Budget

A definition for Budget

Economist want simple models to start to model things. One of the things we will model here is how do you describe what you can afford about some goods.

Budget Set

$$ \text{Budget Constraint inequality}:p^{T}x \leq m $$

We define Budget Set to be $P_{p, m} = \left\{ x \in \mathbb{R}^{d} : p^{T}x \leq m\right\}$

Composite Goods

$$ p_{1}x_{1} + x_{2} \leq m $$

Where $x_{2} = \sum_{i = 2}^{n} p_{i}x_{i}$. In this case, $x_{2}$ is called composite good, which is a sort of abstraction to simplify some calculus.

Opportunity Cost

Introducing this budget set enables us to introduce one important concept in economy. Suppose we just have two goods. Then, if you buy more goods of one, you need to buy less of the other. This is called opportunity cost. The opportunity cost of buying one more unit of good $1$ is the amount of good $2$ you need to give up to buy that unit. In this case, this is given by the slope of the budget constraint.

The opportunity cost is a more general concept that can be applied everywhere in life. Time is a clear budgeted item: if you choose an activity, you are giving up the opportunity to do another, so you say, there is a cost in choosing that specific activity.

Comparative Analysis

Budget Changes

Given a budget set $P_{p, m}$, we can analyze how a change in the budget affects the goods you can buy. Suppose we have a budget set $P_{p, m}$ and a budget set $P_{x, m'}$ where $m' > m$. Then, the budget set $P_{x, m'}$ is a superset of $P_{p, m}$, i.e. $P_{p, m} \subseteq P_{p, m'}$. This is because if you can buy something with a budget $m$, then you can buy the same thing with a budget $m'$. If you plot the budget set in 2D, you will see that the budget line is shifted outward. Exactly the opposite happens when the budget has shrank.

Changes in Prices

$$ tp^{T}x \leq m \implies p^{T}x \leq \frac{m}{t} $$

Which is then equivalent to a budget change. This notion is useful when you want to model inflation -> the effect of inflation is as if every person has less money to spend (i.e. budget).

The Numeraire price

$$ p_{1}x_{1} + p_{2}x_{2} \leq m $$$$ \frac{p_{1}}{p_{2}} x_{1} + x_{2} \leq \frac{m}{p_{2}} $$

For the sake of budgeting, this change has no effect on the quantity of goods, but may make the math a little bit easier. This is called the numeraire price: the price relative to which we are measuring the other price and income.

Tax, Subsidies and Rationing

Types Taxes and Subsidies

A tax is a payment to the government that is proportional to the quantity of goods you buy. A subsidy is a payment from the government that is proportional to the quantity of goods you buy. Tax and subsidies are mathematically the same thing, just one positive the other negative. They can be easily modeled:

$$t\in \mathbb{R}^{d}:p^{T}x \leq m \mp t^{T}x = (p \pm t)^{T} \leq m$$$$t \in \mathbb{R} : p^{T}x \leq m \mp tp^{T}x = (1 \pm t)p^{T}x \leq m$$$$ t \in \mathbb{R} : p^{T}x \leq m \mp t = p^{T}x \leq m \mp t $$

Rationing

$$ p^{T}x \leq m \land x \preceq \bar{x} $$

Where the vector $\bar{x}$ defines the maximum of each entry of $x$. If you visualize the budget graph, it will be a box with sides parallel to some of the axis.

Preferences

In the previous section, we have described what a consumer can afford. In this section, we describe what he/she prefers. Originally, the preferences where described by utility functions. These functions can be seen as one way to encode preferences. The only important thing about utility was the ease of comparison, not their magnitude, which had not a clear interpretation. Another problem with these kinds of functions was the difficulty of assigning a number to them. Note that if it was only their relative ordering that matters, then one can apply every monotonic function to them, and it would still hold!

Given sufficient observations on consumer choices it will often be possible to estimate the utility function that generated those choices.

One can see that finding the perfect choice is quite similar to Optimization methods.

Consumption bundles

What is a Consumption Bundle?

A bundle is just a set of Goods.

Axioms of Preferences

In this subsection, we will define the vocabulary for preferences. Given two consumption bundles $x$ and $y$ we define them to be:

  • Strictly preferred: if $x \prec y$
  • Weakly preferred if $x \preceq y$

There are some axioms that this preference relation needs to satisfy:

  1. Anti-symmetry: $x \preceq y \land y \preceq x \implies x \sim y$ meaning the two are indifferent.
  2. Reflexive: $x \sim x$
  3. Transitive: $x \preceq y \land y \preceq z \implies x \preceq z$
  4. Complete: $x \preceq y \lor y \preceq x$

The third axiom is sometimes problematic. For example, if you have a preference relation that is not transitive, then you can have a cycle of preferences.

$$ x \preceq y \implies \forall \lambda \in \mathbb{R}^{+} : \lambda x \preceq \lambda y $$

It’s income offer curve and Engel Curves are a straight line, which is simple to analyze.

