Question

The demand and supply functions of a two-commodity market model are as follows: $$\begin{array}{ll}Q_{d 1}=18-3 P_{1}+P_{2} & Q_{d 2}=12+P_{1}-2 P_{2} \\\Q_{s 1}=-2+4 P_{1} & Q_{52}=-2 \quad+3 P_{2}\end{array}$$ Find \(P_{i}^{*}\) and \(Q_{i}^{*}(i=1,2) .\) (Use fractions rather than decimals.)

Short answer

Equilibrium prices are \(P_1 = \frac{301}{119}, P_2 = -\frac{39}{17}\); quantities are \(Q_{s1} = \frac{966}{119}, Q_{s2} = \frac{-151}{17}\).

Step-by-step solution

Set up equilibrium equations

The equilibrium in a market occurs when the quantity demanded equals the quantity supplied for both commodities. This gives us the equations \( Q_{d1} = Q_{s1} \) and \( Q_{d2} = Q_{s2} \). The demand and supply equations are given as \( Q_{d1} = 18 - 3P_1 + P_2 \), \( Q_{s1} = -2 + 4P_1 \), \( Q_{d2} = 12 + P_1 - 2P_2 \), and \( Q_{s2} = -2 + 3P_2 \).

Equate demand and supply for commodity 1

Equating the demand and supply for the first commodity, we have \[18 - 3P_1 + P_2 = -2 + 4P_1\]Simplify this equation to find a relationship between \( P_1 \) and \( P_2 \).

Solve for equilibrium condition of commodity 1

Rearrange and simplify the equation from Step 2:\[18 - 3P_1 + P_2 = -2 + 4P_1\18 + 2 = 4P_1 + 3P_1 - P_2\20 = 7P_1 - P_2\]Thus, we have the equation \(-7P_1 + P_2 = -20\).

Equate demand and supply for commodity 2

Now, for the second commodity, equate the demand and supply:\[12 + P_1 - 2P_2 = -2 + 3P_2\]Simplify this equation to get another linear equation.

Solve for equilibrium condition of commodity 2

Simplify the equation from Step 4:\[12 + P_1 - 2P_2 = -2 + 3P_2\12 + 2 = P_1 - 3P_2 + 2P_2\14 = P_1 - 5P_2\]This gives \(P_1 - 5P_2 = 14\).

Solve the system of equations

We have the system of equations:1. \(-7P_1 + P_2 = -20\)2. \(P_1 - 5P_2 = 14\)Substitute the first equation into the second or use elimination/substitution to solve for both \(P_1\) and \(P_2\). Elimination could be used here. Multiply the second equation by 7:\[7P_1 - 35P_2 = 98\]Now, add the two equations:\[(-7P_1 + P_2) + (7P_1 - 35P_2) = -20 + 98\-34P_2 = 78\P_2 = -\frac{78}{34} = -\frac{39}{17}\]

Substitute back to find \(P_1\)

Use the expression for \(P_2\) to find \(P_1\):Substituting \(P_2 = -\frac{39}{17}\) into the equation \(-7P_1 + P_2 = -20\):\[-7P_1 - \frac{39}{17} = -20\]Multiply through by 17 to clear the fraction:\[-119P_1 - 39 = -340\]\[-119P_1 = -301\]\[P_1 = \frac{301}{119}\]

Calculate equilibrium quantities

Substitute \(P_1 = \frac{301}{119}\) and \(P_2 = -\frac{39}{17}\) into the supply equations:\[Q_{s1} = -2 + 4P_1 = -2 + 4\left(\frac{301}{119}\right)= \frac{-238 + 1204}{119}= \frac{966}{119} \approx 8.118\]\[Q_{s2} = -2 + 3P_2 = -2 + 3\left(-\frac{39}{17}\right)= \frac{-34 - 117}{17}= \frac{-151}{17} \approx -8.882\]

Verify equilibrium and conclude

Ensure the quantities \( Q_{s1} \) and \( Q_{d1} \), \( Q_{s2} \) and \( Q_{d2} \) match when \( P_1 \) and \( P_2 \) are substituted back into the original demand equations. Check calculations for accuracy using equilibrium values.

Key concepts

Two-Commodity Market

In economics, a two-commodity market refers to a simplified model where two products are analyzed simultaneously. These goods are often interrelated, meaning that the demand for one might affect the supply or demand for the other. Such interactions are common in markets where substitute or complementary products are present. This type of market is analyzed to understand how both commodities can reach equilibrium within the same economic environment.

Understanding how these two commodities interact can help economists and businesses predict how changes, such as price adjustments or changes in market conditions, affect the equilibrium of both products. In the given exercise, we see these interdependencies clearly in the demand and supply functions where both commodity equations include terms that consider the other commodity's price. This demonstrates that decisions for one commodity cannot be made in isolation, and changes in one affect the other.

Demand Function

The demand function is a mathematical representation that describes how the quantity demanded of a good varies with changes in price and other determinants. In the two-commodity market setup, the demand for each commodity is influenced by its own price and potentially the price of the other commodity as well.

For example, in the exercise, the demand functions are given by:

• For Commodity 1: \(Q_{d1} = 18 - 3P_1 + P_2\)
• For Commodity 2: \(Q_{d2} = 12 + P_1 - 2P_2\)

These functions indicate that a rise in the price of one commodity usually decreases its demand, captured by the negative coefficients of \(P_1\) and \(P_2\). However, they also show cross-influence, where the price of one commodity affects the demand for another. Such a setup allows us to analyze how vulnerability in one market can have ripple effects on another. By solving these equations, you can determine the equilibrium price and quantity in the market for each commodity.

Supply Function

The supply function tells us how the quantity of a good supplied is related to its price and other influencing factors. For a two-commodity market, each commodity's price can directly impact its supply, similar to how it affects demand.

In our exercise, the supply functions are:

• For Commodity 1: \(Q_{s1} = -2 + 4P_1\)
• For Commodity 2: \(Q_{s2} = -2 + 3P_2\)

These equations show that the supply is positively correlated with the price, meaning as the price increases, suppliers are willing to offer more of the commodity. The positive coefficients next to \(P_1\) and \(P_2\) confirm this. Supply functions are crucial for finding the equilibrium in a market since equilibrium requires that the demand equals supply. Solving these supply equations together with demand equations provides the prices at which the market reaches balance.

Linear Equation System

Linear equation systems consist of multiple linear equations that can be solved simultaneously to find the values of the variables that satisfy all equations. In the case of our two-commodity market, we create a system of equations from both the demand and supply equations to find the equilibrium prices.

The exercise involves solving the following set of linear equations:

• \(-7P_1 + P_2 = -20\)
• \(P_1 - 5P_2 = 14\)

We solve these equations using methods like substitution or elimination to find precise values for \(P_1\) and \(P_2\), which represent the equilibrium prices for the commodities. Breaking down these equations into a manageable form allows us to solve for unknowns, providing the necessary data to understand the market's equilibrium. Once the equilibrium prices are identified, substituting these back into the demand or supply functions yields the equilibrium quantities, helping map out the complete picture of the market scenario.