#
Utility, Substitution and Demand

### 3: A Scenario

Let’s apply the aforementioned principle to a particular choice that I
faced on Friday night: buying savoury snacks before a party. I’ve got a
five pound budget for these snacks, and the off-licence presents me with
a choice of crisps and nachos (we’ll ignore the different flavours and
brands for the purpose of this analysis). My choice is how many of each
pack to buy.

At this point, to make it a well-formed problem, I need to define the
utilities. I will do so using two assumptions.

The first is the assumption of *diminishing marginal utility*:
the additional utility due to one extra item is smaller the greater the
quantity of that product you already have. Desirable as a sports car is,
it is usually less desirable once you have won the lottery and bought ten
of them. Applying this to our scenario, the tenth pack of crisps, for example,
does not provide as much additional utility as the second pack of crisps.

The second is an *independence* assumption; that my utility of
having a certain number of packs of nachos does not depend on how many
packets of crisps I have, and vice versa. This is less realistic but simplifies
the mathematics, in that total utility is just the sum of the utilities
due to each good.

It does not matter which particular utility function we use,
so long as it obeys these constraints. Call the number of packs of crisps purchased *n*_{1} and
the number of packs of nachos *n*_{2}. We will arbitrarily choose
the following function:

Here we are considering only positive roots, ignoring the fact that
a negative number squared is positive.

The numbers *n*_{1} and *n*_{2} are related
in that, as mentioned earlier, I have a five pound budget for both kind
of snack.

By increasing *n*_{1} or *n*_{2} in the first
equation, you can see that buying a pack always has a positive effect on
utility, so I will buy as many packs as I can. So *n*_{2}
is the largest whole number which satisfies this rearrangement of the above:

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