In this R Markdown notebook I draw some figures illustrating the basic ideas in the monocentric city model (Alonso-Muth-Mills model). To keep things simple, I use a model without a housing construction industry and where consumers just consume land directly. The city is a line and the amount of land at every point is equal to 1.
Solving the Model
In equilibrium all consumers have the same (endogenous) utility, \(\bar{u}\), and the land rent at the fringe of the city is equal to the agricultural rent, \(r(\bar{x})=r_A\). Further, there is one unit of land at every location, \(L(x)=1\) for \(x=0\) to \(x=\bar{x}\), and everyone is housed between the center of the city and the fringe.
From Cobb-Douglas demand we know that \(z(x)=(1-\alpha)(y-\tau*x)\) and \(l(x)=\frac{\alpha(y-\tau*x)}{r(x)}\). Substituting these into the spatial equilibrium condition (equal utility) gives:
\[U(l,z)=\left[\frac{\alpha(y-\tau*x)}{r(x)}\right]^{\alpha}\left[(1-\alpha)(y-\tau*x)\right]^{1-\alpha}=\bar{u}\tag{3}\] We can rewrite this equation to express land rent at \(x\) as a function of \(y\), \(\tau\), \(\bar{u}\), and \(\alpha\). \[r(x)=\left(\frac{y-\tau*x}{\bar{u}}\right)^{\frac{1}{\alpha}}\alpha(1-\alpha)^{\frac{1-\alpha}{\alpha}}\tag{4}\] The Alonso-Muth condition tells us the gradient of the land rent, which we could also derive by taking the derivative of the above equation: \[\frac{dr(x)}{dx}=-\tau((1-\alpha)(y-\tau*x))^{\frac{1-\alpha}{\alpha}}\bar{u}^{\frac{-1}{\alpha}}=\frac{-\tau}{l(x)}\tag{5}\] There are \(1/l(x)\) people living at every location \(x\). Combining the Alonso-Muth condition with the requirement that everyone is housed in the city implies that: \[\displaystyle\int_{x=0}^{\bar{x}}\frac{1}{l(x)}dx=\frac{1}{-\tau}\displaystyle\int_{x=0}^{\bar{x}}\frac{dr(x)}{dx}dx=\frac{r(\bar{x})-r(0)}{-\tau}=N\tag{6}\] Using \(r(\bar{x})=r_A\) gives \(r(0)=\tau*N+r(\bar{x})\). We can then find the utility of a consumer at the center by inserting this expression for \(r(0)\) into the indirect utility function (eq 3), thus giving us \(\bar{u}\). Finally, with \(\bar{u}\) we can solve for \(\bar{x}\) by the land rent equation (eq 4) equal to \(r_A\).
Solving for equilibrium
First, let’s set the exogenous parameters: \(\alpha\), \(y\), \(\tau\), \(r_A\), and \(N\).
alpha<-0.5
y<-100
tau<-1
r.A<-50
N<-1000
Next, let’s write functions for the indirect utility and land rent, so that we can solve for the equilibrium values of \(\bar{u}\) and \(\bar{x}\).
V<-function(x,r.x,alpha,y,tau){
l.x<-alpha*(y-tau*x)/r.x #demand for land
z.x<-(1-alpha)*(y-tau*x) #demand for numeraire
l.x^alpha*z.x^(1-alpha) #returns the utility at x
}
R<-function(x,alpha,y,tau,u.bar) {
inner<-(y-tau*x)/u.bar
const<-alpha*(1-alpha)^((1-alpha)/alpha)
inner^(1/alpha)*const #returns the rent at x
}
Now let’s solve for the equilibrium values. To find the fringe we will look for the value \(\bar{x}\) such that \(R(\bar{x})-r_A=0\).
r.0<-r.A+tau*N
