Gossen's second law


Gossen's Second Law, named for Hermann Heinrich Gossen, asserts that an economic agent will allocate their expenditures so that the ratio of the marginal utility of each good or service to its price is equal to that for every other good or service. Formally,
where
  • is utility
  • is quantity of the -th good or service
  • is the price of the -th good or service

Informal derivation

Imagine that an agent has spent money on various sorts of goods or services. If the last unit of currency spent on goods or services of one sort bought a quantity with less marginal utility than that which would have been associated with the quantity of another sort that could have been bought with the money, then the agent would have been better off instead buying more of that other good or service. Assuming that goods and services are continuously divisible, the only way that it is possible that the marginal expenditure on one good or service should not yield more utility than the marginal expenditure on the other is if the marginal expenditures yield equal utility.

Formal derivation

Assume that utility, goods, and services have the requisite properties so that is well defined for each good or service. An agent then optimizes
subject to a budget constraint
where is the total available sum of money.
Using the method of Lagrange multipliers, one constructs the function
and finds the first-order conditions for optimization as
and
so that
which is algebraically equivalent to
Since every such ratio is equal to, the ratios are all equal one to another: