6c: The consumer’s optimum
In this section, we will introduce the idea of consumer optimization. Every consumer has utility. Because the utility measure the happiness of a person, every consumer will try to maximize their happiness. However, each individual only has limited amount of I dollars to allocate. Therefore, we need to formulate the budget constraint as the following with n goods with price. The amount of money to buy a good with price is. Since the total amount of spending on goods must less than or equal to income I, then the budget constraint is:
.(6.c.1)
To simplify our analysis, we only consider with two variables now. Then (6.c.1) become
Fig 6.c.1: The Utility maximization by using graphical method
From figure 6.c.1, we can imagine the indifferent curve moving outward. Then the utility increases by that process. As the indifferent curve moving outward, the total expenditure also increases. This process continues until the total expenditure exactly equal to the total Income. Hence, the maximum utility only achieve whenever the budget constraint is tangent of the indifferent curve
In some special case, the maximum point only achieve at the corner of the budget constraint because of the interior property.
Fig 6.c.2: The corner solution of utility optimization
The optimum point is not (X*, Y*). The utility of U1 is lower than that of U2. Therefore, it is impossible to achieve the maximum utility. Let us consider the point with (X**,0). The utility is U2 of this point is the largest utility that can exist. Any higher utility never achieves due to insufficient of income.
6d: Calculus of optimization
In the pervious section, we talked about the graphical method to optimize the utility of a individual. The graphical method is a good way to show how a individual achieves optimum utility. In this section, we will use mathematical method to introduce a more formal way to solve this problem.Let us go back the n goods case. The utility function becomes a multi-variable function. In order to maximize the utility function, we use Lagrange Multiplier.
Then take partial derivative of with respect to variables, and set it equal to zero.
To find the optimize combination of, we need to solve the system of n+1 equation with n+1 unknowns. If we solve each equation for, we get
Let us go back to the simplest two variable case.
Example 1:
Find the combination of X and Y such that the utility function will be maximize with the , and what is the corresponding utility?