Shephard's lemma

Well, Shephard's Lemma is like when you want to buy some toys from the store. Let's say you have $10 to spend, and you want to buy either a toy car or a toy train. If the toy car costs $5 and the toy train costs $3, how many of each can you buy with your $10?

To figure this out using Shephard's Lemma, we need to do some math. We want to find out how much of each toy we can buy based on the money we have. So we can write an equation like this:

$10 = 5q + 3t

Here, "q" represents the number of toy cars we can buy, and "t" represents the number of toy trains we can buy. We multiply the cost of each toy by the number we can buy to get the total amount we spend, and that has to equal the total money we have.

To solve this equation, we can use Shephard's Lemma. This tells us that we can find the amount of each toy based on the price ratio of the toys. In this case, the ratio is 5:3, because the car costs $5 and the train costs $3.

So we take the price ratio, or 5/3, and multiply it by the total money we have ($10). That gives us:

5/3 * 10 = 16.67

This means we could spend $16.67 on the toys if we wanted to. But we only have $10, so we need to find out how much of each toy we can buy with that amount.

To do that, we take the total money we have ($10) and divide it by the price of each toy. So for the car, we would do:

10 / 5 = 2

This means we can buy 2 toy cars with our $10. We do the same thing for the train:

10 / 3 = 3.33

We can't buy a fractional part of a toy, so we round down to the nearest whole number. This means we can buy 3 toy trains.

So using Shephard's Lemma, we were able to figure out how much of each toy we could buy based on the price ratio, without having to do too much complicated math.