Upward-sloping Demand curves for Labour and Housing

I have a simple thought-experiment ("model") that helps me think about the relationship between: migration; planning restrictions; wages; and house prices (or rents).

Assume all workers are identical, all houses are identical, and it's strictly one worker lives in one house. The government of a small city/state controls both immigration and the number of houses that are built. Because the city/state is small, relative to the world, it has no effect on the rest of the world. There's a perfectly elastic supply of labour from the rest of the world at a real wage w*.

There is some sort of economies of scale or agglomeration effect that causes labour productivity in the city/state to be an increasing function of the number of workers who live there. And self-employed workers earn the Average Product of Labour (there are no other inputs).

There are four types of equilibrium: 1. Where the migration constraint is binding because the government allows in fewer workers than it allows houses to be built; 2. Where the housing constraint is binding because the government allows fewer houses to be built than it allows workers to enter; 3. A kife-edge equilibrium where the government allows exactly the same number of houses and workers; 4. An equilibrium where the government allows free entry and free building and the limited land of the city/state is the binding constraint.

Here is what the four types of equilibrium look like:

1. House prices and rents are the same as in the rest of the world (houses can be imported from abroad and erected on the city/state) but wages are higher. Small relaxations on entry (not big enough to make the building constraint binding) cause wages to rise (because productivity rises due to agglomeration effects) but do not affect rents.
2. House prices and rents are higher than in the rest of the world, and so are wages. But the wage differential exactly equals the rent differential, so that real wages (adjusted for the higher cost of housing) are exactly the same as in the rest of the world (otherwise it can't be an equilibrium because more workers would try to enter, pushing up house prices and rents). Small relaxations on building cause house prices and rents to rise and wages to rise but real wages to stay the same because both rise by the same amount.
3. An indeterminate mix of 1 and 2 above.
4. Exactly like 2 above, except that it is landowners who capture the benefits from the high rents, rather than the government auctioning off the building permits (or the government's friends who get given the new building permits for free). Real wages (adjusted for the higher cost of housing) are exactly the same as in the rest of the world.

(I would like to relax the simplifying assumption of one worker one house, but it's too hard for me.)

This post is for Open Borders YIMBYs. Your stories about whether standard Econ 101 supply and demand apply to labour and to housing have got to be consistent.

Comments

[Kevin Erdmann]

I think the assumption that higher wages are coming from agglomeration is doing a lot of work here. What if wages are also higher because of geographically captured production networks? Think of 20th century Detroit. Auto manufactures were captured and entry into that labor market was controlled by unionization.
Similarly, maybe tech firms are captured by Silicon Valley, and entry into that labor market is controlled by housing limits. If that is the constraint pushing wages up, then excess wages are still claimed by rent but demand slopes down.

[Nick Rowe]

Kevin: Aren't "geographically captured production networks" the same as (or one example of) "agglomeration effects"? (I may be using the words wrongly, but that's the sort of thing I meant.)

[Kevin Erdmann]

I don't think so. There are benefits to location, and maybe even some productivity boosts from increased size. But, if, say, we had a rule that where Vidalia onions are grown, half the acreage must lay fallow each year, that would raise the price of the acreage that could be planted. Getting rid of that rule might in some ways help harvesters be more productive, but the increase in supply would dominate, and the value of the acreage that could be planted would fall as other acreage was brought on line. It seems like you are simply assuming that the productivity factor dominates. Maybe that assumption isn't accurate.

[Nick Rowe]

Kevin: agreed. Demand curves usually slope down for all the standard reasons. But I hear Open Borders people saying that increased immigration will cause wages to increase, because of scale/agglomeration effects. So here I am assuming they are right, and asking what that means for house prices.