u.bar<-V(0,r.0,alpha,y,tau)
#Define a function so that we can find x.bar
find.xbar<-function(x,alpha,y,tau,r.A,u.bar) {
R(x,alpha,y,tau,u.bar)-r.A
}
x.bar<-as.numeric(uniroot(find.xbar,lower=0,upper=y/tau,alpha=alpha,y=y,tau=tau,r.A=r.A,u.bar=u.bar)[1])
The equilibrium values are \(\bar{u}\)=1.5430335 and \(\bar{x}\)=78.178211. Is this correct? Let’s check if integrating the population at each location is equal to the total, \(N\).
l.demand<-function(x,alpha,y,tau,r.x) { #demand for land
alpha*(y-tau*x)/r.x
}
pop<-function(x,alpha,y,tau,u.bar) {
r.x<-R(x,alpha,y,tau,u.bar)
1/l.demand(x,alpha,y,tau,r.x)
}
N.check<-integrate(pop,lower=0,upper=x.bar,alpha=alpha,y=y,tau=tau,u.bar=u.bar) #this should be equal to N
Integrating the population over the city yields a total population of N.check=1000
Graphing
First, let’s graph the equilibrium rent curve
curve(R(x,alpha,y,tau,u.bar),from=0,to=x.bar,xlab="Distance from center (x)",ylab="Land rent, r(x)",main="Land rent over the city")
![](data:image/png;base64,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)
Next we can look at land consumption:
curve(l.demand(x,alpha,y,tau,R(x,alpha,y,tau,u.bar)),from=0,to=x.bar,xlab="Distance from center (x)",ylab="Land consumption, l(x)",main="Land consumption over the city")
![](data:image/png;base64,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)
Comparative Statics
Now let’s look at some comparative statics.
y.1<-500
tau.1<-0.7
r.A.1<-100
N.1<-5000
How does a change in the transporation cost \(\tau\) from \(\tau_0\)=1 to \(\tau_1\)=0.7 affect land consumption at every point?
r.0.1<-r.A+tau.1*N
u.bar.1<-V(0,r.0.1,alpha,y,tau.1)
x.bar.1<-as.numeric(uniroot(find.xbar,lower=0,upper=y/tau.1,alpha=alpha,y=y,tau=tau.1,r.A=r.A,u.bar=u.bar.1)[1])
curve(R(x,alpha,y,tau.1,u.bar.1),from=0,to=x.bar.1,xlab="Distance from center (x)",ylab="Land rent, r(x)",main="Land rent over the city",col="red",ylim=c(0,max(r.0,r.0.1)),xlim=c(0,max(x.bar,x.bar.1)))
curve(R(x,alpha,y,tau,u.bar),from=0,to=x.bar,add=TRUE,col="black")
text(x.bar,l.demand(x.bar,alpha,y,tau,r.A),"tau.0")
text(x.bar.1,l.demand(x.bar.1,alpha,y,tau.1,r.A),"tau.1",col="red")
![](data:image/png;base64,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)
curve(l.demand(x,alpha,y,tau.1,R(x,alpha,y,tau.1,u.bar.1)),from=0,to=x.bar.1,xlab="Distance from center (x)",ylab="Land consumption, l(x)",main="Land consumption over the city",col="red",ylim=c(0,0.3),xlim=c(0,max(x.bar,x.bar.1)))
curve(l.demand(x,alpha,y,tau,R(x,alpha,y,tau,u.bar)),from=0,to=x.bar,add=TRUE,col="black")
text(x.bar,l.demand(x.bar,alpha,y,tau,r.A),"tau.0")
text(x.bar.1,l.demand(x.bar.1,alpha,y,tau.1,r.A),"tau.1",col="red")
![](data:image/png;base64,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)
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---
title: "Simple Monocentric City: No Housing, Land in Utility Function"
header-includes: \usepackage{amsmath}
output: html_notebook
---

