Economics 201 question set 2

CAS EC 201. Intermediate Microeconomic Analysis.

Problem set 2 (due on Wednesday September 23, 2020)

1. This exercise reviews income and substitution effects.

1.1 Figure 1 shows the income and the substitution effect for a consumer with utility U(x1,x2) = x1x2 and income m = 4. Initially, the price of each of goods is one, p1 = p2 = 1. Then, the price of good decreases to p

′ 1 = 1/2. Using this

information, compute the income and the substitution effects.

m

p′1

m/p2

income effect

total effect

subs. effect

x1

x2

Figure 1: Income and Substitution Effect

2. 2.1 Draw the indifference curves for the utility function U(x1,x2) = x1 + 3x2.

2.2 What is the marginal rate of substitution evaluated at an arbitrary consumption bundle (x1,x2)?

2.3 Suppose that p1 = 5, p2 = 2, and m = 10. Find the utility-maximizing consump- tion bundle (among those that satisfy the budge constraint) for this agent. You should be able to do this without using any calculus: it should be clear from your indifference curves.

2.4 Given any pair of prices p1, p2 and income m, what is (are) the utility-maximizing consumption bundle(s)? (Assume that p1 > 0, p2 > 0, m > 0.) Consider the following three cases separately: p1 > p2/3, p1 < p2/3, p1 = p2/3.

2.5 How would your answer to 2.1-2.4 change if U(x1,x2) = 1 + 2 ln (x1 + 3x2)?

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3. Some economists have proposed to increase the tax on gasoline, then rebate the tax revenue to consumers through a payroll tax reduction or some other means.1 In this exercise you explore whether such a policy is effective in reducing the consumption of gasoline and whether it increases or decreases consumers’ utility.

Let an agent’s utility function be U(x1,x2) = ln x1 + αx2. Interpret good 1 as gallons of gasoline and good 2 as consumption on other goods. Let p2 = 1.

3.1 Form the Lagrangian function, find the first-order necessary conditions for utility maximization, and solve these first-order conditions to find the demand functions for goods 1 and 2, as a function of the price p1 and the income m.

3.2 Now suppose the government introduces a $1-a-gallon gasoline tax. Hence, the consumer now pays p1 + 1 for each gallon of gasoline, and $1 is collected by the government in taxes for each gallon the consumer purchases. The consumer’s budget constraint is now (p1 + 1)x1 + x2 = m. Given this budget constraint, find the consumer’s demand functions, and explain how the imposition of the tax affects the consumption of good 1.

3.3 Assume p1 = 1. Compute the total revenue that the tax raises. Denote the tax revenue by R.

3.4 Suppose now that the tax revenue is given back to consumers: the consumer income is now m + R. Find the consumer’s demand functions after this raise in income, maintaining the assumption that a gallon of gasoline costs p1 + 1 to the consumer.

3.5 Assume α = 1/4, m = 10. Does the consumption of gasoline decrease with the introduction of the policy? What is the effect on the consumer’s utility?

Critics of tax-cum-rebate argue that paying the revenue raised by the tax back to the consumers would have no effect on demand since they could just use the rebated money to purchase more gasoline. You showed that this argument is actually incorrect.

In general, the consumer is made worse off. However, we are ignoring important considerations such as the fact that other consumers could benefit from better quality of the air they breath.

1See http://www.nytimes.com/2006/02/16/business/a-way-to-cut-fuel-consumption-that-everyone-likes-except-the. html

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