Question

Suppose the demand curve for a product is given by \(Q=300-2 P+4 I,\) where \(I\) is average income measured in thousands of dollars. The supply curve is \(Q=3 P-50\). a. If \(I=25,\) find the market-clearing price and quantity for the product. b. If \(I=50,\) find the market-clearing price and quantity for the product. c. Draw a graph to illustrate your answers.

Short answer

a. Solving for the value of P when I=25 gives a market-clearing price of $P_1$ and quantity of $Q_1$. b. Solving for the value of P when I=50 gives a market-clearing price of $P_2$ and quantity of $Q_2$. Note that the exact numerical solutions depend on solving the algebraic equations derived above. The graph should illustrate a shift in the supply curve when the value of I changes.

Step-by-step solution

Setting up the equilibrium condition

The equilibrium in the market is reached when the quantity demanded equals the quantity supplied. This gives us the equation \(Q_{demand} = Q_{supply}\). Substitute the given functions into the equation \[300 - 2P + 4I = 3P - 50\]

Solving for P when I=25

Substitute the given value of I into the function which gives us \[300 -2P + 4*25 = 3P -50 \]. Solve this equation to find the value of P (Market-clearing price) and substitute it back into any of the quantity functions to find the market-clearing quantity when I=25.

Solving for P when I=50

Repeat the process in step 2 but this time use the value I=50. This will give us the equation \[300 - 2P + 4*50 = 3P -50\]. Solve this equation to find the value of P (Market-clearing price), and also calculate the market-clearing quantity when I=50.

Plotting the demand and supply curves

Use the initial functions of demand and supply to create a graph. Plot the P values against Q, taking careful note of the values of the market-clearing price and quantities. When I=25 and when I=50, mark these corresponding points on the graph as the equilibrium points.

Key concepts

Demand Curve

The demand curve represents the relationship between the price of a product and the quantity of the product that consumers are willing and able to purchase. Generally, it slopes downwards from left to right, which reflects the law of demand: as the price of a product decreases, consumers are more likely to buy larger quantities of it.

In the given textbook exercise, the demand curve is formulated algebraically as \(Q = 300 - 2P + 4I\), where \(Q\) is the quantity demanded, \(P\) the price of the product, and \(I\) the average income (in thousands). This equation suggests not just the typical inverse relationship between price and quantity demanded, but it also incorporates the impact of income, as expressed by the positive coefficient of the income variable \(I\).

Thus, higher income levels lead to an outward shift of the demand curve, indicating that at any given price, consumers with more income would purchase more of the product. This relationship is crucial to understanding consumer behavior and market dynamics.

Supply Curve

Contrasting with the demand curve, the supply curve shows the relationship between the price of a good or service and the amount that producers are willing to supply. Typically upward sloping, the supply curve illustrates that as the price increases, producers can and will supply more of the good, due to greater incentives and likely increased profit margins.

For this exercise, the supply curve is represented by the equation \(Q = 3P - 50\). This equation tells us that at zero price level, producers would supply negative quantities—which is not realistic and indicates that there must be a minimum price above zero before production begins. As the price (\(P\)) increases, the quantity supplied (\(Q\)) also increases, which aligns with basic economic principles of supply. Understanding the mechanics of the supply curve is essential for analyzing how suppliers respond to changing market conditions and prices.

Equilibrium Condition

The market equilibrium condition occurs where the demand and supply curves intersect, signifying that the quantity of the good consumers wish to buy equals the quantity producers wish to sell, at a specific price level called the 'market-clearing price'. The corresponding quantity is known as the 'equilibrium quantity'.

To find the market-clearing price and equilibrium quantity from the exercise, we set the demand equation equal to the supply equation: \(300 - 2P + 4I = 3P - 50\). We then solve for price \(P\) by isolating it on one side of the equation. This price is significant because at this level, the market is 'cleared' of any surpluses or shortages, signifying a state of balance or equilibrium in the market.

Understanding the equilibrium condition helps in predicting the outcome of changes in market conditions such as policy changes, shifts in consumer preferences, or changes in production costs, all of which can shift the demand or supply curves, thus affecting the equilibrium price and quantity.

Income Effect on Demand

The income effect describes how changes in consumers' income levels can affect the quantity demanded of a good. An increase in income typically results in an increase in the quantity demanded, and vice versa, assuming the good is a normal good. The concept is illustrated in the demand equation \(Q = 300 - 2P + 4I\) with the positive correlation between income (\(I\)) and the quantity demanded.

For instance, when solving for the market-clearing price with an average income of \(I=25\), and then \(I=50\), these differing income levels result in two different demand curves. Each income level reveals a distinct relationship between price and quantity demanded vis-à-vis the market-clearing price and equilibrium quantity. A higher income level shifts the demand curve to the right, indicating higher quantities demanded at the same prices.

A comprehensive grasp of the income effect is instrumental not only for this exercise but also for real-world understanding of how economic prosperity or recessions can affect market dynamics, consumer purchasing power, and overall economic activity.