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Demand with agglomeration effects

In this video, Glaeser sets up a demand curve for housing in a city, where there are agglomeration effects: bigger cities have higher wages, so demand (willingness-to-pay) is increasing in city size.

First, he derives the aggregate inverse demand curve for non-financial benefits, starting from individual willingness-to-pay. (These amenity benefits are constant and do not vary with city size.) Note that the 'good' here is binary: living in the city vs. not. So each person demands one unit, and we sum across people to get aggregate demand. (He skips over this step, going from rank to total size.) Next, we introduce agglomeration effects: the financial benefit (wages) are increasing in city size. Then to get total demand, we simply sum up the two sources of benefits.

Is this correct? I have questions about several steps:

  • The inverse demand curve for non-financial benefits is not technically a function of total quantity (city size), so it seems wrong to put it and the wage function on the same graph and sum them. Ie. we interpret the demand curve as the quantity demanded for a given price, so we can't add it to something that's not a function of price.
  • The proper method would be to work with the individual demand functions: add the wage benefit to the amenity benefit for each person, then sum to get aggregate demand. How does this approach work when the wage benefit is a function of total quantity?
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