The algebra of demand and supply
Dr. Amy McCormick Diduch
Concepts:
To describe the demand function symbolically, we could write it as QD = f {price, incomes, preferences, related goods prices, expectations, population}. (This reads as “quantity demanded is a function of price, incomes……”).
A simple linear equation for demand might be QD = 30 – 1/3 P, where the intercept (here, 30) accounts for the current values of all of those determinants other than the product’s price (i.e. incomes, preferences, etc.). Thus, a demand equation assumes all other things are held constant except price and quantity demanded (our “ceteris paribus” assumption).
This equation exhibits the expected negative relationship between price and quantity demanded: as price increases, quantity demanded will fall. This is an accurate way to create a mathematical formula for the demand relationship but it is hard to graph. Why? The problem with our simple linear equation is that to plot it, we would place Quantity on the vertical axis (with an intercept of 30) and Price on the horizontal axis. This is the opposite of how we sketch demand curves in economics! We need to solve this equation for Price so that we can graph it in its more common form (with Price on the vertical axis).
QD = 30 – 1/3 P
1/3 P = 30 - QD
P = 90 – 3QD
Note that we still have the expected negative relationship between price and quantity demanded. The slope of this demand curve is -3 and the y-axis intercept is 90. To graph it, begin by marking the vertical intercept (where P = 90 and Q = 0) and then either (a) find the horizontal intercept by setting P = 0 and solving for Q (here, it is 30) or, in a pinch, (b) plug in a few different values for Q and “connect the dots” to make a line.
Dr. Amy McCormick Diduch
Concepts:
- Expressing linear demand and supply functions algebraically
- Plotting the demand and supply functions; identifying the y-intercept and slope
- Finding equilibrium price and quantity from supply and demand equations
- Determining effects of changes in demand or supply on equilibrium price and quantity
- The demand function
To describe the demand function symbolically, we could write it as QD = f {price, incomes, preferences, related goods prices, expectations, population}. (This reads as “quantity demanded is a function of price, incomes……”).
A simple linear equation for demand might be QD = 30 – 1/3 P, where the intercept (here, 30) accounts for the current values of all of those determinants other than the product’s price (i.e. incomes, preferences, etc.). Thus, a demand equation assumes all other things are held constant except price and quantity demanded (our “ceteris paribus” assumption).
This equation exhibits the expected negative relationship between price and quantity demanded: as price increases, quantity demanded will fall. This is an accurate way to create a mathematical formula for the demand relationship but it is hard to graph. Why? The problem with our simple linear equation is that to plot it, we would place Quantity on the vertical axis (with an intercept of 30) and Price on the horizontal axis. This is the opposite of how we sketch demand curves in economics! We need to solve this equation for Price so that we can graph it in its more common form (with Price on the vertical axis).
QD = 30 – 1/3 P
1/3 P = 30 - QD
P = 90 – 3QD
Note that we still have the expected negative relationship between price and quantity demanded. The slope of this demand curve is -3 and the y-axis intercept is 90. To graph it, begin by marking the vertical intercept (where P = 90 and Q = 0) and then either (a) find the horizontal intercept by setting P = 0 and solving for Q (here, it is 30) or, in a pinch, (b) plug in a few different values for Q and “connect the dots” to make a line.
2. The supply function
To describe the supply function symbolically, we could write it as QS = f{price, technology, input costs, weather, number of sellers}. (This reads as “quantity supplied is a function of price, technology, ……”). The “law of supply” states that quantity supplied is a positive function of price; a linear supply function might take the form QS = 1/2 P – 10.
Since we want to graph price on the vertical axis, we need to rewrite the equation in terms of price:
QS = 1/2 P – 10
½ P = Qs + 10
P = 20 + 2QS
The slope of this supply curve is 2 and the vertical intercept is 20. To graph it, begin by marking the vertical intercept (20) and then plug in a larger value for Q (such as 30). Sketch a line connecting these points.
To describe the supply function symbolically, we could write it as QS = f{price, technology, input costs, weather, number of sellers}. (This reads as “quantity supplied is a function of price, technology, ……”). The “law of supply” states that quantity supplied is a positive function of price; a linear supply function might take the form QS = 1/2 P – 10.
Since we want to graph price on the vertical axis, we need to rewrite the equation in terms of price:
QS = 1/2 P – 10
½ P = Qs + 10
P = 20 + 2QS
The slope of this supply curve is 2 and the vertical intercept is 20. To graph it, begin by marking the vertical intercept (20) and then plug in a larger value for Q (such as 30). Sketch a line connecting these points.
3. Equilibrium
Equilibrium is defined as the price at which quantity supplied equals quantity demanded. We have a demand function, : P = 90 – 3QD, and a supply function P = 20 + 2QS. In equilibrium, QS = QD; there is one unique price at which this occurs. We will solve for the equilibrium quantity, Q*, by setting these equations equal to each other since the equilibrium price, P*, is the same in each.
