# Demand function

The demand function is an equation that explains how the quantity demanded of a good is determined. This, in relation to market prices and consumer income.

In mathematical terms, we can express this function as follows:

Being

The left side of each equation represents the quantity demanded of the respective good. Meanwhile, the right side is a mathematical function where the variables are prices (assuming there are two commodities) and the buyer's budget.

For example, a demand function could be the following:

At this point, it should be mentioned that the quantity demanded of a product is almost always inversely related to its price. This, due to the substitution effect, when when the cost of a product rises, the consumer replaces it with a similar one.

Likewise, another factor that contributes to the price and quantity demanded varying in opposite directions is the income effect. This means that raising the cost of a commodity will reduce the purchasing power of the buyer.

However, it should be clarified that in the case of a good Giffen, all of the above is not true. On the other hand, if the price of the good increases, so will the quantity demanded and vice versa.

## Relationship between demand function and demand curve

To explain the relationship between the demand function and the demand curve, we must remember that the former represents mathematically how the purchase decision is obtained in the consumer's equilibrium. This, in turn, occurs at the intersection between the budget constraint and the indifference curve.

Thus, we can see a graph like the following one, where the different optimal baskets are detailed according to the different budgets of the buyer.

However, if instead of modifying the budget constraint, we were to vary the price of good 1, for example, we would have the following:

Then, joining the different equilibrium points for various prices of good 1, we could graph the demand curve.

## Particular cases of demand functions

There are some particular cases of demand functions:

- Substitute goods: In the demand function there can be two scenarios

In other words, the consumer will only purchase the cheapest merchandise. In case the prices are the same, it will be indifferent between one or another product.

- Complementary goods: The demand function meets the following conditions:

The first equation represents the relationship between both goods, one having to be acquired based on the quantity of the other.

For example, if a and b are 1 and 2, respectively, it means that you always need twice as much good x1 as good x2.

To find the demand function, for example, of good 1, we would only have to solve for x1 as a function of x2 in the budget constraint.

- Cobb Douglas utility function: The consumer's utility function would be as follows:

Then we can express it in its logarithmic form

Likewise, we know that the individual is limited by his budget constraint:

To find the optimal basket, we must first find the Marginal Substitution Ratio (RMS):

As a next step, we set the MSY equal to the slope of the budget constraint:

Finally, we solve for x2 as a function of x1 in the budget constraint and replace it in the above equation:

Therefore, the demand function for x1 would be:

It should be noted that for practical purposes it is assumed that:

Therefore, the demand function for good 1 would be:

At this point, it should be noted that in a Cobb Douglas utility function the coefficient can be interpreted* to *as the portion of the budget that is allocated to good 1. Likewise, it is assumed that the coefficient *b *is the percentage assigned to good 2.

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