1. Theory
Consider a consumer with 100 dollars of income and a utility function: \(U(X,Y,c) = ln(X) + Y\)
ln(\(\cdot\)) is the natural logarithm.
The derivative of the natural logarithm is \(\frac{dln(x)}{d(x)} = \frac{1}{X}\)
1.1 Standard Utility Maximization
1.1.1 The price of both X and Y is 1 but X is taxed at 10 percent. What is the consumer’s budget constraint?
The consumer’s budget constraint is:
\(100 = 1.1*X + 1*Y\)
\(100 - 1.1*X = Y\)
The slope of the budget constraint is -1.1.
1.1.2 How much of X and Y does the consumer buy?
Taking \(U(X,Y,c) = ln(X) + Y\), we can find the marginal utilities and the marginal rate of substitution by taking the partial derivatives.
\(MU_{x} = \frac{d(U)}{d(X)} = \frac{1}{x}\)
\(MU_{y} = \frac{d(U)}{d(Y)} = 1\)
Now we have the \(\frac{MU_{x}}{MU_{y}} = \frac{\frac{1}{X}}{1} = \frac{1}{X}\)
We will now set the MRS equal to the slope of the budget constraint.
\(\frac{MU_{x}}{MU_{y}} = \frac{p_{x}}{p_{y}} = \frac{1}{X} = 1.1\)
\(1 = 1.1X\)
For every one unit of Y, we give up 1.1 units of X.
\(\frac{1}{1.1} = X\)
Plugging this into the budget constraint:
\(100 - 1.1 * \frac{1}{1.1} = Y\)
\(100 - 1 = Y\)
\(99 = Y\)
The consumer buys \(\frac{1}{1.1}\) units of X and 99 units of Y.
1.1.3 U(X,Y ) = ln(X) + Y is called a quasi-linear utility function. Often, it is used when good Y is money an X is a product like tacos or t-shirts. What property of this utility function makes it a good one to use when one of the goods is money?
The property of this utility function that makes this a good one to use when one of the goods is money is that the number of units purchased for the good is not dependent on the amount of wealth that specific player has. If quasi linear utility was not used, the marginal utility for a given good would depend on the level at which the other good is at. The quasi linear function allows money to be fungible so that as wealth increases, the utility for another good increases at a constant rate.
1.2. Utility Maximization with Limited Attention
Now, suppose the consumer’s utility function is given by:
\(U(X,Y, c) = ln(X) + Y - vc\)
Where c is 0 when the consumer does not pay attention to the sales tax and 1 if the consumer does pay attention. Therefore, vc is the psychic cost of attention.
1.2.1. Suppose the consumer pays attention to the tax c = 1. How much X and Y should the consumer purchase?
For the marginal utilities:
\(MU_{x} = \frac{1}{x}\)
\(MU_{y} = 1\)
MRS \(= \frac{1}{x}\)
Our new budget constraint is:
\(100 = (1 + \tau c)X + Y\)
\(100 = (1 + .1 * 1) X + Y\)
\(100 = 1.1X + Y\)
Which is the same as the one above so we will have the same slopes for the budget constraint.
\(\frac{1}{1.1} = X\)
\(99 = Y\)
MRS \(= \frac{1}{x} = 1.1 = \frac{p_{x}}{p_{y}}\)
The consumer will buy 99 units of Y and \(\frac{1}{1.1}\) units of X.
1.2.2. What is the consumer’s utility depending on v?
Our new Utility function is:
\(U(X,Y, c) = ln(X) + Y - vc\)
\(U(\frac{1}{1.1}, 99, 1) = ln(\frac{1}{1.1}) + 99 - v\)
\(U(\frac{1}{1.1}, 99, 1) = 98.9046898202 - v\)
1.2.3. Suppose the consumer does not pay attention to the tax and c =0. Solve the optimization problem in two steps:
First, calculate the optimal relationship between X and Y assuming the prices are both 1.
Solve the optimization problem for budget:
\(100 = (1 + .1*0)X + Y\)
\(100 = X + Y\)
\(Y = -X + 100\)
For the budget constraint there is a -1 slope on the budget constraint.
For marginal utilities for X and y, they are the same as the first example.
\(MU_{x} = 1/X\)
\(MU_{y} = 1\)
Setting this equal to the slope of the budget constraint:
\(\frac{1}{X} = \frac{1}{1}\)
\(X = 1\)
This is different as compared to before where we had \(x = \frac{1}{1.1}\)
\(100 = 1 + Y\)
\(99 = Y\)
\(U(1,99, 0) = ln(1) + 99\)
Second, use this derived relationship and the actual budget constraint to solve for the quantities. What is the consumer’s utility?
