Consider a consumer who has an experienced utility function given by . Let denote the market price of good , and assume that it remains fixed throughout the problem.

Question

Consider a consumer who has an experienced utility function given by . Let denote the market price of good , and assume that it remains fixed throughout the problem.

The company selling the good starts an advertisement campagin that has the following effect on the consumer: he makes decisions as if maximizing a decision utility function given by .

QUESTION. Write down the expression for the difference between the consumer demand and his optimal level of consumption (as a function of ).

Answer 1

To answer this question, we need to find the consumer's optimal level of consumption with and without the advertisement campaign.

Let's start with the consumer's optimal level of consumption without the advertisement campaign. The consumer maximizes their experienced utility function U(x), subject to their budget constraint, which can be written as:

px = I

where p is the price of good x and I is the consumer's income. The consumer's demand function Dx(p) represents the quantity of good x the consumer will purchase at price p.

Now, with the advertisement campaign, the consumer makes decisions based on a decision utility function D(x). To find the consumer's optimal level of consumption with the advertisement campaign, we again need to maximize the decision utility function subject to the budget constraint:

px = I

The difference between the consumer's demand and their optimal level of consumption can be expressed as:

Dx(p) - Dx'(p)

where Dx'(p) represents the quantity of good x the consumer will purchase at price p under the advertisement campaign.

Therefore, the expression for the difference between the consumer demand and their optimal level of consumption as a function of price p is:

Dx(p) - Dx'(p)

Answer 2

To find the expression for the difference between the consumer demand and his optimal level of consumption, we need to compare the consumer's experienced utility function with the decision utility function after the advertisement campaign.

Let's denote the consumer's initial experienced utility function as U(X) and the decision utility function after the advertisement campaign as DU(X). The consumer maximizes his experienced utility function U(X) and makes decisions according to it.

The difference between the consumer demand and his optimal level of consumption can be calculated as:

Difference = DU(X) - U(X)

This expression represents the deviation from the consumer's optimal consumption level due to the effect of the advertisement campaign.