# How do you formulate a distance constraint and a budget constraint?

Everybody knows about budget constraints and how they are represented:

but what if I want to represent a distance constrain from the shop you buy the goods? How can I build that?

• Depends on what kind of situation you are trying to capture/describe in your model. You can assume that shops locate along some geometric shape, line or circle or full blown grid, then people randomly populate the shape/area you can further then model some disutility from travelling or if you want explicitly limit distance of travel. Consider being bit more specific what kind of situation you are even trying to model.
– 1muflon1
Oct 22 '20 at 11:00
• In order to simplify i was thinking about the KM from their homes Oct 22 '20 at 11:10
• but that does not help at all those are just units. Against you can have line, circle or grid where distances are measured in kilometers. You can just add units to elementary cartesian coordinate system and impose some restriction on absolute distance $\sqrt{x^2+y^2}$ traveled - to be clear I am not saying you should do that that is just an example. Since I know nothing about the context of situation because it was not provided in your Q. Depending on different context different assumptions might be appropriate
– 1muflon1
Oct 22 '20 at 11:12
• thanks for the help! i will see Oct 22 '20 at 11:32

Because you are talking about constraints, it appears you do not consider the case of inserting such "access costs" (because this is what they are) in the utility function. It implies that they do not create disutility directly, only direct or indirect monetary costs.

Let $$d$$ be the distance in some units. Let $$C_d = c_dd$$ be a linear (for simplicity) distance cost function, where $$c_d$$ is the average cost per unit of distance. This cost may include fuel consumed, or opportunity costs of spending the time to go to the shop. It is realistic to also postulate the existence of a strictly non-zero minimum distance, $$d_{min}$$, because the shops are not inside your house.

Your budget constraint now looks like (the undefined symbols have the usual meaning)

$$\sum p_ix_i + c_dd \leq I,$$

while you also have the additional constraint $$d \geq d_{min}$$. So the Lagrangian is here

$$U(\mathbf x) + \lambda \left(I-\sum p_ix_i - c_dd\right) + \mu(d- d_{min}),$$

with $$\lambda, \mu$$ being the non-negative Karush-Kuhn-Tucker multipliers. Your optimization problem involves maximizing the Lagrangian with respect to the $$x_i$$s and with respect to $$d$$.

One could specify reasonably a convex non-linear distance cost function. Finally, one could include $$d$$ in the utility function as generating directly disutility.

I guess you can take it from here.

Just note that with the formulation as above, you should obtain that at the solution, $$\mu^* >0$$ meaning that the constraint will be binding and so you will always choose to go to the nearest shop... which aligns, at least in this case, with intuition (don't start thinking about shop/product differentiation, that would be a whole different situation).

• Thanks a lot! I tried to build something very similar since I was working with perfect substitutes. Oct 25 '20 at 20:23
• but should we use di since there could be many shops ? Oct 25 '20 at 21:45
• @kala123 Yes we could. Oct 25 '20 at 22:44