Price Indepence Between Products: Lagragian Based Optimization of Utility Functions


Consumers face choices everyday about what to spend their income on.  Given that most consumers have a relatively fixed income there is a budget constraint imposed which gives rise to trade-offs that involve a persons sense of well being as they purchase a mix of goods in the market. Even if one is an extremely wealthy individual, a constraint would still exist in the form of time spent choosing what goods to consume.  The idea is that scarcity drives consumer decision making, where scarcity is a fluid term that is different for different people.  There is scarcity in the form of: time, storage space, prohibitively large search cost, and for most people income.  These  limitations place sufficiently large constraints to ensure trade-off are present in the consumption environment for every consumer.

Consumer theory assumes people will attempt to maximize utility subject to income constraints when deciding what goods to purchase. Key factors in determining  the quantity and mix of goods a consumer buys are of course income, prices of goods, and the relative appeal of these good in the consumer’s mind now and in the future as is the case with durable goods. This post will set up a basic Lagragian optimization problem to determine whether or not two products are substitutes or complements to each other.The final case is when goods are independent of each other, which is when they are neither substitutes nor compliments.  Pricing strategy should not consider any substitutions or complementary effects of good x on good y.  Reducing the price of good x may increase both the demand of good x and it’s compliment, but may reduce sales of substitute products as consumers transfer expenditures towards a cheaper product which is similar in their minds.

Mathematical Theory

Given a utility function for goods x and y the Lagrangian is set up in with the income constraint:

Next, the partial derivative with respect to x, y, and lambda are taken of the Lagrangian equation and set equal to zero as the optimization condition requires.

Solving the first two equations for lambda and solving for price of x* x and substituting into the third equation yields:

Notice how the derivative of good y decreases as the price of it goes up.  Given that income and the price of good x haven’t changed one would assume that more would be consumed of good x since all income must be spent on good x and good y. This would imply that imply that goods x and y are gross substitutes for each other, on the other hand notice that an increase in good x does not increase consumption of good y.

This illustrates the ambiguous case where goods are neither gross complements nor gross substitutes for each other.  These two goods are independent of each other thus pricing should not consider the substitution or complementary effects of goods x and y.  This is a special case, it has been mathematically proven that most goods are substitutes in the complete market model, next in the order of density are substitutes, and finally are goods which are independent of each other.