Question

In Example 2.8 we examined the effect of a 20 -percent decline in copper demand on the price of copper, using the linear supply and demand curves developed in Section $2.6.$ Suppose the long-run price elasticity of copper demand were -0.75 instead of -0.5 a. Assuming, as before, that the equilibrium price and quantity are $P^{\ast }=\mathrm{\$}3$ per pound and $Q^{\ast }=18\mathrm{mil}$ lion metric tons per year, derive the linear demand curve consistent with the smaller elasticity. b. Using this demand curve, recalculate the effect of a 55-percent decline in copper demand on the price of copper.

Short answer

The new demand function with an elasticity of -0.75 will be $P=0.1111Q+1$. The new price of copper following a 55% decrease in demand, according to the new demand function, is $1.9 per pound.

Step-by-step solution

Compute the new slope of the demand curve

Given that the price elasticity of copper demand $E_p$ is -0.75, the formula for price elasticity of demand is denoted as $E_p=(dQ/Q)/(dP/P)$, where $dQ/Q$ is the percentage change in quantity demanded and $dP/P$ is the percentage change in price. We can rearrange the formula for the slope of the demand curve which features the price $P$, quantity $Q$, and the price elasticity of demand $E_p$: $slope=-(P/E_p\ast Q)$. Substituting the given values we get: $slope=-(3/-0.75\ast 18)=0.1111$

Find the y-intercept of the demand curve

To find the y-intercept, we'll use the equilibrium point in the line equation $P=slope\ast Q+intercept$. By rearranging, the intercept equals to $P-slope\ast Q$. Substituting the values we get: $intercept=3-0.1111\ast 18=1$

Write down the new demand function

Now we have the slope and y-intercept, we can create the new linear demand curve which is $P=0.1111\ast Q+1$

Calculate the new price given a 55% decrease in demand

A 55% decline in demand means that the new quantity demanded will be $Q=18\ast (1-0.55)=8.1$ million metric tons. By inserting the new quantity in our derived demand function, we get a new price $P=0.1111\ast 8.1+1=1.9$ dollars. Therefore, the price of copper following a 55% decrease in demand, given the new price elasticity of -0.75, is 1.9 dollars.

Key concepts

Linear Supply and Demand Curves

Understanding the basics of linear supply and demand curves is essential for analyzing market dynamics. These curves visually represent the relationship between the price of a good and the quantity supplied or demanded at that price. For demand, the curve typically slopes downwards, indicating that as the price decreases, the quantity demanded increases. Conversely, for supply, the curve slopes upwards, suggesting that as the price increases, suppliers are willing to offer more of the good.

When we speak of a 'linear' relationship, it means that the graph of the supply or demand curve is a straight line. This implies a constant rate of change — the slope — which is rare in real-life scenarios but simplifies calculations and conceptual understanding. For example, if the demand curve for copper is linear, a consistent reduction in the price will lead to proportional increases in the quantity demanded.

To enhance this concept for students, demonstrating how changes in various factors shift these lines or affect their slopes provides practical insight into market behavior, such as how a subsidy affects supply or how consumer preference changes impact demand.

Equilibrium Price and Quantity

The point where the supply and demand curves intersect is known as the equilibrium price and quantity. At this point, the amount of goods that consumers are willing to buy equals the amount that producers are willing to sell. This is a state of balance where there is no inherent tendency for change—unless external factors disrupt the equilibrium.

In our example, the equilibrium price of copper is initially $3 per pound, and the equilibrium quantity is 18 million metric tons per year. These figures are crucial as they reflect the market's consensus on the value of copper. If there's a disturbance, like a change in demand, equilibrium will shift, changing both the price and quantity. A detailed understanding of this concept can help students predict how various market forces will influence prices and production levels over time.

Slope of the Demand Curve

The slope of the demand curve is a critical element in understanding consumer behavior. It measures the rate at which quantity demanded changes in response to a change in price. Mathematically, the slope is the ratio of the change in the price to the change in the quantity demanded.

In simplified terms, imagine you're climbing down a hill—the steeper the hill (or the curve), the more significant the change in elevation (or quantity demanded) for every step you take (change in price). A steeper demand curve means that a small change in price leads to a large change in the quantity demanded. In the provided solution, calculating the new slope of the demand curve is vital to determining both the demand curve's equation and how responsive consumers are to price changes.

Y-intercept of the Demand Curve

The y-intercept of the demand curve is the point on the graph where the demand curve crosses the y-axis. This intercept represents the price that consumers would pay if the quantity demanded was zero—a theoretical construct since it's uncommon for the quantity demanded to be zero.

The y-intercept is a starting point for plotting the demand curve and holds importance in the equation of the demand curve. For instance, let's say you're setting up a lemonade stand. Initially, regardless of price, you haven't sold any lemonade—this would be represented by the y-intercept. In the exercise, we derive a y-intercept value of 1. This value corresponds to the scenario without the effects of changes in the quantity—providing us with the baseline price from which to build our demand equation.