ECN741: Urban EconomicsThe Basic Urban Model 3: Comparative Statics

The Basic Urban Model

Class Outline

The point of comparative statics

Open model comparative statics results

Closed model comparative statics results

Comparative statics tables

Comparative statics graphs

The Basic Urban Model

Why Comparative Statics?

Urban models describe urban residential structure when (simplified!) models of 6 markets are combined.

A key set of questions involves the way urban residential structure changes when one of the parameters of the model changes.

CS involves finding the derivative of the variables in the model with respect to key parameters—accounting for interactions across markets.

The Basic Urban Model

Why Comparative Statics, 2?

Key parameters include Y, t, , U* (open), and N (closed).

With the basic model we can ask what happens to urban residential structure when

incomes rise over time; transportation innovation or investment takes place; the cost of non-urban activity (e.g. agriculture) changes; the situation in one city changes (or the opportunities in

other cities change); or population increases everywhere (or in one city where

out migration does not occur).

R

The Basic Urban Model

How to Do Comparative Statics

Comparative statics are based on total derivatives, not partial derivatives.

CS results must account for all the variables in one of the equations we have derived.

For example, many of the equations in the model, including equations for R{u}, P{u}, and D{u}, can be expressed as a function of the parameters and only one variable, namely, .u

The Basic Urban Model

How to Do Comparative Statics, 2

In these cases, CS derivations are based on equation like this one for R{u}:

In this equation, δ can be any of the model’s parameters (at least any of the ones that appear in the R{u} equation!).

A key to finding many CS derivatives, therefore, is to find the impact of the relevant parameter on .

dR u R u R u du

d du

u

The Basic Urban Model

Open Model Comparative Statics

Most CS results are relatively easy to obtain with an open model because of the form taken by the indirect utility function:

or

*

a

k Y tu k Y tuU

P RC

* *aP U R U

Y Yk C ku

t t

The Basic Urban Model

Open Model Comparative Statics, 2

In this formulation, is on the left side of the equation and all the key parameters are on the right.

So it is straightforward to derive CS results for the impact of all these parameters on .

These results can then be inserted into the formula given earlier to find CS results for R{u} and other variables.

u

u

The Basic Urban Model

Open Model Comparative Statics, 3

From

We can see that

10 ; 0

du du u

dY t dt t

1*

*0 ; 0

a ad u P d u a U R

d U kt k t Cd R

* *aP U R U

Y Yk C ku

t t

The Basic Urban Model

Open Model Comparative Statics, 4

So the physical size of an urban area:

Increases when income rises.

Decreases when commuting costs rise.

Decreases when opportunities improve elsewhere.

Decreases when agriculture becomes more profitable.

The Basic Urban Model

Open Model Comparative Statics, 5

Now take the expression for R{u}

To find the CS derivative for Y, differentiate with respect to Y, recognizing that Y affects , and plug in the above result for d /dY. The result:

1 a

Y tuR u R

Y tu

uu

0d R u R u

d Y a Y tu

The Basic Urban Model

Open Model Comparative Statics, 5

Similarly, we can start with the expression for N

Then we can differentiate with respect to Y, recognizing that Y affects , and plug in the above result for d /dY. This yields:

u u

1

11

b

b

R Y Y tuN u

t t bt b Y tu

21 0

b

b

d N R Y

d Y t Y tu

The Basic Urban Model

Open Model Comparative Statics Table

  Parameter

Variable Y t U*

+ - - -

R{u}or P{u} or D{u}

+ - 0 -

N + - - -

R

u

The Basic Urban Model

Closed Model Comparative Statics

In a closed model, the derivatives of with respect to the parameters come from the population equation, now with a bar over the N:

This equation is messier than the one for an open model, but its nonlinearity does not get in the way of CS as it does for solving the model.

1

11

b

b

R Y Y tuN u

t t bt b Y tu

u

The Basic Urban Model

Closed Model Comparative Statics, 2

For example, after a little algebra one can show that:

Not surprisingly, increasing agricultural rents shrinks the urban area.

Recall: b = 1/aα

12

1 10

1b

t b Ndu

dR R b Y

Y tu

The Basic Urban Model

Closed Model Comparative Statics, 3

To find the CS derivative , we must substitute this result into:

The result (derive as an exercise) indicates, not surprisingly, that more competition for land squeezes an urban area and pushes up rents (and density) until there is enough room for the population in a smaller space.

{ } /dR u dR

1

d R u R u tbR du

d R R dRY tu

The Basic Urban Model

Closed Model Comparative Statics Table

  Parameter

Variable Y t N

+ - - +

R{u}or P{u} or D{u}

small u:-large u:+

small u:+ large u: -

+ +

U* + - - -

R

u

The Basic Urban Model

Comparative Statics Intuition

We can develop an intuition for these results with some simple graphs for R{u} (or P{u} or D{u}).

To interpret these graphs note that

Population depends on density and urban size (= )

A change in Y flattens P{u}and R{u} (remember, Pʹ{u}= -t/H and H depends on Y).

A change in t steepens R{u}.

u

The Basic Urban Model

CS Result for Y

Open Model Closed Model

R{u} R{u}

u u2u

1u

R

1u

R

2u

With city size increase, density cannot increase at all locations

R must rise at u=0 because utility is fixed

Density declines near center

Density increases in suburbs, which grow

The Basic Urban Model

CS Result for t

Open Model Closed Model

R{u} R{u}

u u1u 1u

R

2u

R

2u

With city size decrease, density cannot decrease at all locations

R does not change at u=0 because tu=0.

Density goes up in the center and down in the suburbs, which shrink.

The Basic Urban Model

CS Result for

Open Model Closed Model

R{u} R{u}

u u1u 1u

R

2R

1R 1R

2u

2R

2u

City size and density cannot both move in the same direction

Utility level (indexed by height of R{u}) cannot change

Implies higher densityevery-where

The Basic Urban Model

Other CS Results

Open Model (U*) Closed Model (N)

R{u} R{u}

u u

1u 1u

R

2u

R

2u

City grows and becomes more dense

City shrinks and becomes less dense

The Basic Urban Model

Informal Tests of CS Results

These results predict that cities will get less dense in the center, more dense in the suburbs, and larger as incomes rise and transportation costs fall.

Many estimates of population density functions for cities around the world find this to be true.

But the models have only one worksite and many other simplifications.

Is this just a lucky coincidence or do the models capture something fundamental?