Reservation Price

The reservation price is the maximum price that a consumer is willing to pay for a certain quantity good. It measures the increment in utility necessary to induce the consumer to choose an additional unit of the good.

$$ RP_{x_{1}} = u(x_{1} + 1, x_{2}) - u(x_{1}, x_{2}) $$

Indifference Curves

Given a consumption bundle $x$, the indifference curve plots all consumption bundles that are equivalent to $x$. I have no idea why we assume consumption bundles are considered to be continuous. Intuitively, I would say that two different goods are hardly comparable with each other. Probably it makes sense to thing about indifference curves when we are talking about a quantity of some specific goods. This is the case that is intended here. The drawback is that we cannot compare different set of goods, but only different assignments of the same set of goods.

Indifference Curves do not Cross

If two indifference curves cross, then you can find a bundle that is both strictly preferred to another and indifferent, which is a contradiction to the axioms of preferences.

Well-Behaved Preferences

We say that preferences are well-behaved if the indifference curves are:

  1. Convex: The curve is always concave to the origin: $$ \forall x, y \in \mathcal{X}, x \sim y, \forall \lambda \in [0, 1]: \lambda x + (1 - \lambda)y \preceq x $$ Some bundles could be convex, or neither of those. But often goods are consumed together, which makes the convex case more applicable.
  2. Monotonic: The curve is always increasing: $$ \forall x, y \in \mathcal{\mathbb{R}^{d}}, \forall i \in \left\{ 1, .. d \right\} : x_{i} \leq y_{i} \implies x \preceq y $$ Meaning $x$ is preferred than $y$ if every component of $x$ is less than the corresponding component of $y$.
$$ u(x) = x_{1}^{\alpha}x_{2}^{\beta} $$

Marginal Rate of Substitution

$$ MRS_{x_{1} \rightarrow x_{2}} = -\frac{d x_{1}}{d x_{2}} $$

Which is the rate at which the customer is willing to substitute good $x_{1}$ for good $x_{2}$ at a point $x_{1}$ of the indifference curve.

This value is sometimes useful to model the trade propensity for a certain customer. If the rate of exchange is exactly the MRS, then the customer is sayd to be on the margin of being willing to substitue the goods, this can be also interpreted as a transaction for good 2 by paying with good 1.

Convexity of indifference curves seems very natural: it says that the more you have of one good, the more willing you are to give some of it up in exchange for the other good.

If the goods have prices, then the ratio between these prices should be the MRS.

Marginal Utility

$$ MU_{x_{1}} = \frac{d u(x)}{d x_{1}} $$

This value can be used to compute the Marginal Rate of Substitution presented before, that is the slope of the indifference curve at a certain point:

$$ MRS_{x_{1} \rightarrow x_{2}} = -\frac{MU_{x_{1}}}{MU_{x_{2}}} = -\frac{d x_{1}}{d x_{2}} $$

Types of Indifference Curves

Note: the slope of these indifference curves will be important: The direction of growth of the utility should be perpendicular to the indifference curve for it to be an optimal point.

Perfect Substitutes

$$ u(x) = \alpha x_{1} + \beta x_{2} $$$$ \frac{d x_{1}}{d p_{2}} > 0 $$

This means that if the price of good 2 increases, then the quantity of good 1 should increase.

Perfect Complements

If the indifference curve is a right angle, parallel to the axis, then the two goods are perfect complements. This is for modeling cases where the goods are always paired together, as for the case of some shoes.

$$ u(x) = \min\left\{ \alpha x_{1}, \beta x_{2} \right\} $$$$ \frac{d x_{1}}{d p_{2}} < 0 $$

This means that if the price of good 2 increases, then the quantity of good 1 should decrease. Intuitively, the change of price in the second good affects also the demand on the first good.

Bads

This is quite similar tot he substitutes, but has a positive slope. It can be used to model for bad commodities that need to be balanced by another good commodity. The preference is usually always towards having less of the bad commodity.

Neutrals

If the indifference curve is orthogonal to one dimension, that dimension is a neutral one, meaning the customer doesn’t care about that good.

Satiation

In this case, we assume there is a point in the commodity space that is the best possible, then the indifference curves are some closed curve that has this point in its interior. For example, humans have a satiation point for food, after which they don’t want to eat more: If we have two foods, we might want to have some of both, but after a certain point, we don’t want to eat more of one of them. And eating only one of them could be quite bad.

Consumer Surplus

Gross and Net Surplus

$$ GCS = \int_{0}^{q} p(q) dq $$$$ NCS = GCS - p(q) \cdot q $$

Similar things can be defined for producer’s.