# Managerial Economics/Budgeting Simple

## Assumptions

A simple example of budgeting requires a few assumptions:

1. There is only one asset to allocate: income.
2. There are two items on which income can be spent: w(a)ter and (b)read
3. water and bread are infinitely divisible.
4. The utility function is completely specified.
5. The prices of water and bread are known.

## Definitions

The price of water is represented as ${\displaystyle P_{a}}$ and the price of bread as ${\displaystyle P_{b}}$.

The quantity of water acquired by the individual is represented as ${\displaystyle Q_{a}}$, and of bread as ${\displaystyle Q_{b}}$

The total income of the individual is given by ${\displaystyle M}$.

## Discussion

### Budget Line

If the individual will spend all of her income (budget constraint) on either one item or the other, we can say that

${\displaystyle M=P_{a}Q_{a}+P_{b}Q_{b}}$


If we were to plot this line on a graph that has ${\displaystyle Q_{a}}$ as the vertical axis and ${\displaystyle Q_{b}}$ as the horizontal axis, the line traced would be the budget line of the individual.

The budget line is a set of points where the combination of water and bread purchased exhausts the individual's income.

The place where the budget line crosses the horizontal axis represents a combination including no water; all income would be spent on bread. Therefore, our formula could be rearranged to read

${\displaystyle Q_{b}={\frac {M}{Pb}}}$


And if the individual were instead to spend all of her income on water we would have

${\displaystyle Q_{a}={\frac {M}{Pa}}}$


Using this information we can derive the equation for the slope of the budget line, which gives us

${\displaystyle M=P_{a}Q_{a}+P_{b}Q_{b}}$
And ...
${\displaystyle Q_{a}={\frac {M}{P_{a}}}-{\frac {P_{b}}{P_{s}}}Q_{b}}$
And therefore the slope is ${\displaystyle -{\frac {P_{b}}{P_{a}}}}$


### Utility

To know which combination of water and bread a person will choose, it is useful to refer to utility, which is the economists' term for satisfaction. Given the individual's tastes and desires, she will feel more or less satisfied about a particular experience. A generic utility function is described mathematically as follows

${\displaystyle U=f(X_{1},DX_{2},...,X_{n})}$


meanint that an individual's utility is a function of the items ${\displaystyle X_{1},X_{2},...}$ she has available.

In this case, the utility function would be described as

${\displaystyle U=f(Q_{a},Q_{b})}$


Utility is measured in utils, which are a made-up measure with no use beyond that of making comparisons.

### Convergence of Utility and the Budget Line

Every combination of water and bread on the individual's budget line will afford her a certain amount of utility. Since all individuals are assumed to be maximizing their utility (or satisfaction), it is easy to see that an individual would search for the combination that gives the most satisfaction. Such a point on the budget line is called a solution to the economic problem.

An individual's solution is not necessarily a state of 'bliss as an individual may have a lot of unsatisfied desires; what a solution means is that an individual has done the best that she can do under the constraints she lives in. In economic terms, no one ever achieves a state of bliss in this life.

### Indifference Maps

If we were to calculate the utility generated by every possible combination of water and bread and plotted it on a graph having ${\displaystyle Q_{a}}$ in the Y-axis and ${\displaystyle Q_{b}}$ in the X-axis, we would trace an individual's indifference map.

If we then took all the combinations that yielded the same utility, for example 50 utils, we would notice that they form a rather smooth curve called an indifference curve. In other words, since an individual receives the same utility from every combination in the curve, she is indifferent as whether she is consuming one combination or another.

All the indifference curves of an indifference map trace a sort of topographical map where the peaks are bliss points.

### Diminishing marginal utility

The shape of the indifference curves is given because the utility function obeys the law of diminishing marginal utility.

In the formula ${\displaystyle U=f(Q_{a},Q_{b})}$, U is the total utility of the individual for a given combination of ${\displaystyle a}$ and ${\displaystyle b}$. Suppose now that we plot a point X that represents 8 units of water and 6 units of bread; furthermore, suppose that such a combination gives the individual 50 utils of satisfaction. If by acquiring one more unit of bread we discover the individual now enjoys 55 utils, we say that the marginal utility of the 9th unit of bread is five utils because a change from 8 units to 9 resuted in an increase of 5 utils.

Note that we didn't change the quantity of water in the previous example. This is essential because otherwise we cannot tell what caused the change in utility. When analyzing economics, we often hold all things equal except one, a situation termed ceteris paribus.

TODO: finish definition of diminishing marginal utility tie it back to the shape of the indifference curves, define the equimarginal principle