**Consumer Theory**

In the previous lectures, we simply made use of given supply and demand curves. Now we are going to start looking at how the curves are made. We shall start by focusing on the demand curves in the study of consumer theory.

All consumer-related behavior in economics revolves around utility maximization. In order to do these, we need to take into account consumer preferences, budget constraints, and perform constraint optimization. In the most basic sense, we will look at the tradeoff between two goods and how the consumer will choose between them.

In summary, we will need to:

1) Make assumptions about preferences

2) Translate these assumptions into utility functions

3) Add budget constraints and perform maximization

In order to model preferences across goods, we will need to impose the following assumptions to make our modeling simpler:

1)

*Completeness*- When we compare two bundles of goods, we would prefer one or the other but never equally. In other words, you can always make a choice between them.

2)

*Transitivity*- If we prefer x to y, and y to z, then we will prefer x to z.

3)

*Non-Satiation*- More is always better and we will never be satisfied. We will never turn down having more.

**Indifference Curves**

These curves are somewhat like preference maps. These are the graphical representation of people's preferences.

Suppose that your parents gave you some money and you want to decide between pizza or movies. We can have the following choices.

Assume that we are indifferent between two pizzas and one movie, and one pizza and two movies. Clearly we would prefer two pizzas and two movies better than either of them. An indifference curve is a curve which shows all combinations of assumptions that the consumer is indifferent. Example curves based on the above statements can be shown as such:

From the assumptions of consumers above, we have the following properties of indifference curves:

1) Due to the non-satiation assumption, consumers will prefer higher indifference curves.

2) Due to non-satiation, we can also say that indifference curves are always downward sloping. If it is upward sloping, then there can be a case where we are indifferent between 2 pizza and 2 movies and 1 pizza and 2 movies.

3) Indifference curves cannot cross. If the curves cross, there would be a case where we are indifferent between two choices even though one of it has more, which violates non-satiation.

4) Completeness also implies that we cannot have more than one indifference curves through a point.

**Utility**

Utility functions are just a mathematical representation of the preference maps. We simply need to maximize the utility function to see what the user would choose.

An example utility function can be U = sqrt(M*P). This is something that is empirical that we come up with and this is not the only utility function we can come up with. However, this is consistent with the indifference maps.

Marginal utility refers to the amount utility changes with each change of the unit. In other words, it is the (partial) derivative of the utility function.

An example of a diminishing marginal utility is shown below:

We can see that each additional movie improves the utility, but it increases at a diminishing rate. Marginal utility would usually be diminishing, but at different rates for different goods. However, marginal utility will always be positive due to non-satiation.

The Delta Marginal Utility graph can show us the actual contributions of each pizza.

**Marginal Rate of Substitution**

The MRS links the utility to the preference map. The MRS is the slope of the indifference curve, which is dP/dM. It is the rate at which we are willing to trade off the y-axis for the x-axis. For example, it is how many pizzas we are willing to trade off to get another movie. It purely comes from the preferences.

We will refer to the following diagram and attempt to compute the MRS for each segment.

From the first segment to the second, we have an MRS of -2, because we are willing to give up 2 movies for 1 pizzas. However, the MRS of the second to the third segment is -0.5. This is because when we have 4 pizzas, the last pizza has a very low marginal utility. However its, marginal utility is increasing the lower number of pizzas we have.

In general, we have:

Marginal utility is a negative function of quantity. The lower quantity we have, the higher the marginal utility, which is why we flip the X and Y axis.

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