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I was thinking about quantity discounts and if there is a possibility to model them not as bundles (as is typical for second price discrimination) but rather as prices being some continous functions of quantity demanded, meaning that consumer pays some autonomous price $P^a$ per purchased quantity as well as some quantity-dependent price $P^v$.

What I have in mind especially are these four budget constraints:

$$ BC_1: \sum_{i=1}^n \left( P_i^a+ \frac{P_i^v}{x_i} \right) x_i \leq M $$

$$ BC_2: \sum_{i=1}^n \left( P_i^a+ \frac{P_i^v}{x_i^2} \right) x_i \leq M $$

$$ BC_3: \sum_{i=1}^n \left( P_i^a+ \frac{P_i^v}{(x_i +\alpha_i)} \right) x_i \leq M $$

$$ BC_4: \sum_{i=1}^n \left( P_i^a+ \frac{P_i^v}{(x_i^2 + \alpha_i)} \right) x_i \leq M $$

The questions are:

  • Do these constraints make sense to investigate?
  • Are there any mathematical hardships when solving for basic utility functions trio (cobb-doug; subst; compl)?
  • Which one of these would be better in sense of: easy/effect?
  • Is there some more efficient alternative than what I tried here?

I know that $BC_1$ is nothing more than linear constraint, but still I am curious if this might be considered as quantity discount: The price is lower depending on how much I have bought. Then, $BC_2$ formulates something kinda as a squished circle so it does not allow not-buying some goods (similarly to previous case) which makes little to no sense. I also do not get why $BC_1$ and $BC_3$ look so different when visualising them in geogebra. The same case for $BC_2$ and $BC_4$.

Thank you very much.

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Nice question, a nice effort of imagination.

Just some general observations and suggestions, as the answer could be prohibitively long and time-consuming, if one had to made all the calculations and verify all the cases.

1) All your budget constraints describe a price discount on quantities, as the overall price paid for each unit of a good will decrease as the quantity of the good increases. This is true also for case $1$.

Actually, the overall price of one unit of good $i$, let's call it $P_i^O$ is given by: $$P_i^O= P_i^a+ \frac{P_i^v}{x_i}\;\;\;\;\;\;\;(1)$$

which evidently decreases as the quantity $x_i$ increases.

From an intuitive point of view, how can this case be considered? Let's write the budget constraint $BC_1$ in the following way:

$$\sum_{i=1}^n \left( P_i^a+ \frac{P_i^v}{x_i} \right) x_i = \sum_{i=1}^n P_i^a x_i+ \sum_{i=1}^n P_i^v \leq M$$

The last sum is a term independent of the quantity $x_i$, an amount paid independently of the quantity of the good.

I suggest that it can be seen as a 'ticket', as if a fixed sum paid when you enter the shop of the good you want to buy, and obviously this is a price (per unit) that is lesser the greater the quantity you buy.

From a formal, mathematical, point of view this case is completely analogous to the usual linear budget constraint, so the formal treatment is the same.

2) The budget constraint $BC_2$ of course doesn't allow solutions in which some good is not bought. In $BC_2$, $0$ solutions must be excluded, and this can be immediately seen observing that there is the term

$$\frac{P_i^v}{x_i^2}x_i= \frac{P_i^v}{x_i}$$

which is not defined for $x_i=0$.

3) The budget constraints $BC_1$ and $BC_3$ look very different because the first one gives you a linear budget constraint, as $x_i$ in the denominator in the second term in the sum simplifies with the $x_i$ that multiplies it, in $BC_3$ there isn't a simplification like that, so we have a very different, not linear function, the overall price is not a linear function of the quantity as in $BC_1$.

And analogously for $BC_2$ and $BC_4$, they use very different functions.

4) You wrote

  • Are there any mathematical hardships when solving for basic utility functions trio (cobb-doug; subst; compl)?
  • Is there some more efficient alternative than what I tried here?