[Chris Auld]

Quibble: I don't think there are upward-sloping demand curves in this model. All else equal, an increase in the price of housing has no effect on quantity of housing demanded (up until we hit budget constraints, at which point people... leave the country?) So demand for housing is weakly downward sloping. If we think of each person as being a firm supply and demanding a unit of labor, then demand for labor is perfectly inelastic. The objects which slope up--wages and house prices as functions of population levels--are locii of equilibria for the economy, not demand schedules.

The framework reminds of David Albouy's papers on recovering city amenities from data on housing prices and wages, eg,

http://www.fernandoleibovici.com/canada_qolprod.pdf

(which I like to mention because Victoria #1!)

[Kevin Erdmann]

In my onion example, the harvesters could have increased productivity and wages while the value of harvested land declines.

Isn't your equilibrium #4 basically Texas? Can't an awful lot of wage growth happen before land constraints create rents?

[Ralph Musgrave]

The agglomeration effects are feeble. New Zealand has a similar land mass to the UK. It’s culture (language, legal system, educational system) is very similar. It’s population is less than a tenth that of the UK, but it’s GDP per head is slightly higher.

[Nick Rowe]

Chris: "Quibble: I don't think there are upward-sloping demand curves in this model. All else equal, an increase in the price of housing has no effect on quantity of housing demanded (up until we hit budget constraints, at which point people... leave the country?)"

I want to pick a nerdy fight with you over this! We can't really say "*all* else equal". Whenever we draw a (labour for example) demand curve, we are holding some things constant, but varying some other things as we move along it. For example, if we have a traditional (diminishing returns) production function where labour and land produce wheat, do we hold the quantity of land constant when we move along the labour demand curve because the supply of labour increases (in which case land rents increase as we move along it); or do we hold land rents constant (in which case the quantity of land employed increases as we move along it)? It all depends on the model, which things stay constant and which things change when labour supply increases.

And even though we normally (following Walras(?)) read demand curves from P to Qd, in this case (where they slope up) it is easier to follow Marshall and read them from Q to Pd. "Demand price (Willingness To Pay) is a function of Quantity". Take the example of the demand curve for telephones. One telephone has a demand price of $0, because there's nobody you can call. Two telephones have a higher marginal demand price, because there's 1 person you can call. And so on, so Pd is an increasing function of Q (up to a point, presumably, unless everyone is identical). Now you could say that the quality of phones is an increasing function of Q, and that *shifts* the demand curve right. But the whole point of drawing a demand curve is that it stays put and tells us what happens when Supply shifts. It's supposed to show us the loci of equilibria.

Interesting idea about estimating amenities from P and W data. If amenities are an increasing function of city size (because each amenity has a fixed cost) that would be another reason why housing demand curves might "slope up".

[Nick Rowe]

Kevin and Ralph: imagine a purely agricultural economy, where land and labour produce food, with diminishing returns. Just like in Malthus or Ricardo. In this case the labour demand curve slopes down (adding more labour to a fixed stock of land reduces average and marginal product of labour, and wages). The real world is probably a mix of both diminishing returns and increasing returns (which is why cities exist, rather than the population being spread out evenly, only being denser on the best land). I have deliberately rigged my model here to ignore diminishing returns to try to capture what I hear the Open Borders people say about migration increasing wages. And I'm saying that, even if that is true, we need to think about house prices.

[Chris Auld]

Nerd fight it is! Let's make the discussion more abstract, and let's stick with the (accepted, I'd say) definition of demand as giving quantity choices as functions of exogenously given prices.

Suppose demand and supply are both functions of some third variable, say, population: D=D(p, N) and S=S(p, N), and that the partial derivatives of these functions with respect to price are negative and positive, respectively. Now consider the objects Q*(N) and P*(N), the sets of equilibrium quantities and prices as we vary N. Suppose Q* and P* are both increasing functions of N. Do we say, then, that demand slopes up? Nope, demand still slopes down, but in equilibrium higher prices are associated with higher quantities. This is what I'm asserting is going on in your model.

