# 3.3. The Supply Function and the Supply Curve

#cfa #cfa-level-1 #economics #microeconomics #reading-13-demand-and-supply-analysis-introduction #study-session-4

The willingness and ability to sell a good or service is called supply. In general, producers are willing to sell their product for a price as long as that price is at least as high as the cost to produce an additional unit of the product. It follows that the willingness to supply, called the supply function, depends on the price at which the good can be sold as well as the cost of production for an additional unit of the good. The greater the difference between those two values, the greater is the willingness of producers to supply the good.

In another reading, we will explore the cost of production in greater detail. At this point, we need to understand only the basics of cost. At its simplest level, production of a good consists of transforming inputs, or factors of production (such as land, labor, capital, and materials) into finished goods and services. Economists refer to the “rules” that govern this transformation as the technology of production. Because producers have to purchase inputs in factor markets, the cost of production depends on both the technology and the price of those factors. Clearly, willingness to supply is dependent on not only the price of a producer’s output, but also additionally on the prices (i.e., costs) of the inputs necessary to produce it. For simplicity, we can assume that the only input in a production process is labor that must be purchased in the labor market. The price of an hour of labor is the wage rate, or W. Hence, we can say that (for any given level of technology) the willingness to supply a good depends on the price of that good and the wage rate. This concept is captured in the following equation, which represents an individual seller’s supply function:

Equation (7)

Qsx=f(Px,W,…)

where Qsx is the quantity supplied of some good X, such as gasoline, Px is the price per unit of good X, and W is the wage rate of labor in, say, dollars per hour. It would be read, “The quantity supplied of good X depends on (is a function of) the price of X (its “own” price), the wage rate paid to labor, etc.”

Just as with the demand function, we can consider a simple hypothetical example of a seller’s supply function. As mentioned earlier, economists often will simplify their analysis by using linear functions, although that is not to say that all demand and supply functions are necessarily linear. One hypothetical example of an individual seller’s supply function for gasoline is given in Equation 8:

Equation (8)

Qsx=−175+250Px−5W

Notice that this supply function says that for every increase in price of $1, this seller would be willing to supply an additional 250 units of the good. Additionally, for every$1 increase in wage rate that it must pay its laborers, this seller would experience an increase in marginal cost and would be willing to supply five fewer units of the good.

We might be interested in the relationship between only two of these variables, price and quantity supplied. Just as we did in the case of the demand function, we use the assumption of ceteris paribus and hold everything except own-price and quantity constant. In our example, we accomplish this by setting W to some value, say, $15. The result is Equation 9: Equation (9) Qsx=−175+250Px−5(15)=−250+250Px in which only the two variables Qsx and Px appear. Once again, we can solve this equation for Px in terms of Qsx , which yields the inverse supply function in Equation 10: Equation (10) Px = 1 + 0.004Qx The graph of the inverse supply function is called the supply curve, and it shows simultaneously the highest quantity willingly supplied at each price and the lowest price willingly accepted for each quantity. For example, if the price of gasoline were$3 per gallon, Equation 9 implies that this seller would be willing to sell 500 gallons per week. Alternatively, the lowest price she would accept and still be willing to sell 500 gallons per week would be $3. Exhibit 3 represents our hypothetical example of an individual seller’s supply curve of gasoline. Exhibit 3. Individual Seller’s Supply Curve for Gasoline What does our supply function tell us will happen if the retail price of gasoline rises by$1? We insert the new higher price of $4 into Equation 8 and find that quantity supplied would rise to 750 gallons per week. The increase in price has enticed the seller to supply a greater quantity of gasoline per week than at the lower price. ## 3.4. Changes in Supply vs. Movements along the Supply Curve As we saw earlier, a change in the (own) price of a product causes a change in the quantity of that good willingly supplied. A rise in price typically results in a greater quantity supplied, and a lower price results in a lower quantity supplied. Hence, the supply curve has a positive slope, in contrast to the negative slope of a demand curve. This positive relationship is often referred to as the law of supply. What happens when a variable other than own-price takes on different values? We could answer this question in our example by assuming a different value for wage rate, say,$20 instead of \$15. Recalling Equation 9, we would simply put in the higher wage rate and solve, yielding Equation 11.

Equation (11)

Qsx=−175+250Px−5(20)=−275+250Px

This equation, too, can be solved for Px, yielding the inverse supply function:

Equation (12)

Px = 1.1 + 0.004Qx

Notice that the constant term has changed, but the slope has remained the same. The result is a shift in the entire supply curve, as illustrated in Exhibit 4:

Exhibit 4. Individual Seller’s Supply Curve for Gasoline before and after Increase in Wage Rate

Notice that the supply curve has shifted both vertically upward and horizontally leftward as a result of the rise in the wage rate paid to labor. This change is referred to as a change in supply, as contrasted with a change in quantity supplied that would result only from a change in this product’s own price. Now, at a price of 3, a lower quantity will be supplied: 475 instead of 500. Alternatively, in order to entice this seller to offer the same 500 gallons per week, the price would now have to be 3.1, up from 3 before the change. This increase in lowest acceptable price reflects the now higher marginal cost of production resulting from the increased input price the firm now must pay for labor.

To summarize, a change in the price of a good itself will result in a movement along the supply curve and a change in quantity supplied. A change in any variable other than own-price will cause a shift in the supply curve, called a change in supply. This distinction is identical to the case of demand curves.

EXAMPLE 3

# Representing Seller Behavior with a Supply Function and Supply Curve

An individual seller’s monthly supply of downloadable e-books is given by the equation

Qseb=−64.5+37.5Peb−7.5W

where Qseb is number of e-books supplied each month, Peb is price of e-books in euros, and W is the hourly wage rate in euros paid by e-book sellers to workers. Assume that the price of e-books is €10.68 and the hourly wage is €10.

1. Determine the number of e-books supplied each month.

2. Determine the inverse supply function for an individual seller.

3. Determine the slope of the supply curve for e-books.

4. Determine the new vertical intercept of the individual e-book supply curve if the hourly wage were to rise to €15 from €10.

## Solution to 1:

Insert given values into the supply function and calculate the number of e-books:

Qseb=−64.5+37.5(10.68)−7.5(10)=261

Hence, each seller would be willing to supply e-books at the rate of 261 per month.

## Solution to 2:

Holding all other things constant, the wage rate is constant at €10, so we have

Qseb=−64.5+37.5Peb−7.5(10)=−139.5+37.5Peb

We now solve this for Peb:

Peb = 3.72 + 0.0267Qeb

## Solution to 3:

Note that when Qeb rises by one unit, Peb rises by 0.0267 euros, so the slope of the supply curve is 0.0267, which is the coefficient on Qeb in the inverse supply function. Note that it is not 37.5.

## Solution to 4:

In the supply function, increase the value of W to €15 from €10:

Qseb=−64.5+37.5Peb−7.5(15)=−177+37.5Peb

and invert by solving for Peb:

Peb = 4.72 + 0.0267Qeb

The vertical intercept is now 4.72. Thus, an increase in the wage rate shifts the supply curve upward and to the left. This change is known as a decrease in supply because at each price the seller would be willing now to supply fewer e-books than before the increase in labor cost.