In this [R Markdown](http://rmarkdown.rstudio.com) notebook I draw some figures illustrating the basic ideas in the monocentric city model (Alonso-Muth-Mills model). To keep things simple, I use a model without a housing construction industry and where consumers just consume land directly. The city is a line and the amount of land at every point is equal to 1.

## Assumptions and Functional Forms
Consumers have Cobb-Douglas utility over *land* $l$ and a numeraire good $z$. 

$$U(l,z)=l^{\alpha}z^{1-\alpha}\tag{1}$$

For a consumer living at distance $x$ from the center, the budget constraint is:
$$y=z+r(x)l(x)-\tau*x\tag{2}$$
The city is closed, with population $N$, and  agricultural land rent $r_A$.

## Solving the Model
In equilibrium all consumers have the same (endogenous) utility, $\bar{u}$, and the land rent at the fringe of the city is equal to the agricultural rent, $r(\bar{x})=r_A$. Further, there is one unit of land at every location, $L(x)=1$ for $x=0$ to $x=\bar{x}$, and everyone is housed between the center of the city and the fringe.

From Cobb-Douglas demand we know that $z(x)=(1-\alpha)(y-\tau*x)$ and $l(x)=\frac{\alpha(y-\tau*x)}{r(x)}$. Substituting these into the spatial equilibrium condition (equal utility) gives:

$$U(l,z)=\left[\frac{\alpha(y-\tau*x)}{r(x)}\right]^{\alpha}\left[(1-\alpha)(y-\tau*x)\right]^{1-\alpha}=\bar{u}\tag{3}$$
We can rewrite this equation to express land rent at $x$ as a function of $y$, $\tau$, $\bar{u}$, and $\alpha$.
$$r(x)=\left(\frac{y-\tau*x}{\bar{u}}\right)^{\frac{1}{\alpha}}\alpha(1-\alpha)^{\frac{1-\alpha}{\alpha}}\tag{4}$$
The Alonso-Muth condition tells us the gradient of the land rent, which we could also derive by taking the derivative of the above equation:
$$\frac{dr(x)}{dx}=-\tau((1-\alpha)(y-\tau*x))^{\frac{1-\alpha}{\alpha}}\bar{u}^{\frac{-1}{\alpha}}=\frac{-\tau}{l(x)}\tag{5}$$
There are $1/l(x)$ people living at every location $x$. Combining the Alonso-Muth condition with the requirement that everyone is housed in the city implies that:
$$\displaystyle\int_{x=0}^{\bar{x}}\frac{1}{l(x)}dx=\frac{1}{-\tau}\displaystyle\int_{x=0}^{\bar{x}}\frac{dr(x)}{dx}dx=\frac{r(\bar{x})-r(0)}{-\tau}=N\tag{6}$$
Using $r(\bar{x})=r_A$ gives $r(0)=\tau*N+r(\bar{x})$. We can then find the utility of a consumer at the center by inserting this expression for $r(0)$ into the indirect utility function (eq 3), thus giving us $\bar{u}$. Finally, with $\bar{u}$ we can solve for $\bar{x}$ by the land rent equation (eq 4) equal to $r_A$.

