A Simple Application of Lagrange Multiplier on Location Choice
My previous article Yiu (2019) introducing Lagrange Multiplier receives a very good view number. This article further illustrates its application in a simple residential location choice issue.
Lagrange Multiplier for Optimizing the Residential Location Choice
It is a common struggle to choose between living closer to the city centre but at a smaller flat or living at a bigger house but far from the centre. There are, of course, many other factors, but just for illustration's sake, let’s fix on these 2 factors only. The decision depends very much on the different considerations of each household, some may treasure more on housing size, but some may find the accessibility of utmost importance. The following examples show how to apply Lagrange Multiplier to find the optimal residential location and size when the considerations are different.
If households are choosing between housing size and distance from the city centre in their residential location choice, i.e. the closer to the city centre, the smaller the housing size because of the higher rent. Assuming that the choice is subject to the household’s budget line, say g = x+y-c, where
· x sm is the housing size of living floor area in square meters,
· y km is the distance from the city boundary (assuming a circular city of radius 100km from the centre to the boundary), and
· c is a constant
Assuming that the household’s indifferent curve is convex to the origin, i.e. the household is willing to sacrifice more of one variable when the other variable is too little. Let’s say the equation of the curve is f=xy.
In other words, the optimization task is to maximize the objective fn, f=xy; subject to the constraint fn, g=x+y-c.
Applying the Lagrange Fn Z: xy + L(x+y-c),
put dZ/dx = y+L = 0; put dZ/dy = x+L = 0; put dZ/dL = x+y-c = 0;
thus L = -c/2, x=y= -L
Graphically, it is to find the tangent point of the budget line (g=x+y-c) to the outermost indifference curve (f=xy).
3. Example 1
If the household’s weekly income is only $1000, thus the max housing size affordable is 100 sm (every 1 sm bigger would cost $10 more) located at the city boundary (i.e. 0km from the boundary, or 100km from the centre), or the affordable housing size would be 0 sm if it is located at the city centre. (every 1km closer to the city centre would cost $10 more)
i.e. g = x+y+100
thus, the optimal location and size are 50 km from the city centre and 50 sm in size (middle size, the middle between centre and boundary).
But for low-income households, living in the city centre provides more job opportunities, their jobs require very long working hours and the traveling cost is very expensive, so they are more willing to sacrifice their housing size than the distance to the city centre.
Let’s say their indifference curve becomes f2 = x(y-50)
Z: x(y-50) + L(x+y-100)
put dZ/dx = y-50+L = 0
put dZ/dy = x+L = 0
put dZ/dL = x+y-100 = 0
thus x=-L, y=-L+50, i.e. -2L+50–100 = 0, therefore, L=-25
These low-income households prefer living at a smaller sized home (25sm), but closer to the city centre (at 25 km from the city centre, or 75 km from the city boundary).
Yiu, C.Y. (2019) An Introduction to Lagrange Multiplier on Solving Optimization Questions under Economic Constraints, Medium, Mar 20. https://medium.com/me/stats/post/f9bc9b439169