We already saw many government policies that affect housing, such as land-use regulations and the tax subsidy to homeownership

Now, we analyze additional public policies in the housing market: rent-control laws and housing-subsidy programs

Rent control benefits current tenants in the short-term but has pervasive long-run effects on overall welfare

A less distortionary policy is to offer a subsidy to specific groups. However, each program involves opposing interests between the ones who finance the assistance (taxpayers) and the targeted group

The graph on the right represents the flow supply curve $S_{\Delta H}$ and the net flow of new housing into the market $\Delta H$

When the price is higher than the equilibrium $p_{e}$, there will be new construction, and the net flow $\Delta H$ is positive. When the price is lower than $p_{e}$, the net flow is negative: the supply of rental apartments is shrinking

The new price $p'$ indicates scarcity and, as time passes, m new constructions would be added to the former $H_{e}$ stock of houses, bringing down the price until it again reaches $p_{e}$

During the transition, existing residents suffer from higher housing costs. As a consequence, they may persuade the local government to impose a rent-control law immediately after the demand shock, limiting price to $p_{c}$

By keeping the price artificially low under conditions of high demand, the rent control prevents the developers from responding with an appropriately massive burst of new construction

Eventually, the slow-growing housing stock reaches a size of $\hat{H}$. At that point, rent control does not bite anymore and the rental price, at last, reaches the equilibriu $p_{e}$

One justification for rent control is to help low-income renters. However, rent-control benefits are usually distributed in a highly unsystematic fashion

Another way of reducing housing costs without interfering directly in prices is to subsidize consumption explicitly

Besides, housing-subsidy programs may serve two goals: raise the standard of living of the targeted group and curb the negative externality associated with "slums"

Given that twin focus, the following analysis considers two effects of subsidy programs: the increase in the utility of the targeted group (the rise in the standard of living), and the slum-reduction effect, i.e., the rise in housing consumption generated by the subsidy

The slope of the budget constraint is $\frac{dq}{dc}=-\frac{1}{p}$

The low-income consumer maximizes his utility at the tangency point between his indifference curve and the budget line (points $q_{0}$, $c_{0}$)

Assume that the government pays a fraction $\beta$ of the household's rent bill $pq$. Now, the household pays only a fraction of the former bill, and his new budget constraint is $y=c+(1-\beta)pq$. The respective new slope is $\frac{dq}{dc}=-\frac{1}{(1-\beta)p}$

The household can reach a higher utility level - effectively, housing prices are now lower (the government is subsidizing a fraction of the rent)

Besides, the PRS program achieves slum reduction by increasing the household's consumption of housing from $q_{0}$ to $q_{PRS}$

The budget line is now $y+G=c+pq$, and it shifts in a parallel fashion, without changing the slope

Both the consumption of housing $q$ and bread $c$ are now higher, and the consumer can reach a higher indifference curve

The IG program yields to a higher utility level - households are free to choose the optimal quantities of $q$ and $c$. On the other hand, the slum-reduction effect is higher under the PRS program

The resulted optimal consumption point is shown in the figure $(q_{HV})$. When the household is free to choose his optimal consumption levels of $c$ and $q$, he would like to consume $q_{IG}, c_{IG}$. Using the housing voucher, he cannot choose more bread than $y$, preventing him from moving onto the dashed segment

In this scenario, the $HV$ program generates an outcome that lies between the $PRS$ and the $IG$ outcomes

Assume that the household pays the same rent to the government as he/she paid for his original unit where the housing consumption was $q_{0}$. Also, suppose that the government invested $G=HV=\beta pq$ (same as the other subsidies) in building better dwellings

Since the consumer is paying the same rent $pq$ and getting the same income $y$, he will consume the same amount of $c$ as in the pre-subsidy situation.

Because the government spends $G$ per household in those new dwellings, the vertical movement of the budget line is the same as in the Income Grant or Housing Voucher programs. As a result, the household consumes the same old $c$ and a new $q_{PH}$

One can see that the Public Housing program yields the most considerable slum-reduction effect of all the plans - $q_{PH}>q_{PRS}>q_{HV}>q_{IG}$ -, while generating the smallest increase in utility level

On the other hand, in terms of utility, $U_{IG}>U_{HV}>U_{PRS}>U_{PH}$

Society's choice of a particular program would thus involve opposing interests