Question

Give a numerical example that illustrates how a tax placed on the purchase of good \(X\) can change the relative price of good \(X\) in terms of good \(Y\).

Short answer

The relative price of good \(X\) in terms of good \(Y\) changed from 2 to 2.4 after the tax imposition on good \(X\).

Step-by-step solution

Setup of the problem

Before applying the tax, assume the current price of good \(X\) is $10 per unit and the price of good \(Y\) is $5 per unit. Here, the relative price of \(X\) in terms of \(Y\) is 2, since a unit of \(X\) is worth 2 units of \(Y\). In mathematical terms, this is computed as $\frac{Price( X )}{Price( Y )}=\frac{10}{5}=2.$

Introduce a tax on good X

Let's say a tax of $2 is imposed on good \(X\). This translates to a new price of $12 for good \(X\) which now incorporates the tax imposed. Note that the price of good \(Y\) remains unaffected by this tax.

Compute the new relative price

Now, we recalculate the relative price of \(X\) in terms of \(Y\) under the new tax regime. This involves recalculating the ratio in the similar manner as before: $\frac{Price( X )}{Price( Y )}=\frac{12}{5}=2.4$ This means, under the new tax regime, a unit of \(X\) is now worth 2.4 units of \(Y\). Hence the introduction of the tax on good \(X\) increased the relative price of \(X\) in terms of \(Y\) from 2 to 2.4.

Key concepts

Relative Price

The term "relative price" refers to the price of a good in comparison to another good. It is an important concept in economics because it determines how much of one good needs to be given up to obtain another. This relative comparison can guide consumer choices and reflect opportunity costs.

When we talk about relative price, we usually describe how many units of good Y one would need to exchange for a unit of good X. This is calculated by dividing the price of good X by the price of good Y. For example, if good X costs $10 and good Y costs $5, the relative price of X in terms of Y is 2. This means one unit of good X is worth two units of good Y. Understanding these concepts aids in figuring out how consumers might react to price changes due to external factors like taxes.

Numerical Example

A simple numerical example can help illustrate the impact of taxes on relative prices. Let's assume the initial prices of two goods, X and Y, are \(10 and \)5 respectively. Without any taxes, the relative price of X in terms of Y can be calculated as \( \frac{10}{5} = 2 \). This signifies that one must trade two units of good Y for one unit of good X.

Next, imagine a scenario where a \(2 tax is introduced on good X. This tax raises the price of X from \)10 to $12. To find out how this affects the relative price, we recalculate: \( \frac{12}{5} = 2.4 \). Now, you would need to give up 2.4 units of Y to acquire one unit of X. This illustrates how a tax on one good affects its value relative to another.

Goods Pricing

In the context of goods pricing, understanding how taxes affect prices is crucial. Prices are the signals that help buyers and sellers make decisions in the market. When a tax is added to a product, it can directly influence the final selling price of that product.

For instance, before any taxation, the market price of a product may simply reflect its production costs, competition, and demand. However, once a tax is applied, this adds to the final price paid by the consumer, as seen in our numerical example where good X's price increased from $10 to $12. This increase could potentially alter demand for good X and shift consumer preferences towards goods that remain untaxed or cheaper, like good Y in our example.

Tax and Economics

Taxes are vital in economics as they can influence market behavior and the distribution of resources. When a government imposes a tax on a good, it effectively increases its price, which can decrease demand for that good. This concept was demonstrated when the tax on good X increased its price relative to good Y.

Furthermore, this interplay affects not only consumer behavior but also can cause producers to alter production to focus on less-taxed goods. On a broader scale, taxes can be used as tools to discourage negative externalities, such as pollution, by placing them on goods producing such effects.

In our example, the imposed tax shifted the relative price from 2 to 2.4, potentially driving consumers away from good X towards good Y. This shift showcases how taxation can nudge economic participants toward alternative choices and resource allocations.