You have tried not linear forms of functions relating negatively the overall price $P_i^O$ to quantity, as in equation $(1)$. Why not to try the simplest linear form?

$$P_i^O= P_i^a- P_i^vx_i.\;\;\;\;(2)$$

Anyway, of course, the possible analytical intricacies of the various budget constraints can be ascertained only making the calculations with various cases of utility functions, verifying if nothing prohibitively cumbersome occurs.

What I could suggest to frame the problem is to use a generic form for the quantity discount, that is a general differentiable function $p_i$ relating negatively the overall price $P_i^O$ of a good $i$ to its quantity, that is

$$P_i^O= p_i(x_i)$$

with $p'_i <0,$ for all $i=1,…,n$.

(Your price functions in your $BC$ are particular cases of this $p_i$).

Let's formulate the problem of consumer's maximization with these $p_i$ using Lagrange multipliers (the budget constraint taken as an equality). I limit myself to two variables.

The problem is

$$max \;\;\;\;U(x_1,x_2)$$

s. t.

$$p_1 (x_1) x_1+p_2(x_2)x_2=M,$$

where $U$ is a differentiable utility function with the usual properties.

The Lagrangian is:

$$\mathcal {L}= U(x_1,x_2)+ \lambda (p_1 (x_1) x_1+p_2(x_2)x_2-M)$$

This first order conditions are:

$$\frac {\partial U(x_1,x_2)}{\partial x_1}+\lambda (p_1(x_1)+p’(x_1) x_1=0.\;\;\;\;(3)$$

$$\frac {\partial U(x_1,x_2)}{\partial x_2}+\lambda (p_2(x_2)+p’(x_2) x_2=0.\;\;\;\;(4)$$

$$p_1 (x_1) x_1+p_2(x_2)x_2-M=0$$.

Eliminating $\lambda$ from $(3)$ and $(4)$ we have:

$$\frac {\frac {\partial (U(x_1,x_2)}{\partial x_1}}{\frac {\partial (U(x_1,x_2)}{\partial x_2}}=\frac{p_1(x_1)+p’(x_1) x_1}{p_2(x_2)+p’(x_2) x_2}\;\;\;\; (5)$$

The conditions $(5)$ are not very different from the usual first order conditions

$$\frac {\frac {\partial (U(x_1,x_2)}{\partial x_1}}{\frac {\partial (U(x_1,x_2)}{\partial x_2}}=\frac{p_1}{p_2}\;\;\;\; (6)$$

the difference is that the second side of equation $(5)$ is not the ratio of prices as in $(6)$, but the ratio of the derivatives of $p_i (x_i) x_i$ with respect to quantities $x_i$ (but nothing surprising, except the fact that these conditions are not so elegant as usual).

The first side of $(5)$ is the same as in the classical problem of consumer’s maximization, the ratio of marginal utilities, so I can’t see analytical difficulties in writing these first order conditions in the case of differentiable functions as a Cobb Douglas.

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    $\begingroup$ Thank you very much! This was really great! I must admit that there are so many "eye-openers" there... The thought of quantity discount as being connected to the tickets was initially so alien to me but right now I can see the lesson from it, because it incentivizes consumers to buy as much as they can such that the relative price of entering the shop is minimal. I really appreciate the idea with solving it in general form (I did not think of it up till now) but I must say it makes all problems dramatically easier to solve. So thank you very much once again! $\endgroup$
    – Athaeneus
    Jan 20, 2023 at 19:27
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    $\begingroup$ You are welcome! I appreciated your ideas and your imagination, because this is the way to become a reseacher! Try new ideas and ask possible questions. The idea of the ticket is mine, it is just an idea to give some intuitive content to your first budget constraint, but it is possible to imagine a big supermarket where you pay a ticket to enter ... because is particularly good and so you have the incentive to buy a lot of goods... $\endgroup$ Jan 20, 2023 at 20:11

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