Consider a consumer who has an experienced utility function given by UEU(x,m)=2sqrtx+m. Let q denote the market price of good x, and assume that it remains fixed throughout the problem.
The company selling the good x starts an advertisement campagin that has the following effect on the consumer: he makes decisions as if maximizing a decision utility function given by UDU(x,m)=4sqrtx+m.
QUESTION. Write down the expression for the difference between the consumer demand and his optimal level of consumption (as a function of q).
Well, well, well, looks like we have a consumer who is easily swayed by advertisements. Let's calculate the difference between their demand and their optimal level of consumption.
The consumer's demand is given by the inverse of the marginal utility of consumption, which is obtained by differentiating the utility function with respect to x. So, for the experienced utility function UEU(x,m) = 2√x + m:
MU(x) = (d/dx)UEU(x,m) = (d/dx)(2√x + m) = 1/√x
Now, let's move on to the decision utility function UDU(x,m) = 4√x + m. To find the optimal level of consumption, we need to differentiate it with respect to x and set it equal to the price q:
MU(x) = (d/dx)UDU(x,m) = (d/dx)(4√x + m) = 2/√x
Setting this equal to q, we get:
2/√x = q
Now, let's solve for x:
√x = 2/q
x = (2/q)^2
x = 4/q^2
So, the optimal level of consumption is given by x = 4/q^2.
Now, let's find the difference between the consumer demand and the optimal level of consumption:
Demand - Optimal Consumption = (1/√x) - (2/√x)
= (1 - 2)/√x
= -1/√x
But we have the expression for x in terms of q, so let's substitute it in:
Demand - Optimal Consumption = -1/√(4/q^2)
= -1/(2/q)
= -q/2
So, the difference between the consumer demand and the optimal level of consumption is -q/2.
Now, that's some serious difference! Looks like the consumer's decisions are far from optimal. Maybe they should stop falling for those advertisements and start prioritizing their own well-being.
To find the difference between the consumer demand and his optimal level of consumption, we need to compare the utilities obtained from the experienced utility function UEU(x,m) and the decision utility function UDU(x,m).
The consumer's experienced utility function is given by UEU(x,m) = 2sqrt(x) + m, where x represents the quantity of good x consumed and m represents the consumer's income.
The consumer's decision utility function is given by UDU(x,m) = 4sqrt(x) + m, where x represents the quantity of good x consumed and m represents the consumer's income.
To find the difference between the consumer demand and his optimal level of consumption, we need to subtract the experienced utility from the decision utility. Therefore, the expression for the difference is:
Difference = UDU(x,m) - UEU(x,m)
Substituting the expressions for UDU(x,m) and UEU(x,m), we get:
Difference = (4sqrt(x) + m) - (2sqrt(x) + m)
Simplifying the equation:
Difference = 4sqrt(x) + m - 2sqrt(x) - m
The m terms cancels out, and we are left with:
Difference = 4sqrt(x) - 2sqrt(x)
Simplifying further:
Difference = 2sqrt(x)
Therefore, the expression for the difference between the consumer demand and his optimal level of consumption (as a function of q) is given by:
Difference = 2sqrt(q)
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 (UEU) with the decision utility function (UDU).
The consumer's optimal level of consumption is determined by maximizing utility, which is the derivative of the utility function with respect to x equal to zero.
Let's start by finding the consumer's demand under the experienced utility (UED), which is the quantity demanded at a given price q. We can do this by solving the following optimization problem:
Maximize UEU(x, m) subject to the budget constraint q*x = m, where x is the quantity of good x and m is the consumer's income.
Taking the derivative of the objective function UEU(x, m) with respect to x and setting it equal to zero gives us:
d/dx (2√x + m) = 1/√x = q
Solving for x, we have:
√x = 1/q
x = 1/q^2
Therefore, the consumer demand under the experienced utility (UED) is:
D(q) = 1/q^2
Now let's find the consumer's optimal level of consumption under the decision utility (UDU). We can follow the same procedure by maximizing UDU(x, m) subject to the budget constraint q*x = m.
Taking the derivative of the objective function UDU(x, m) with respect to x and setting it equal to zero gives us:
d/dx (4√x + m) = 2/√x = q
Solving for x, we have:
√x = 2/q
x = 4/q^2
Therefore, the consumer's optimal level of consumption under the decision utility (UDU) is:
O(q) = 4/q^2
The difference between the consumer demand and his optimal level of consumption is given by:
D(q) - O(q) = 1/q^2 - 4/q^2 = (1 - 4)/q^2 = -3/q^2
So, the expression for the difference between the consumer demand and his optimal level of consumption as a function of q is:
D(q) - O(q) = -3/q^2