In your example we hold both land rents *and* the quantity of land constant as we move along the demand curve for labor. It doesn't matter that that can't actually happen: the demand curve for labor is a theoretical object that answers a specific causal question: what is the ceteris paribus effect of a change in price on quantity demanded? We hold all other prices and inputs constant when conducting this experiment even though in reality other prices and inputs might, or even, must, change along with price. For example, we conventionally hold consumers' incomes constant as we vary the price of some good in the standard partial equilibrium setup, even though incomes are in reality determined simultaneously along with prices. Or in your telephone example, if demand for telephones is, say, T( p , R), where p is price and R is the proportion of the population who own phones, then T_p still gives the demand curve for phones even though in general R is also a function of p.

Finally, I'm not sure I agree that the "whole point of drawing a demand curve is that it stays put and tells us what happens when Supply shifts." In my example above, there is only one exogenously given variable, N, and both supply and demand shift when N shifts. Nonetheless, the supply and demand schedules answer well-posed causal questions and the model predicts how the observables should move as exogenous shocks strike.

[Nick Rowe]

Chris: "Suppose demand and supply are both functions of some third variable, say, population..." Stop right there! It's not really a third variable.

Take the standard assumption that Q = min{Qs,Qd} (Quantity actually traded is whichever is less, Quantity supplied or Quantity demanded.) Now suppose that Qd = D(P,Q). (This is exactly like the Keynesian Consumption Function (in a model where consumption is the only source of demand, for simplicity.) And it's also like the demand for telephones.) Linearise it for simplicity, so Qd = a -bP + cQ. I can set Qd=Q and rewrite it as Qd = [1/(1-c)][a-bP]. (And if c > 1, then that curve slopes up.)

Now I see your point. You want to call that second function an equilibrium condition (which it is) rather than a demand function. OK. But it's not like Q is some third variable that merely happens, as a matter of contingent fact, to be related to Qd or Qs. It's far more closely related than that. It's inherent in the very idea of voluntary exchange that Q=min{Qd,Qs}.

Stepping outside the nerdy fight, I'm actually comfortable either way, and it's more a matter of convenience. What we call the "Aggregate Demand curve" in macro is an equilibrium condition in exactly the same sense, and not strictly a demand curve in your sense. But then the demand curve is an equilibrium condition too.

[Chris Auld]

Sure, that your mechanism is called a social multiplier, following Manski, who was highlighting the analytical equivalance to the Keynesian multiplier. But that's not what I mean: in your model, there is a third variable driving the results: population.

Consider housing. Again, you assume every person presents inelastic unit demand for housing, so demand doesn't slope up, right? If a house costs p and a person's income is w, then each individual presents demand q(p,w) = 1 for all p < w, and for p > w, q(p,w)=0. If we sum those to get market demand Q(p), we get a weakly downward-sloping function. Demand for housing does not slope up. Note neither N nor Q is an argument in the demand function: any effect of population on demand for housing is mediated by prices and incomes. Also, we recover the demand schedule for housing without imposing the equilibrium condition Q=N or the equilibrium relationship between incomes and wages.

The equilibrium price for a house, p*, depends on the population size, N, and also equilibrium wages w* depend on N, as you describe in your cases 1 through 4. And due to the externality, p*(N) and w*(N) are both weakly increasing in their arguments. So for a population of size N, we will observe Q*=f(p*(N), w*(N))=N, assuming everyone can buy a house. Then Q* and p* move in the same direction, but that locus of housing prices and quantities isn't a demand curve.

I'm repeating myself. I suppose the shorter version of my claim is: population is too really is a third variable! I think you're claiming that it's not due to equilibrium relationships between population and other variables in your model, but I don't agree that we impose those relationships when deriving conventionally-defined demand for housing or labor.

[Chris Auld]

Typepad seems to have swallowed some symbols: that claim in the second paragraph is q(p,w) = 1 if p is less than w, and q(p,w) = 0 otherwise.

