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I want to know under what preferences relation will I not want to consume all of my budget. Because if my preferences are strictly monotonic, strictly convex or convex, even LNS or continuous. I would consume all of my budget. Are they any situations where I would not?

Edit: when I meant preferences I am treating each individual case i.e. if it is LNS it is not convex (necessarily)

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Yes. Suppose the consumer has the following utility function:

$$u(x) = - \sum_{i=1}^n x_i$$

Here's a different example, where commodities are goods instead of bads:

$$v(x,y) = -(x-a)^2 - (y-b)^2$$

When the consumer's budget $m$ is strictly greater than $p_x a + p_y b$, then the consumer will not consume all of their budget.

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  • $\begingroup$ I don't know who the downvote belongs to (OP?) but would they care to explain? This function is strictly monotonic, convex, LNS and continuous yet the budget would not be consumed. So it is a perfect example. $\endgroup$ – Giskard Dec 1 '16 at 21:55
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    $\begingroup$ @denesp Downvote aside, we are talking strictly about "bads" here, not goods. Mathematically we are fine, but how useful is this if we want to maintain some rapport with the reality we try to model? $\endgroup$ – Alecos Papadopoulos Dec 1 '16 at 22:22
  • $\begingroup$ I think "bads" are realistic. If complete realism is essential, then we probably need to throw out our standard models in microeconomics. $\endgroup$ – Theoretical Economist Dec 1 '16 at 22:27
  • $\begingroup$ @TheoreticalEconomist with bads why would they be classed in goods would it not be a cost? so you effectively cost minimise? like a production set? $\endgroup$ – Dashmone Dec 2 '16 at 20:39
  • $\begingroup$ @Dashmone that's sort of what the discussion above is about. Typically we think of consumption commodities as 'goods', but it's not clear to me why that should necessarily be the case. I'm not really sure what you mean by what you say next -- I personally don't think there's a meaningful distinction between costs and benefits, at least in standard models. A cost is just a negative benefit, and a benefit is just a negative cost. Note however that any utility function that exhibits the property that a consumer (with sufficient budget) would not exhaust their budget must have -- to be continued $\endgroup$ – Theoretical Economist Dec 2 '16 at 20:46
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Here is a case where the non-exhaustion of the budget can happen without it being preference-driven.

Assuming totally standard preferences, the budget most likely won't be fully consumed if we have some degree of indivisibility in goods. Note that the resulting "gaps" in the set of feasible consumption bundles has nothing to do with preferences: we assume that the consumer can order etc all conceivable consumption bundles, even those that are not available to him.

Note that by postulating the existence of a single price per good we unavoidably assume a single unit of measurement for the good. This allows us to reduce the divergence to the neighborhood integers.

If $(x^*, y^*)$ is the optimal bundle and (under standard preferences) $p_xx^* + p_yy^* = M$ then the candidate expenditure will be

$$E_1 = p_x\lfloor\tilde x\rfloor + p_y \lfloor\tilde y\rfloor < M $$ or $$E_2 = p_x\lfloor\tilde x\rfloor + p_y \lceil\tilde y\rceil $$ or $$E_3 = p_x\lceil\tilde x\rceil + p_y \lfloor\tilde y\rfloor $$

$E_1$ is certainly smaller than the budget, while $E2$ and $E3$ may be larger equal or smaller, depending on how big are the prices. If either $E1$ or $E2$ are within budget, they certainly dominate $E_1$ in terms of utility (given standard preferences and assuming both are goods and not bads), so one of them will be observed if they are feasible, while if both are feasibe, the one providing greater utility will be observed. Otherwise $E_1$ will be observed.

If one thinks that this is some artificial situation, with negligible deviation, it is not: it is one of the main drives behind the observed variety in packaging the same good (and a trend towards smaller packages), which is an attempt to reduce the degree of indivisibility and cover some of the gaps in the feasible consumption bundles set. Essentially by reducing the implied physical quantity that corresponds to "one unit", we lower the corresponding prices, and so make deviations from the optimal bundle $(x^*, y^*)$ smaller.

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Marginal utility would need to turn negative with more goods. It is trivial that if your utility function is monotonic, the budget will be exhausted, because spending increases utility. Questions to think: Why should they not exhaust the budget?

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  • $\begingroup$ That's what I was basically asking $\endgroup$ – Dashmone Dec 2 '16 at 20:35
  • $\begingroup$ You need to somehow value savings. First way: Turn the problem to a multiperiod one. Investing the money may make profit, but is not necessary. Then you need to look into discounting. It becomes a dynamic optimisation problem in which for every period you have the allocation vector. Second way: you simply take the saved money as one of the utility creating products. The utility needs to be somehow rational. It could be something like risk-free minus discount. $\endgroup$ – user3644640 Dec 5 '16 at 7:18

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