Choice and Utility Maximization:

We have now described in detail what the consumer can do and we have also acquired a numerical function representing the consumer’s preferences and we can therefore proceed to analyze the consumer’s optimal choice behavior, using these concepts. The utility function is convenient since we can assume that the consumer make choices as if she tried to maximize this function. We’ve also seen that we can define the consumer’s external rate of exchange as the price ratio (p₁/p₂), and the consumer’s internal rate of exchange as the ratio of the marginal utilities, or MRS₁₂, (−MU₁/MU₂). Figure below shows the solution to the consumer’s utility maximization problem, or optimal choice problem. It is obvious from the figure that the optimal condition is such that the external- and internal rate of exchange are equal, or,

MRS₁₂ = - MU₁/MU₂ = - p₁/p₂

Example:

As before we assume that the utility function is given by, U(Q₁,Q₂) = Q₁ . Q₂. The next step is to find MU₁ and MU₂. We first define our initial bundle as Q⁰ = Q⁰₁ . Q⁰₂ and the new bundle as Q₁ = (Q⁰₁ + ΔQ₁) Q⁰₂, the marginal utility of good 1 is then,

Similarly, MU₂ = Q⁰₁. Hence, MRS₁₂ is equal to,

MRS₁₂(Q⁰) = MU₁/MU₂ = Q⁰₂/Q⁰₁.

We proceed by setting this ratio equal to the price ratio,

Q₂/Q₁ = p₁/p₂,

Q₂ = p₁/p₂Q₁

This equation shows how they actually consumed quantities of the two goods are related to each other and the goods prices. But it doesn’t help us to pin down what these quantities are, this depends, of course, on how much money the consumer has available, or the budget equation,

p₁Q₁ + p₂Q₂ = Y

By using above equation we rewrite the budget equation so that we only have one unknown variable (Q₁),

p₁Q₁ + p₂ [(p₁/p₂) Q₁] = Y,
p₁Q₁ + p₁Q₁ = Y,
2p₁Q₁ = Y

Solving for Q1 gives us the demand function for good 1:

Q₁ (Y, p₁) = Y/2p₁

To find the demand function for good 2 we substitute this expression into equation, which gives,

Q₂ (Y, p₂) = (p₁/p₂) (Y/2p₁)
= Y/2p₂

For example if p₁ = 1, p₂ = 2 and Y = 15, the optimal quantities are Q^∗₁ = 7.5 and Q^∗₂ = 15/ 2.2 = 3.75. Note that the optimal bundle Q^∗ = {7.5, 3.75} cost the consumer 7.5 + 2 x  3.75 = 15, i.e., exactly equal to the consumer’s budget.

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