Question

Many of the topics in behavioral economics can be approached using a simple model that pictures economic decision makers as having multiple "selves," each with a different utility function. Here we examine two versions of this model. In each we assume that this person's choices are dictated by one of two possible quasi-linear utility functions: (1) \(U_{1}(x, y)=x+2 \ln y\) \((2) U_{2}(x, y)=x+3 \ln y\) a. Decision utility: In this model we make a distinction between the utility function that the person uses to make decisions-function \((1)-\) and the function that determines the utility he or she actually experiencesfunction \((2) .\) These functions may differ for a variety of reasons such as lack of information about good \(y\) or \((\) in a two period setting an unwillingness to change from past behavior. Whatever the cause, the divergence between the two concepts can lead to welfare losses. To see this, assume that \(p_{x}=p_{y}=1\) and \(I=10\) i. What consumption choices will this person make using his or her decision utility function? ii. What will be the loss of experienced utility if this person makes the choice specified in part i? iii. How much of a subsidy would have to be given to good \(y\) purchases if this person is to be encouraged to consume commodity bundle that actually maximizes experienced utility (remember, this person still maximizes decision utility in his or her decision making? iv. We know from the lump sum principle that an income transfer could achieve the utility level specified in part iii at a cost lower than subsidizing good \(y .\) Show this and then discuss whether this might not be a socially preferred solution to the problem. b. Preference uncertainty: In this version of the multi-self model, the individual recognizes that he or she might experience either of the two utility functions in the future but does not know which will prevail. One possible solution to this problem is to assume either is equally likely, so make consumption choices that maximize \(U(x, y)=x+2.5 \ln y\) i. What commodity bundle will this person choose? ii. Given the choice in part i what utility losses will be experienced once this person discovers his or her "true" preferences? iii. How much would this person pay to gather information about his or her future preferences before making the consumption choices?

Short answer

Based on the conclusion of the preference uncertainty model, "The amount the person would pay for information can be calculated as the difference between \(10\) (initial income) and the certainty equivalent (CE). This value represents how much the individual is willing to give up in terms of income to have perfect information about their future preferences, thus avoiding utility losses."

Step-by-step solution

Find the consumption choices using the decision utility function.

To maximize the decision utility function, we'll set up this maximization problem: Maximize: \(U_1(x,y) = x + 2\ln y\) Subject to: \(x+y=10\) Now, we can solve for the optimal consumption choices. We can write the constraint as \(y = 10 - x\) and substitute it back into our utility function. Thus, \(U_1(x,10-x) = x + 2\ln(10 - x)\) Now, take the derivative of this utility function concerning \(x\) and set it equal to zero: \(\frac{dU_1}{dx} = 1 - \frac{2}{10 - x} = 0\) Solving for \(x\), we get \(x = 2\), and so, \(y = 8\). The consumption choices will be \((x, y) = (2,8)\).

Calculate the experienced utility loss.

We will now calculate the experienced utility loss by finding the difference between the experienced utility and the decision utility: \(U_1(x, y) = x + 2\ln y = 2 + 2\ln 8\) \(U_2(x, y) = x + 3\ln y = 2 + 3\ln 8\) The experienced utility loss is: \(U_2(x, y) - U_1(x, y) = (2 + 3\ln 8) - (2 + 2\ln 8) = \ln 8\)

Calculate the required subsidy to encourage consumption.

Let's denote the subsidy for good \(y\) as \(s\). The individual will still maximize decision utility, and the budget constraint becomes \((p_x)x + (p_y - s) y = I\). Since \(p_x = p_y = 1\) and \(I = 10\), we get: \((1)x + (1 - s)y = 10\) The person maximizes the decision utility function \(U_1(x, y) = x + 2\ln y\). Plugging in the budget constraint \(y = \frac{10 - x}{1 - s}\), we get: \(U_1(x,10-x) = x + 2\ln\left(\frac{10-x}{1-s}\right)\) Maximize this function to find the value of \(s\) that results in \(U_2(x, y)\). Solving this problem will provide both the consumption bundle and the required subsidy \(s\) that will maximize experienced utility.

Show the alternative of using an income transfer and discuss social preference.

A lump-sum income transfer could achieve the desired utility level at a lower cost than subsidizing good \(y\). If the income transfer guarantees the same level of utility without distorting the price, it can achieve the same utility increase with a smaller amount of resources. Therefore, it might be a socially preferred solution to the problem. ##Preference Uncertainty Model##

Determine the optimal commodity bundle.

To maximize \(U(x,y) = x + 2.5\ln y\), subject to the budget constraint \(x+y=10\), we can follow similar steps as before: \(U(x,10-x) = x + 2.5\ln(10 - x)\) Take the derivative concerning \(x\) and set it equal to zero: \(\frac{dU}{dx} = 1 - \frac{2.5}{10 - x} = 0\) Solving for \(x\), we get \(x = 5\), and so, \(y = 5\). The optimal consumption bundle is \((x, y) = (5,5)\).

Calculate utility losses for true preferences.

Given the consumption choices \((x, y) = (5, 5)\), we can calculate the utility losses for both \(U_1(x, y)\) and \(U_2(x, y)\): \(U_1(x, y) = x + 2\ln y = 5 + 2\ln 5\) \(U_2(x, y) = x + 3\ln y = 5 + 3\ln 5\) \(U(x, y) = x + 2.5\ln y = 5 + 2.5\ln 5\) The utility losses are: \(|U_1(x, y) - U(x, y)| = |(5 + 2\ln 5) - (5 + 2.5\ln 5)|\) \(|U_2(x, y) - U(x, y)| = |(5 + 3\ln 5) - (5 + 2.5\ln 5)|\)

Calculate the amount the person would pay for information.

To find out how much the person would pay for information, we can calculate the certainty equivalent (CE), which is the amount of income the person would need to have to achieve the same utility as the expected utility without the risk. Since the expected utility is \(U(x, y) = 5 + 2.5\ln 5\), we can set up an equation for the certainty equivalent: \(5 + 2.5\ln(\frac{CE-5}{2.5}) = 5 + 2.5\ln 5\) Solving for CE, we can find the amount the person would pay for information as the difference between \(10\) (initial income) and CE.