Demand Supply
P = 90 – 3QD = P = 20 + 2QS
90 – 3Q = 20 + 2Q
70 = 5Q
70/5 = Q = 14
We can plug this equilibrium value for Q into either equation to find price:
P = 90 – 3Q = 90 – 42 = 48
P = 20 + 2Q = 20 + 28 = 48
Graphing the supply and demand curves on the same diagram, we can check our answers:
Equilibrium is defined as the price at which quantity supplied equals quantity demanded. We have a demand function, : P = 90 – 3QD, and a supply function P = 20 + 2QS. In equilibrium, QS = QD; there is one unique price at which this occurs. We will solve for the equilibrium quantity, Q*, by setting these equations equal to each other since the equilibrium price, P*, is the same in each.
Demand Supply
P = 90 – 3QD = P = 20 + 2QS
90 – 3Q = 20 + 2Q
70 = 5Q
70/5 = Q = 14
We can plug this equilibrium value for Q into either equation to find price:
P = 90 – 3Q = 90 – 42 = 48
P = 20 + 2Q = 20 + 28 = 48
Graphing the supply and demand curves on the same diagram, we can check our answers:
4. Change in demand
When sketching a “comparative statics” graph (in which a determinant of supply or demand changes), we illustrate the old and new equilibrium prices and quantities and indicate the direction a curve has shifted. For example, if incomes increase and a good is “normal,” we would shift the demand curve to the right and mark a higher price and higher quantity. How do we convey this information using an equation?
Our original demand equation was P = 90 – 3QD and our supply equation is P = 20 + 2QS. Assume that supply conditions remain constant but the increase in incomes (for a normal good) results in the new demand equation P = 120 – 3QD.
This increase in incomes shows up in the equation as a new (higher) y-intercept. Does this change match the way we draw demand shifts when we work with comparative statics problems? Yes! If you sketch a general increase in demand in the market equilibrium graph above, you’ll notice that the new vertical demand curve intercept would be higher. (It’s certainly possible that the slope of the demand curve could also change but we won’t work with that scenario here).
We can solve for the new equilibrium P and Q:
Demand Supply
120 – 3Q = 20 + 2Q
120-20 = 3Q + 5Q
100 = 5Q
Q = 20
Find price using either the supply or demand equation. Here's the calculation with the demand equation:
P = 120 – 3*20 = 60
Therefore, the increase in demand has resulted in a higher price and a higher quantity demanded. The graph below shows this change:
When sketching a “comparative statics” graph (in which a determinant of supply or demand changes), we illustrate the old and new equilibrium prices and quantities and indicate the direction a curve has shifted. For example, if incomes increase and a good is “normal,” we would shift the demand curve to the right and mark a higher price and higher quantity. How do we convey this information using an equation?
Our original demand equation was P = 90 – 3QD and our supply equation is P = 20 + 2QS. Assume that supply conditions remain constant but the increase in incomes (for a normal good) results in the new demand equation P = 120 – 3QD.
This increase in incomes shows up in the equation as a new (higher) y-intercept. Does this change match the way we draw demand shifts when we work with comparative statics problems? Yes! If you sketch a general increase in demand in the market equilibrium graph above, you’ll notice that the new vertical demand curve intercept would be higher. (It’s certainly possible that the slope of the demand curve could also change but we won’t work with that scenario here).
We can solve for the new equilibrium P and Q:
Demand Supply
120 – 3Q = 20 + 2Q
120-20 = 3Q + 5Q
100 = 5Q
Q = 20
Find price using either the supply or demand equation. Here's the calculation with the demand equation:
P = 120 – 3*20 = 60
Therefore, the increase in demand has resulted in a higher price and a higher quantity demanded. The graph below shows this change:
5. Change in supply
Suppose there is an increase in inputs costs (maybe oil prices increase). How does this affect market supply? In a “comparative statics” graph we would shift the supply curve to the left and mark a higher equilibrium price and lower equilibrium quantity. In shifting the supply curve to the left, we are effectively working with a higher vertical intercept.
Suppose our new supply equation is P = 25 + 2QS and we are working with our original demand equation P = 90 – 3QD.
Solving for the new equilibrium P and Q:
90 – 3Q = 25 + 2Q
90-25 = 2Q + 2Q
65 = 5Q
Q = 13
The new equilibrium price is
P = 25 + 2(13) = 51
Thus, the decrease in supply (due to the higher input costs) causes an increase in price and decrease in quantity, as predicted by our comparative statics analysis.
Suppose there is an increase in inputs costs (maybe oil prices increase). How does this affect market supply? In a “comparative statics” graph we would shift the supply curve to the left and mark a higher equilibrium price and lower equilibrium quantity. In shifting the supply curve to the left, we are effectively working with a higher vertical intercept.
Suppose our new supply equation is P = 25 + 2QS and we are working with our original demand equation P = 90 – 3QD.
Solving for the new equilibrium P and Q:
90 – 3Q = 25 + 2Q
90-25 = 2Q + 2Q
65 = 5Q
Q = 13
The new equilibrium price is
P = 25 + 2(13) = 51
Thus, the decrease in supply (due to the higher input costs) causes an increase in price and decrease in quantity, as predicted by our comparative statics analysis.
The file below contains practice problems using algebra.
algebra_of_supply_and_demand_-_practice_problems.pdf |
Prefer to see these concepts demonstrated step-by-step? The videos below cover the same material.