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In economics, the demand function is a mathematical expression that describes how the quantity demanded of a good changes in response to changes in price. It tells us how much of a product consumers want to buy at different price levels. To understand this, let's consider the college newspaper example. If the papers are free, 2,000 copies are demanded each week. When a price of 10 cents per copy is introduced, the demand drops to 1,000 copies each week.
To create a linear demand function, we use the slope-intercept form from algebra, which is essential for defining this relationship. We identify two key points from the problem:

• Point 1: Price = 0, Demand = 2,000.
• Point 2: Price = 10, Demand = 1,000.

The slope of the demand curve is calculated as the change in demand over the change in price. In this example, the slope is -100, indicating that for every increase of 1 unit in price, the demand decreases by 100 copies. The demand function is thus calculated as:\[ D(p) = 2000 - 100p \]This equation shows that the demand decreases as the price increases, reflecting consumer behavior that is common in real-world scenarios. Understanding the demand function is crucial for businesses to set optimal prices and predict sales.

The supply function in economics is similar to the demand function but focuses on the relationship between price and the quantity of a good that producers are willing to supply. Generally, as the price of a good increases, producers are willing to supply more of it. In our college newspaper scenario, when the newspapers are free, the university supplies 600 copies per week. However, if the price is set at 20 cents per copy, the supply increases to 1,400 copies per week. To form the supply function, we examine these two points:

• Point 1: Price = 0, Supply = 600.
• Point 2: Price = 20, Supply = 1,400.

The slope of the supply curve is calculated as the change in quantity supplied over the change in price, resulting in a slope of 40.Using this information, the linear supply function is derived as:\[ S(p) = 40p + 600 \]This means that for each increase of 1 unit in price, the supply increases by 40 copies. The supply function is essential for understanding how producers respond to price changes, enabling them to make informed production decisions.

The equilibrium price occurs when the quantity of a good demanded by consumers equals the quantity supplied by producers. This concept is crucial in economics as it represents a stable point where the market is in balance, with no pressure for the price to change.To find the equilibrium price in our exercise, we equate the demand and supply functions:\[ 2000 - 100p = 40p + 600 \]Solving this equation yields an equilibrium price of 10 cents per copy. At this price, the number of newspapers demanded by students is equal to the number supplied by the university, ensuring that there is no excess supply or unmet demand.Understanding how to find the equilibrium price helps in predicting market behavior and in strategic decision-making, ensuring resources are allocated efficiently without wastage or scarcity.