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 (QG= 75 andQS= 300) and that the demandsfor gold and silver are given by the following

equations:

PG= 975 -QG+ 0.5PSandPS= 600 -QS+ 0.5PG.

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 for gold and silver will be $1400 and $1000, respectively.

b) The increase in the quantity of gold from 75 to 150 will decrease the prices of both gold and silver to $1300 and $950, respectively.

Step-by-step solution

The equilibrium prices of gold and silver

To find the equilibrium prices of silver, you need to solve for the equation of gold given as:

$P_G=975-Q_G+0.5P_S$

The supply of gold in the short run is 75, which is fixed; substitute this in the given equation:

$$\begin{gathered} P_G=975-75+0.5P_S \\ =900-0.5P_S \end{gathered}$$

Now since the supply of silver is fixed at 300 in the short run, substitute it in the equation:

$$\begin{gathered} P_S=600-Q_S+0.5P_G \\ =600-300+0.5P_G \\ =300-+0.5P_G \end{gathered}$$

There are two equations now, substitute the price of gold into the silver demand equation to get the price of silver:

$$\begin{gathered} P_S=300+\left(0:5\right)\left(900+0.5P_S\right) \\ {P}_S=1000 \end{gathered}$$

Therefore, price of silver = $1000

Putting the price of silver in the gold equation will give the price of gold.

$P_G=900+\left(0.5\right)\left(1000\right)=\$1400$

Therefore the price of gold is $1400, and the price of silver is $1000.

Increased supply of gold

Suppose the supply of gold increases from 75 to 150 in the short run. Then the prices of both silver and gold will fall. Resolve the gold demand equation to get the new price of gold as:

$$\begin{gathered} P_G=975-150+0.5P_S \\ {P}_G=825+\left(0.5\right)\left(300+0.5P_G\right) \\ {P}_G=\$1300 \end{gathered}$$

Now substitution in the silver demand equation will give prices of silver.

$$\begin{gathered} P_S=300+\left(0.5\right)\left(1300\right) \\ {P}_S=\$950 \end{gathered}$$

Therefore the price of gold will decrease to $1300, and the price of silver will fall to $950.