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

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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