I'm supposed to learn to derive the monocentric city model from these assumptions: everyone has the same preferences, same income, same work location (Central Business District, single point with infinitely small area) and same transportation per mile costs. They get utility from two goods, c (a composite good of everything but housing) and q (housing as measured by size or quality). The price of housing is a function of its distance from the CBD, x, like this p(x). In the end I'm supposed to see that p'(x) < 0, and population density and structural density both decrease as x increases.
1 Answer
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The reason people do not all live on top of each other in the monocentric city is that people can attain the same level of utility further away from the city center by enduring higher transportation costs for the daily commute due to being compensated by cheaper housing and therefore larger housing consumption.
Here is how you derive it assuming preferences given by the Cobb-Douglas utility function. For the more general case consult the references. I'm not going to copy-paste the general proofs from the literature here.
You start by assuming preferences. Individuals have preferences for a composite good $c$ and square meters of housing $h$ given as
\begin{align} U(c,h) = \left(\frac{h}{\alpha} \right)^\alpha\left(\frac{c}{1-\alpha}\right)^{1-\alpha} \end{align}.
Everyone works in the city center earning the same income $I$. However, at location $z$ being the distance from the city center the income net transportation cost is given as
$$W(z) = I - tz.$$
Demand for housing at location $z$ is therefore given as
\begin{align} h(z) = \frac{\alpha(I - tz)}{p(z)}, \end{align} where $p(z)$ is the square meter price of housing at location $z$. This is simply the usual Marshall demand. The only thing special is that price is locations dependent and income is net transportation costs.
Demand for the composite good is given as
\begin{align} c(z) = (1-\alpha)(I - tx). \end{align}
You then insert this into the utility function to get the value function.
$$V(I,p(z)) := U(h(z),c(z)) = \frac{I - tz}{p(z)^\alpha}.$$
You then have to decide whether it is an open or a closed city. The result will be the same so that does not really matter but you still have to make the choice in order to present the argument.
Let us assume that it is an open city so people can enter and leave. We then assume that there is some other place people can live where the utility they can attain is $\bar u$. This is an outside option. If the utility in the city is lower everyone will leave and if it is higher the city will grow infinitely large so in equilibrium the utility attained in the city is $\bar u$.
There we set the value function equal to $\bar u$ to get
\begin{align} V(I,p(z)) = \frac{I - tz}{p(z)^\alpha} = \bar u, \end{align} which implies that the price in equilibrium is given as
$$p^\star(z) = \left(\frac{I - tz}{\bar u} \right)^{1/\alpha}.$$
You can differentiate this with respect to $z$ to see that housing prices are decreasing in distance from the city center. You can insert the solution for $p(z)$ into the Marshall demand for housing to get see individuals consume fewer square meters of housing in the city center
\begin{align} h^\star(z) = \frac{\alpha(I-tz)}{\left(\frac{I - tz}{\bar u} \right)^{1/\alpha}} = \frac{\alpha \bar u^{1/\alpha}}{ (I-tz)^\frac{1-\alpha}{\alpha}}. \end{align}
You then assume that at each $z$ there is and amount of land available $L$. And that housing is produced using land and capital $K$. Because land is fixed you can only produce more housing using more capital at a given location. The marginal cost of housing is therefore increasing. Let us assume we have derived a housing supply which at location $z$ is given as $H(p(z))$ and assumed to be increasing in $p(z)$. Without further ado let us just assumed to functional form is
$$H(p(z)) = B p(z)^\beta \phantom{xxx}, 0 < \beta < 1.$$
You can then argue that the supply of housing at location $z$ in equilibrium must be
$$H(p(z)) = B \left(\frac{I - tz}{\bar u} \right)^{\beta/\alpha},$$
and because each individual consumes $h^\star(z)$ units of housing the number of individuals at $z$ must be
$$D(z) := \frac{H(p^\star(z))}{h^\star(z)}.$$
Closer to the city center - smaller $z$ - results in higher square meter prices of housing which increases equilibrium supply of square meters of housing per unit land. Furthermore, the higher square meter price of housing imply that individuals consumer fewer square meters of housing per individual. Hence, the closer to the city center the larger the population density.
If you want the proof to be general and therefore more abstract there are several good references. The number one reference is no doubt
Brueckner, Jan K. (1987). The Structure of Urban Equilibria: A Unified Treatment of the Muth-Mills Model.
https://www.sciencedirect.com/science/article/abs/pii/S1574008087800068
And for some nice slides and hand outs you can look here:
https://www.nathanschiff.com/webdocs/grad_urban/monocentric_city_lecture_HANDOUT.pdf
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$\begingroup$ You may also find this post of mine interesting: economics.stackexchange.com/questions/43654/… $\endgroup$ Commented Jun 6 at 19:57