Question

Suppose that a budget equation is given by \(p_{1} x_{1}+p_{2} x_{2}=m .\) The government decides to impose a lump-sum tax of \(u,\) a quantity tax on \(\operatorname{good} 1\) of \(t,\) and a quantity subsidy on good 2 of \(s .\) What is the formula for the new budget line?

Short answer

The new budget line is \((p_1 + t)x_1 + (p_2 - s)x_2 = m - u\).

Step-by-step solution

Understand the Original Budget Equation

The original budget constraint is expressed as \( p_1 x_1 + p_2 x_2 = m \), where \( p_1 \) and \( p_2 \) are the prices of goods 1 and 2, \( x_1 \) and \( x_2 \) are the quantities of goods 1 and 2, and \( m \) is the consumer's income.

Incorporate the Lump-Sum Tax

A lump-sum tax \( u \) reduces the consumer's available income. Therefore, the income available for consumption becomes \( m - u \). So, the new income term in the budget equation will be \( m - u \).

Adjust for the Quantity Tax on Good 1

A quantity tax \( t \) on good 1 effectively increases its price. Thus, the price term for good 1 in the budget equation becomes \( (p_1 + t) \). So, the modified term for good 1 becomes \( (p_1 + t)x_1 \).

Adjust for the Quantity Subsidy on Good 2

A quantity subsidy \( s \) on good 2 effectively decreases its price. Therefore, the price term for good 2 in the budget equation becomes \( (p_2 - s) \). So, the modified term for good 2 becomes \( (p_2 - s)x_2 \).

Derive the New Budget Line Equation

Combine the adjusted terms from Steps 2, 3, and 4 to form the new budget equation: \((p_1 + t)x_1 + (p_2 - s)x_2 = m - u\). This is the new budget line that incorporates the lump-sum tax, the quantity tax on good 1, and the subsidy on good 2.

Key concepts

Lump-Sum Tax

A **lump-sum tax** is a fixed amount of tax charged regardless of the taxpayer's situation. In terms of the budget constraint, this type of tax reduces the overall income available for a consumer without directly affecting the prices of goods or services.

For instance, if a person has an income of \( m \) and is subject to a lump-sum tax \( u \), their effective income decreases to \( m - u \). This means that the consumer must reallocate their spending with a reduced amount of money. Lump-sum taxes are often praised in economics for their efficiency, as they do not distort consumer choices among different goods, unlike taxes that alter prices.

Quantity Tax

A **quantity tax** affects the price of a specific good by adding a set tax cost per unit of that good. This effectively raises the price consumers must pay. Take, for example, a quantity tax \( t \) on good 1.

This tax modifies the price of good 1 to \( p_1 + t \). When calculating a new budget line, you understand that the price increase due to the tax affects how much of good 1 a consumer can afford, given their income. This tax impacts purchasing decisions directly by altering the price that consumers see at the point of purchase, often leading to reduced consumption of the more expensive good. Quantity taxes can therefore change market demand for taxed goods significantly.

Subsidy

A **subsidy** operates essentially as the opposite of a tax, by reducing the effective price a consumer pays for a good. The goal is often to encourage higher consumption of a good deemed beneficial or necessary.

In the case of a subsidy \( s \) on good 2, the price in the consumer's budget line is modified to \( p_2 - s \). This indicates that consumers get to purchase good 2 at a lower cost, making it more affordable. Through subsidies, governments aim to incentivize consumers to purchase more of subsidized goods, which might be essential for public welfare or have beneficial externalities. As such, subsidies can influence consumer behavior by making certain goods more appealing through financial relief.