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Chapter 1: Problem 46

For each supply equation, where \(x\) is the quantity supplied in units of 1000 and \(p\) is the unit price in dollars, (a) sketch the supply curve and (b) determine the number of units of the commodity the supplier will make available in the market at the given unit price. $$ p=\frac{1}{2} x+20 ; p=28 $$

Short Answer

Expert verified
When the supply equation is \(p = \frac{1}{2}x + 20\), the supply curve has a slope of \(\frac{1}{2}\) and a y-intercept of 20. At a unit price of \(p = 28\), the supplier will provide \(x = 16\) units (in 1000s) of the commodity in the market.

Step by step solution

01

Rewrite the supply equation

The given supply equation is \(p = \frac{1}{2}x + 20\). To rewrite this in the form of \(y = mx + b\), we will treat the price \(p\) as \(y\) and the quantity \(x\) as \(x\). This gives us the equation: $$ y = \frac{1}{2}x + 20 $$
02

Identify slope and y-intercept

Now that we have rewritten the equation in the familiar form of \(y=mx+b\), we can identify the slope \(m\) and the y-intercept \(b\) of the supply curve. In this equation, the slope \(m\) is the coefficient of \(x\), which is \(\frac{1}{2}\), and \(b\), the y-intercept, is 20. Knowing these values will help us in sketching the graph in the next step.
03

Sketch the supply curve

To sketch the graph, we first note that the y-intercept \(b=20\), which means the curve intersects the y-axis (Price axis) at the point \((0,20)\). Next, we can use the slope value to find another point on the graph. Since the slope \(m=\frac{1}{2}\), we know that for each 2-unit increase in \(x\) (Quantity), there is a 1-unit increase in \(y\) (Price). This means that the graph will go through the point \((2,21)\). Using these two points, we can sketch the supply curve: 1. Draw an X-Y Cartesian plane. 2. Label the X-axis as "Quantity (in 1000s)" and the Y-axis as "Price". 3. Mark the points \((0,20)\) and \((2,21)\) on the graph. 4. Draw a straight line through these points to create the supply curve.
04

Calculate the number of units supplied at p=28

Now that we have the graph, we can determine the quantity supplied when the unit price is \(p=28\). We will substitute this value for \(y\) in our equation and solve for \(x\): $$ 28 = \frac{1}{2}x + 20 $$ To solve for \(x\), follow the steps below: 1. Subtract 20 from both sides of the equation: $$ 8=\frac{1}{2}x $$ 2. Multiply both sides of the equation by 2 to isolate \(x\): $$ 16=x $$ So when the unit price is \(p=28\), the supplier will provide \(x=16\) units (in 1000s) of the commodity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equilibrium Price
The equilibrium price is a key concept in economics representing the price at which the quantity of a good demanded by consumers equals the quantity supplied by producers. It's where the supply and demand curves cross on a graph, indicating a balance in the market. When price is above equilibrium, there's an excess supply, leading to lower prices until equilibrium is reached. Conversely, when the price is below equilibrium, there's excess demand, causing prices to rise. Understanding this concept helps us predict how changes in market conditions can affect price and quantity in the market.
Slope of Supply Curve
The slope of the supply curve represents the relationship between the price of a good and the quantity supplied. In your supply equation, the slope is \(\frac{1}{2}\) which means for every one dollar increase in price, the quantity supplied increases by two units of 1000. The positive slope reflects the law of supply: as the price increases, the quantity producers are willing to supply also increases. This upward slope is a fundamental characteristic of supply curves, depicting producers' willingness to supply more at higher prices.
Quantity Supplied
Quantity supplied refers to the specific amount of a good or service that producers are willing to supply at a given price. It鈥檚 a snapshot of supplier intentions and depends on numerous factors, including production costs and market price. For instance, at the price \(p=28\), your exercise determined that the producer would supply 16 units (in 1000s). This shows how the quantity supplied changes in response to changes in the market price, which is a central aspect of the law of supply.
Price-Quantity Relationship
The price-quantity relationship in the context of the supply curve is a direct relationship, expressed by the equation of the supply curve. In the exercise, this relationship is depicted as \(p = \frac{1}{2}x + 20\). Here, \(p\) corresponds to price, and \(x\) to quantity. As price increases, so does the quantity supplied, which is graphically represented by the upward sloping line on a supply curve graph. This relationship is vital for understanding how suppliers react to market prices and how they decide the quantity of a good to offer for sale.

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Most popular questions from this chapter

At a unit price of \(\$ 55\), the quantity demanded of a certain commodity is 1000 units. At a unit price of \(\$ 85\), the demand drops to 600 units. Given that it is linear, find the demand equation. Above what price will there be no demand? What quantity would be demanded if the commodity were free?

The projected number of navigation systems (in millions) installed in vehicles in North America, Europe, and Japan from 2002 through 2006 are shown in the following table \((x=0\) corresponds to 2002): $$ \begin{array}{lccccc} \hline \text { Year, } \boldsymbol{x} & 0 & 1 & 2 & 3 & 4 \\ \hline \text { Systems Installed, } \boldsymbol{y} & 3.9 & 4.7 & 5.8 & 6.8 & 7.8 \\ \hline \end{array} $$

The following table gives the projected U.S. online banking households as a percentage of all U.S. banking households from \(2001(x=1)\) through \(2007(x=7)\) : $$ \begin{array}{lccccccc} \hline \text { Year, } \boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \begin{array}{l} \text { Percentage of } \\ \text { Households, } \boldsymbol{y} \end{array} & 21.2 & 26.7 & 32.2 & 37.7 & 43.2 & 48.7 & 54.2 \\ \hline \end{array} $$ a. Find an equation of the least-squares line for these data. b. Use the result of part (a) to estimate the projected percentage of U.S. online banking households in 2008 .

The quantity demanded of a certain brand of DVD player is \(3000 /\) wk when the unit price is \(\$ 485\). For each decrease in unit price of \(\$ 20\) below \(\$ 485\), the quantity demanded increases by 250 units. The suppliers will not market any DVD players if the unit price is \(\$ 300\) or lower. But at a unit price of \(\$ 525\), they are willing to make available 2500 units in the market. The supply equation is also known to be linear. a. Find the demand equation. b. Find the supply equation. c. Find the equilibrium quantity and price.

Moody's Corporation is the holding company for Moody's Investors Service, which has a \(40 \%\) share in the world credit-rating market. According to Company Reports, the total revenue (in billions of dollars) of the company is projected to be as follows \((x=0\) correspond to 2004\()\) : $$ \begin{array}{lccccc} \hline \text { Year } & 2004 & 2005 & 2006 & 2007 & 2008 \\ \hline \text { Revenue, } \boldsymbol{y} & 1.42 & 1.73 & 1.98 & 2.32 & 2.65 \\\ \hline \end{array} $$ a. Find an equation of the least-squares line for these data. b. Use the results of part (a) to estimate the rate of change of the revenue of the company for the period in question. c. Use the result of part (a) to estimate the total revenue of the company in 2010 , assuming that the trend continues.
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