## Solving for equilibrium
First, let's set the exogenous parameters: $\alpha$, $y$, $\tau$, $r_A$, and $N$.
```{r}
alpha<-0.5
y<-100
tau<-1
r.A<-50
N<-1000
```
Next, let's write functions for the indirect utility and land rent, so that we can solve for the equilibrium values of $\bar{u}$ and $\bar{x}$.
```{r}
V<-function(x,r.x,alpha,y,tau){
  l.x<-alpha*(y-tau*x)/r.x #demand for land
  z.x<-(1-alpha)*(y-tau*x) #demand for numeraire
  l.x^alpha*z.x^(1-alpha) #returns the utility at x
}
R<-function(x,alpha,y,tau,u.bar) {
  inner<-(y-tau*x)/u.bar
  const<-alpha*(1-alpha)^((1-alpha)/alpha)
  inner^(1/alpha)*const #returns the rent at x
}
```
Now let's solve for the equilibrium values. To find the fringe we will look for the value $\bar{x}$ such that $R(\bar{x})-r_A=0$.
```{r}
r.0<-r.A+tau*N
u.bar<-V(0,r.0,alpha,y,tau)
#Define a function so that we can find x.bar
find.xbar<-function(x,alpha,y,tau,r.A,u.bar) {
  R(x,alpha,y,tau,u.bar)-r.A
}
x.bar<-as.numeric(uniroot(find.xbar,lower=0,upper=y/tau,alpha=alpha,y=y,tau=tau,r.A=r.A,u.bar=u.bar)[1])
```
The equilibrium values are $\bar{u}$=`r u.bar` and $\bar{x}$=`r x.bar`.
Is this correct? Let's check if integrating the population at each location is equal to the total, $N$.
```{r}
l.demand<-function(x,alpha,y,tau,r.x) { #demand for land
  alpha*(y-tau*x)/r.x
}
pop<-function(x,alpha,y,tau,u.bar) {
  r.x<-R(x,alpha,y,tau,u.bar)
  1/l.demand(x,alpha,y,tau,r.x)
}
N.check<-integrate(pop,lower=0,upper=x.bar,alpha=alpha,y=y,tau=tau,u.bar=u.bar) #this should be equal to N
```
Integrating the population over the city yields a total population of *N.check*=`r N.check$value`

## Graphing
First, let's graph the equilibrium rent curve
```{r}
curve(R(x,alpha,y,tau,u.bar),from=0,to=x.bar,xlab="Distance from center (x)",ylab="Land rent, r(x)",main="Land rent over the city")
```
Next we can look at land consumption:
```{r}
curve(l.demand(x,alpha,y,tau,R(x,alpha,y,tau,u.bar)),from=0,to=x.bar,xlab="Distance from center (x)",ylab="Land consumption, l(x)",main="Land consumption over the city")
```

### Comparative Statics
Now let's look at some comparative statics.
```{r}
y.1<-500
tau.1<-0.7
r.A.1<-100
N.1<-5000
```
How does a change  in the transporation cost $\tau$ from $\tau_0$=`r tau` to $\tau_1$=`r tau.1` affect land consumption at every point?
```{r}
r.0.1<-r.A+tau.1*N
u.bar.1<-V(0,r.0.1,alpha,y,tau.1)
x.bar.1<-as.numeric(uniroot(find.xbar,lower=0,upper=y/tau.1,alpha=alpha,y=y,tau=tau.1,r.A=r.A,u.bar=u.bar.1)[1])

curve(R(x,alpha,y,tau.1,u.bar.1),from=0,to=x.bar.1,xlab="Distance from center (x)",ylab="Land rent, r(x)",main="Land rent over the city",col="red",ylim=c(0,max(r.0,r.0.1)),xlim=c(0,max(x.bar,x.bar.1)))
curve(R(x,alpha,y,tau,u.bar),from=0,to=x.bar,add=TRUE,col="black")
text(x.bar,l.demand(x.bar,alpha,y,tau,r.A),"tau.0")
text(x.bar.1,l.demand(x.bar.1,alpha,y,tau.1,r.A),"tau.1",col="red")

curve(l.demand(x,alpha,y,tau.1,R(x,alpha,y,tau.1,u.bar.1)),from=0,to=x.bar.1,xlab="Distance from center (x)",ylab="Land consumption, l(x)",main="Land consumption over the city",col="red",ylim=c(0,0.3),xlim=c(0,max(x.bar,x.bar.1)))
curve(l.demand(x,alpha,y,tau,R(x,alpha,y,tau,u.bar)),from=0,to=x.bar,add=TRUE,col="black")
text(x.bar,l.demand(x.bar,alpha,y,tau,r.A),"tau.0")
text(x.bar.1,l.demand(x.bar.1,alpha,y,tau.1,r.A),"tau.1",col="red")
```


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