Question

Suppose gold (G) and silver (S) are substitutes for each other because both serve as hedges against inflation. Suppose also that the supplies of both are fixed in the short run \(\left(Q_{G}=75 \text { and } Q_{S}=300\right)\) and that the demands for gold and silver are given by the following equations: $$P_{G}=975-Q_{G}+0.5 P_{S} \text { and } P_{S}=600-Q_{S}+0.5 P_{G}$$ a. What are the equilibrium prices of gold and silver? b. What if a new discovery of gold doubles the quantity supplied to \(150 ?\) How will this discovery affect the prices of both gold and silver?

Short answer

The equilibrium prices initially are \(P_{S}=1000\) and \(P_{G}=1400\). After the new discovery, the prices change to \(P_{S}=950\) for silver and \(P_{G}=1300\) for gold.

Step-by-step solution

Establish the Initial Equations

Firstly, replace the fixed quantities of gold and silver into the provided price functions: \(P_{G}=975-75+0.5P_{S}\) and \(P_{S}=600-300+0.5P_{G}\). These can be simplified to be: \(P_{G}=900+0.5P_{S}\) and \(P_{S}=300+0.5P_{G}\).

Solving Simultaneously

Solving these two equations simultaneously, we substitute \(P_{G}\) from the first equation into the second: \(P_{S}=300+0.5(900+0.5P_{S})\). This can be further rearranged to: \(0.75P_{S}=750\). Solving for \(P_{S}\) gives us \(P_{S}=1000\). Then, substitute \(P_{S}=1000\) into the first equation to solve for \(P_{G}\), resulting in \(P_{G}=900+0.5(1000)\) which simplifies to \(P_{G}=1400\).

Quantity Increase Impact

For the second part, replace the increased quantity of gold as \(Q_{G}=150\) into the first equation to calculate the new price of gold: \(P_{G}=975-150+0.5P_{S}\) or \(P_{G}=825+0.5P_{S}\). Substituting this new equation into the second equation, will yield the updated versions of prices: \(P_{S}=300+0.5(825+0.5P_{S})\), and that simplifies to: \(0.75P_{S}=712.5\), which results in \(P_{S}=950\). By substituting this value back into the gold equation we get \(P_{G}=825+0.5(950)\), so the new value of \(P_{G}\) is \(P_{G}=1300\).

Key concepts

Substitute Goods

Substitute goods are products that can be used in place of each other. They satisfy the same basic desire or need of the consumer, meaning that if the price of one good increases, the demand for its substitute may rise if consumers switch to the cheaper alternative. An increased demand for the substitute good can also push up its price.
In the provided exercise, gold (G) and silver (S) are considered substitutes because they both can serve as investments and hedges against inflation. When their prices are linked, changes in the market for one of these goods can greatly impact the market for the other. For instance, if the price of gold increases, some consumers may start buying more silver, boosting its demand and potentially its price as well. This interdependence is mathematically represented in the demand equations by including the price of one good in the equation of the other.

Demand and Supply Equations

In economics, the demand and supply equations model the relationship between the price of a good and the quantity demanded or supplied. Demand equations typically show how price is influenced by quantity, while supply equations are often fixed in the short term or rise with price in the long term.
In our exercise, the demand for gold and silver is governed by the equations:

• For gold: \(P_G = 975 - Q_G + 0.5P_S\)
• For silver: \(P_S = 600 - Q_S + 0.5P_G\)

The inclusion of \(P_S\) in the demand equation for gold and \(P_G\) in that for silver illustrates the concept of substitute goods and how the price of one affects the demand for the other. The fixed short-term supply of gold and silver is also considered, showing the scenario often found in markets where the quantity supplied does not change quickly in response to price changes. The equations provide a basis to calculate the equilibrium prices for these goods.

Market Equilibrium

Market equilibrium occurs when the quantity demanded of a good is equal to the quantity supplied at a particular price, resulting in a stable market where there is no tendency for the price to change. This concept is at the heart of identifying equilibrium prices in the market for goods. The point of equilibrium ensures that the market clears, with no surpluses or shortages.
In the given exercise, solving the demand and supply equations simultaneously helps us find the market equilibrium prices for gold and silver. Initially, for gold, we have \(P_G = 900 + 0.5P_S\) and for silver \(P_S = 300 + 0.5P_G\). These equations need to be solved together because the price of gold depends on the price of silver and vice versa. Once the equilibrium prices are found, they indicate the price point at which the quantity of gold and silver being demanded is exactly met by the short-term fixed supply. Any change in this equilibrium, such as the one caused by an increased quantity of gold (from the new discovery), will have ramifications for the current prices of both goods as the market seeks a new equilibrium.