 # (AT) Real Estate Bid Rent Function: Commuting Cost & Income Affect Housing Demand

November 10, 2019

How do you do? I’m Dr. John Sase. Welcome to Urban Economics, Advanced Topic number five. In this video, we will consider the effects
that changes in commuting costs and changes in income have upon the demand for real estate
in an urban area. Though the bid-rent curve can be modeled alternatively with a quadratic
function or topological algebra, we will use the standard negative exponential rent- density
function introduced by Professor Edwin Mills and other pioneers in this field.
Let’s present our two basic considerations: An increase in household income affects the
slope of the residential bid-rent function. Generally, the square footage of real estate
rented is directly related to income. However, the optimal distance from the Central Business
District (CBD) may vary due to other factors. In addition, commuting costs will impact the
bid-rent function. A higher elasticity of commuting cost compared to the income elasticity
of demand for land leads to a steeper bid-rent function as income rises. Commuting costs
in themselves are a matter of both time-cost and physical-cost. This steepening of the
bid-rent curve suggests that as income increases, residents may prefer to move closer to the
CBD due to the relative value of the time-cost or move further out as other amenities take
precedence. For example, many high-income individuals prefer to reside in central Manhattan
close to Central Park or locate further away in the Hamptons on the eastern end of Long Island. Nevertheless, we generally expect bid-rent
functions to be concave outward, away from the origin. This may explain why the negative
exponential function has been agreeable to use by urban economists for decades. With
this or similar models, we assume that the horizontal axis represents the distance from
point zero at the City Centre (CBD) and that the vertical axis represents the Rent density
as a function of the distance from the CBD. Therefore, we can express our bid-rent function
as Rent equal to R(mu), a function of the distance mu miles from the City Centre.
We may also consider in this model that Rent is a function of radial distance. The curvature
of the function is affected by the commuting cost per mile (which in itself is determined
by a number of other factors) and the quantity of land demanded by a household (which in
turn is determined by various needs and wants). Therefore, we can express the slope of the
bid-rent function as the first derivative taken in respect to the distance mu. This
derivative can be identified as the inverse ratio of commuting cost to the quantity of
land demanded. The slope of the bid-rent function is not
constant. Rather, it decreases at a decreasing rate. This suggests that the individual values
of t (commuting cost) and/or L (demand for land) as well as the ratio of t and L may
vary with distance from the CBD. Generally, we assume that commuting cost t
is a positive function of income such that as income increases, the preference for more
comfortable and swifter transport offering greater privacy will be chosen. Likewise,
the demand for land L is a positive function of income such that a greater quantity and
quality will be chosen. Therefore, the ratio of these two functions
will be negative with a rate of change that decreases at a decreasing rate as the radial
distance from the City Centre increases. As a result, we can rewrite the first derivative
of the bid-rent function as the negative of the commuting cost function times the land
demand function taken to the power of negative one. Correspondingly, the second derivative of our bid-rent function can be expressed as
the negative of the first derivative of the commuting cost function, in respect to income
times the land-demand function taken to the power of negative one, plus, the commuting-cost
function times the land demand function taken to the power of negative two, times the first
derivative of the land-demand function in respect to income.
For simplification, we may factor out the inverse of the land demand function so that
we are left with the second derivative of bid-rent being equal to negative 1 over the
land demand quantity times the quantity represented by the difference between the first derivative
of the commuting cost function and the ratio of the first derivative of land demand times
the ratio of the commuting cost to the quantity of land demanded.
Next, let’s factor out the ratio of commuting cost to income. In turn, this leaves us with
the second derivative of the bid-rent function being equal to the ratio of commuting cost
to the product of income times the quantity of land demanded times the quantity of the
difference between the first derivative of the commuting cost function times the ratio
of income to commuting cost and the product of the first derivative of the land demand
function times the ratio of commuting cost to quantity of land demanded times the ratio
of income to the quantity of land demanded. In short, the curvature of the bid-rent function
changes in respect to individual change in commuting cost and land demand, both in respect
to income, as well as the proportional relationship of the two costs to one another and to income.
Put more simply, this second derivative that expresses the rate of change in respect to
radial distance equals the negative of the ratio of commuting cost to the product of
income times the quantity of land demanded times the quantity of the difference between
the income elasticities for commuting cost and land demanded.
The two terms within the brackets are the elasticities. The first one is the income
elasticity of commuting cost and the second one is the income elasticity of land demanded.
I hope this has helped you to understand this somewhat terse topic. If it has, please Like,
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