Question

Use partial derivatives to determine if the demand functions for two commodities represent a substitute, complement or neither.

D1=2000+p1+2100−25p2
D2=1500+p1+7p2

Answer 1

The demand functions for commodities 1 and 2 are substitutes, as the cross-partial derivative
`(∂D1/∂p2)/(∂D2/∂p1)`is positive.

Step-by-step explanation

The substitution or complementarity of goods can be determined by evaluating the cross-partial derivatives of the demand functions with respect to the prices of the two commodities. If
`(∂D1/∂p2)/(∂D2/∂p1) > 0,`the goods are substitutes; if < 0, they are complements; and if = 0, they are independent.

Now, let's calculate these partial derivatives. The demand functions are:

`\[ D1 = 2000 + p1 + 2100 - 25p2 \]\[ D2 = 1500 + p1 + 7p2 \]`

Taking the partial derivatives:

`\[ (\partial D1)/(\partial p2) = -25 \]\[ (\partial D2)/(\partial p1) = 1 \]`

Therefore,
`\( (\partial D1)/(\partial p2) \) divided by \( (\partial D2)/(\partial p1) \) is -25/1 = -25,`which is less than 0. This implies that the goods are substitutes.

In economic terms, this means that an increase in the price of commodity 2
`(\(p2\)`) leads to an increase in the demand for commodity
`1 (\(D1\)),` indicating a substitutable relationship. The negative sign indicates that the relationship is inverse - as the price of one good goes up, the demand for the other goes down, confirming they are substitutes.