Residential Use of Land in a Monocentric City

Hannah Faith Enriquez

Two steps in the analysis of residential land rent:

Housing-Price Function- indicates how much a household is willing to

pay for dwellings at different locations in the city

Residential Bid-Rent Function- indicates how much housing producers are willing to pay per acre of land at various locations in the city

Assumptions in Residential Land Use:

HOUSING-PRICE FUNCTION

Price of housing – price per square foot of housing per month

Ex. 1,000 square-foot houseHousehold rent – $250 per month

Price of housing – 25 cents/square foot ($250 divided by 1,000 sq. foot)

2 Types of Housing-Price Function:

a. Linear Housing Price Function: No Consumer Substitution

b. Convex Housing Price Function: Consumer Substitution

Linear-Housing Price Function: No Consumer Substitution

Assumptions:

1. Identical dwellings – every dwelling in the city has 1,000 square feet or living space

2. Fixed budget – the household has a fixed budget of $300 per month ( on commuting and housing costs)

3. Commuting cost – monthly commuting costs are $20 per mile/month

1 mile - $20 2 miles - $40

Miles to city center

Price of housing per square foot

6

0.06

0.30

0.18

12 15

City center

Housing-price function

Housing-Price Function for a City with Identical Dwellings (1,000 square-foot): No Consumer Substitution (LINEAR)

A

B

$

$300 - housing

$120 – commuting $180 - housing

Linear Housing Price Function: No Consumer Substitution

a. Households are indifferent among all locations within the city

Why? Because differences in commuting costs are offset by differences in housing costs

Slope: If: t = $20H = 1,000 sq. foot P = 0.02/sq.foott – commuting cost/mile

H – housing consumption/sq. foot P – change in price/sq. foot

Convex Housing Price Function: Consumer Substitution

A more realistic assumption:◦Household obeys the law of demand:

decreases the quantity demanded as price increases

As a household moves toward the city center, it pays a higher price for housing, so it occupies a smaller dwelling

As the relative price of housing increases, the household substitutes nonhousing goods

Miles to city center

Price of housing per square foot

6

0.06

0.30

0.12

12 15

City center

Housing-Price Function for a City with Identical Dwellings : With Consumer Substitution (CONVEX)

$Housing-price function: no

consumer substitution

Housing-price function: with

consumer substitution

93

Distance to city center (miles)

Housing consumpti

on (sq. foot)

3 400

6 600

9 750

12 1,000

Convex Housing Price Function: Consumer Substitution

a. As the household moves toward the city center, housing consumption decreases, thus increasing the slope of the housing price-function.

Slope:

If: u = 9 milest = $20H(u) = 750 sq. foot Slope = 0.0267/sq.foot

t – commuting cost/mileH – housing consumption/sq. foot P – change in price of housingu – distance to city center

How rapid does the price of housing decrease as distance to the city center increases?

Housing-Price Gradient: percentage change in the price of housing per mile

Residential Bid-Rent Function- indicates how much housing producers are willing to pay per acre of land at various locations in the city

2 Types of Residential Bid-Rent Functions:

a. Housing with Fixed-Factor Proportionsb. Housing Firms engage in Factor Substitution

Residential Bid-Rent Function Equation:

P – price of housingu – distance to city centerQ – square feet of housingT – acres of landK –nonland cost

Given:

u – 6 milesT – 50K - 50

Distance to city center

(miles)

Housing consumption (sq. foot)

Price of Housing

3 400 0.24

6 600 0.18

9 750 0.12

12 1,000 0.06

R(u) = 1.98

Since P(u) decreases as u increases, R(u) declines as u increases. The bid rent function is convex since the housing price function is convex.

Bid-Rent Function: Fixed Factor Proportions

The characteristics of the housing industry are as follows:

1. Production - each firm produces Q square feet of housing using land and non-land inputs.Once the firm erects a building, it can be used as a single dwelling (with Q square feet of space) or divided into x units each of which has (Q/x) square feet of living space.

2. Non-land Cost - Firms use (K) worth of non-land inputs for each building.

Miles to city center

u*

Residential Bid-Rent Function: Fixed Factor Proportions

$Total Revenue = P(u) times Q

Cost of nonland inputs

Bid-rent Function

City center

3. Fixed Factor Proportions - Each firm produces its house, regardless of the price of land.

4. Housing Prices - the housing price function is negatively sloped and convex (i.e. slope gets steeper as location approaches city center)

5. Perfect Competition - the housing industry is perfectly competitive so each house builder makes zero economic profits in long run equilibrium.

Bid-Rent Function: Factor Substitution

Involves substituting non-land inputs for land as the price of land increases which means building progressively taller buildings as location approaches city center.

The flexible firm (with factor substitution) is able to produce housing more cheaply than the inflexible firm since the flexible firm uses less of the more expensive input at each location. Thus the flexible firm can always outbid the inflexible firm for land.

Miles to city center

6

Residential Bid-Rent Function: Factor Substitution

$

Bid Rent without factor substitution

City center3 9 12

Bid rent (per acre)

Bid Rent without factor substitution

Residential Density- population density at different locations in the city

a. Consumer substitution – price of housing decreases as distance to the city center increases

b. Factor substitution – price of land decreases as the distance to the city center increases

Monocentric City

2 features:

1. Office firms occupy the central area of CBD (central business district)

2. Employment is concentrated in the CBD, not distributed throughout the city

Land rent per acre

Distance from city center

Why do all manufacturers and office firms locate at the CBD?

There are trade-offs:

1. Higher freight costs (transportation costs)

2. Lower wages

SuburbsCity Center

Low wagesHigher transportation costs

Why? - Horse-drawn wagons are slower and more expensive

High wagesLower transportation costs

Why?- Workers commute by streetcar and they are faster and more efficient

Residential Bid Rent Functions

Manufacturer Bid Rent Functions

Office Bid Rent Functions

Transport Costs Lowest transport costs

Higher transport costs

Highest transport costs

Slope of Bid-rent Function

Flat Steeper Steepest

Distance from City Center

Farthest Closer Closest

END