[Nick Rowe]

Chris: I edited your comment, to add a space before and after the inequality signs (otherwise Typepad thinks it's a link).

[Majromax]

>> House prices and rents are higher than in the rest of the world, and so are wages. But the wage differential exactly equals the rent differential, so that real wages (adjusted for the higher cost of housing) are exactly the same as in the rest of the world

Wait a second, I think you've made an oversimplification in assuming that there's a single "rest of the world". What if there's "the rest of the world" and Toronto and Vancouver and San Francisco?

In that case, I think you have a story analogous to the one you wrote about patents, where a new excludable benefit (there wheat technology, here living in a particular city) can only charge as much as its marginal benefit over the next-best benefit. If you assume that city housing markets set quantities rather than prices, we'd still be in a Cournot-Nash equilibrium that fails to capture the whole nominal wage gain.

[C Trombley]

Another theoretical question is: let's say you had a perfect LVT so rents are completely distributed back to laborers (perhaps directly, perhaps by providing public goods). Then would the system go to a corner solution where everyone is in the city? Yes, I think so. The trouble is caused by the assumption of no feedback and zero cost of migration. Perfect LVT would set the PV of the wage differential at exactly the cost of migration.

Of course, the big problem is the law of one price - wages equilibrating everywhere. With apples, maybe. But wages? This guy (and his handsome first commenter) had some interesting things to say about this:

http://worthwhile.typepad.com/worthwhile_canadian_initi/2017/01/humes-sailing-ships-correcting-samuelsons-icebergs.html

[Nick Rowe]

Majro: I don't think I follow. Maybe it will come to me.

CT: Yes, I think that's right. And people would be desperate to get in, even when there is no more land or housing. (And, in a slightly different model, might even choose to live there homeless, just to collect wages and their share of the Land Value Tax.)

[Nick Rowe]

CT: Hang on. Doesn't the model say that rents will explode to infinity with a redistributed LVT? The higher are rents, the higher the LVT, so rising rents creates no disincentive to try to emigrate to the city?

[C Trombley]

Yeah, the equilibrium is ambigous because lim N-> infinity rent(N)/LVT(N)= infinity/infinity is ambiguous.

Consider the city of Henryton. Henryton explores LVT policy cautiously so that they never accidentally tax wages or capital. As a result they undershoot. In the long run the rate of people coming in equals the rate of increase of the LVT. The accountant rent goes to infinity, but the post-LVT value rent converges by L'Hopital's Theorem.

[Nick Rowe]

CT: (vaguely remembering L'Hopital's rule) I think that's right.

I really need to relax my assumption of one worker to one house. If we stick housing in the Utility function, in the normal way, I think the LVT case would have an (inefficient?) equilibrium with severe overcrowding. Like the common resource problem. But I need to think about it.

[Majromax]

> Majro: I don't think I follow. Maybe it will come to me.

I'm probably being unclear since I'm not a fluent economist. I'll instead make my point in the form of a question:

In your earlier post, you described how a patent-holder could only charge users the marginal benefit over the next-best technology because of competition with other technologies, even if those technologies were defended by their own patents.

How does that situation differ from this one, where you say landowners in a (supply-restricted) city can charge workers their full wage gain and not just the marginal gain over the next-best city?

[Nick Rowe]

Majro: OK, I get you now. Suppose there are several other cities. What my model calls "w* in the rest of the world" would become the wage (net of housing cost) in the *best* of those several other cities *that you can actually migrate to*. So we are talking about the marginal benefit over the next-best (accessible) city.

[Muhammad Anees]

Upward sloping demand curve is mostly due to other factors that shift the demand curve as if we control for the effect of such factors, business will reduce demand for workers with higher wages. It is a simple fact and can be seen everywhere. The fact is, business only pay higher to workers when it see the other factors than productivity of the labour can add more competitive advantage to the business value creation process. For some of the technical issues in such economic models and through econometric modeling here: Econometric Modeling