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The inverse demand function is a crucial concept in understanding demand and revenue. It reverses the conventional demand relationship by expressing price as a function of quantity, rather than quantity as a function of price. For instance, if the demand equation is generally written as \(Q = f(p)\), the inverse demand function flips this to \(p = g(Q)\). This expression gives businesses and economic analysts insight into how much of a product they can sell at different price levels.

Let's take a look at our example: the inverse demand function is given by \(p = 60 - 2Q\). Here, \(p\) represents price and \(Q\) represents quantity. In this function:

• The constant term \(60\) indicates the price intercept when quantity is zero; no amount is supplied, yet buyers value the first unit at a price of \(60\).
• The coefficient \(-2\) shows the impact of quantity on price. As \(Q\) increases by 1 unit, \(p\) decreases by 2 units.

This gives us a linear relationship which will decline linearly as quantity increases, representing a downward sloping line when plotted graphically.

A linear demand curve is represented as a straight line when plotted graphically with price \((p)\) on the vertical axis and quantity \((Q)\) on the horizontal axis. The equation \(p = 60 - 2Q\) describes this specific curve. Let's break down what it means for a demand curve to be linear:

• The constant rate of change, indicated by the slope, shows that each additional unit of quantity decreases the price by a consistent amount, in this case, by 2 for each unit increase in \(Q\).
• This slope, derived from the coefficient of \(Q\) in the demand function, is key to understanding consumer behavior and how price changes impact demand.

To visualize, starting at \(Q = 1\), the corresponding price \(p\) is \(60 - 2(1) = 58\). As you increase \(Q\), you see the price fall steadily following this pattern, creating a straight downward line. Such a curve simplifies analysis as it avoids the complexity of variable elasticity, offering a constant price elasticity over its range.

The marginal revenue curve is fundamental to price-setting and output decisions for firms. It represents the additional revenue generated by selling one more unit of a product. For a linear demand curve like \(p = 60 - 2Q\), the marginal revenue curve has a distinct mathematical relationship.

The marginal revenue is derived from the inverse demand function and follows the formula \(MR = a - 2bQ\), where \(a\) is the price intercept and \(b\) is the slope of the demand curve. In our example:

• We substitute \(a = 60\) and \(b = 2\) into the formula, resulting in \(MR = 60 - 4Q\).
• This new equation means the marginal revenue curve has the same intercept as the demand curve but is twice as steep, indicating a steeper decline.

Graphically, when plotted against \(Q\), the marginal revenue curve lies below the linear demand curve across all quantities. This reflects the basic economic principle that marginal revenue decreases faster than price due to the need to lower prices on all units to sell additional units. The understanding of this relationship helps businesses determine optimal pricing strategies and quantities to maximize profit.