The actual budget constraint is:
\(99 = - 1.1 + 100\)
\(100.1 \neq 100\)
\(U(1, 99, 0) = ln(1) + 99 - v0\)
\(U(1, 99, 0) = 99\)
1.2.4. For what values of v is it optimal for the consumer to pay attention to the taxes?
\(U(1, 99, 0) = 99 = U(\frac{1}{1.1}, 99, 1) = 98.9046898202 - v\)
\(99 = 98.9046898202 - v\)
\(-.0953101798 = v\)
It is optimal for the consumer to pay attention to the taxes if v is less than -.0953101798.
1.3. Who pays for a sales tax the consumer, the store, or someone else?
The consumer pays for the tax. The store sells products at a specific price and sets a margin for the selling of goods compared to what they purchased it for. Consumers can either pay attention to added sales tax to a product or not, and if consumers choose to not pay attention to the tax, they have a lower utility compared to if they paid attention to it.
2. Evidence
This question will be about the paper “Salience and Taxation: Theory and Evidence” by Raj Chetty, Adam Looney, and Kory Kroft. You can find this paper on Canvas under “Course Readings.” You are not expected to read and understand the whole paper.
2.1 What is their research question? Why is it interesting?
The main research question of this paper is how do people internally account for taxes. More specifically, if a store were to incorporate taxes into the sticker price of a good, will people consume the same amount as if taxes were not included on the price tag. This is interesting because it gives insight into consumer behavior. If consumers can incorporate tax into their mental accounting then they are behaving rationally, but as the paper finds, they do not. This means that consumers on average do not follow basic economic theory.
2.2 Describe their experiment. Why is an experiment necessary?
In their experiment, they partnered with a store in order to label goods with a price tag that listed the price of the goods after taxes. They then compared the amount of sales to a control group in which taxes were not included in the price. The experiment was needed because it was impossible to identify if people would follow economic theory without first observing them.
2.3 Interpret the coefficient (−13.12) and standard error (4.89) in Table 4, Column 2. What does this teach us?
The -13.12 is the average treatment effect on revenue per category in dollars. The 4.89 is the standard error of the average treatment effect on revenue per category in dollars. Since |-13.12| > 4.89 * 1.96, for a given alpha of 0.05, we can call -13.12 a statistically significant estimate.
3. Standard Errors
3.1 Draw 10 random variables which are normally distributed with a mean of 5 and a standard deviation of 5. In R, you can use the rnorm function for this. Type “?rnorm” to read the help file.
set.seed(1006)
rand = rnorm(10,5,5)3.2 Calculate the mean of the 10 random variables you just drew. How does it compare to the true population mean. In R, you can use the mean function for this.
mean(rand)## [1] 3.612561The mean of these randomly drawn variables is 3.61 which is much lower than the population mean of 5.
3.3 The formula for the standard error of a mean is \(\hat{SE}\) = hat(\(\sigma\))/\(\sqrt{N}\) where hat(\(\sigma\)) is the standard deviation of the data and N is the number of observations. What is \(\hat{SE}\) based on the data you drew in step 1?
se = (sd(rand))/10
se## [1] 0.6754975The standard error based on the data we drew in step one is 0.675
3.4 Repeat steps 1 and 2 1, 000 times, save the results, and plot a histogram of the mean across the 1, 000 iterations.
set.seed(1006)
results = vector(length=1000)
for (i in 1:1000) {; results[i]<-mean(rnorm(10,mean=5,sd=5)); }
hist(results,
main="Histogram of Results",
xlab="Results",
border="blue",
col="lavender") 
3.5 What is the standard deviation of means across the 1,000 simulations? In R, you can use the “sd” command for this.
sd(results)## [1] 1.583731The standard deviation of means is 1.584.
3.6 How do your answers to parts 3 and 5 compare?
The standard errors from part 3 is smaller than the standard errors in part 5 because as you increase the sample size the mean will become less biased but there will be more variation in the observations leading to a larger standard error.
4.1 Comment on the following statement: “I was at dinner with a bunch of economists. All of us ordered dessert, but I was the only one who finished it. A colleague looked at my finished plate and said it was very likely that I ate more or less than the amount of food I wanted.”
The colleague implies that consuming the exact optimal amount of food is unlikely. If the person was full at dinner and ordered dessert, he overate. If he was full partially through dessert but continued eating, he overate. If he continued to be hungry after eating dessert, he under ate. Realistically, the restaurant’s portions are probably not identical to the person’s ideal amount of food. One extra bite over the optimal amount pushes him into the “ate more” category, and if he were to have wanted just one more bite of dessert, he belongs into the “ate less” category. The amount of food desired is highly specific, so it is quite likely that the person’s ideal serving was more or less than the restaurant portions he consumed.