity Discounts
Suppose p2 is constant at \$1 but that p1=\$2 for 0 £ x1 £ 20 and p1=\$1 for x1>20. Then the constraint’s slope is , for 0 £ x1 £ 20 -p1/p2 = , for x1 > 20 and the constraint is {

Shapes of Budget Constraints with a Quantity Discount
x2 m = \$100 Slope = - 2 / 1 = (p1=2, p2=1) 100 Slope = - 1/ 1 = (p1=1, p2=1) x1 20 50 80

Shapes of Budget Constraints with a Quantity Discount
x2 m = \$100 Slope = - 2 / 1 = (p1=2, p2=1) 100 Slope = - 1/ 1 = (p1=1, p2=1) x1 20 50 80

Shapes of Budget Constraints with a Quantity Discount
x2 m = \$100 100 Budget Constraint Budget Set x1 20 50 80

Shapes of Budget Constraints with a Quantity Penalty
x2 Budget Constraint Budget Set x1

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