Lab 17: Consumer and Producer Surplus Who benefits from rent controls? Who loses with price controls? How do taxes and subsidies affect the economy? Some of these questions can be analyzed using the concepts of consumer and producer surplus and the definite integral. 1. Demand and Supply Curves Open the Surplus tool in the Integration Kit and look at examples of linear and non-linear demand and supply curves. Keep checking the screen as you read about consumer and producer surplus. Economists assume that demand curves are never increasing, whereas supply curves are generally assumed to be upward sloping. For the demand curve shown, we see that when the quantity of a good is increased the unit price decreases, or conversely, when the price is lowered the amount demanded increases. For the supply curve we see that when the quantity supplied to the market is increased, the price to the supplier decreases. We see that when the price to the supplier is increased, the quantity producers are willing to supply is increased. The demand curve and supply curves intersect at the equilibrium point ( x eq, ). The price is also called the market price and x eq is the corresponding market quantity. It is generally assumed that the market will operate at the equilibrium point so that the quantity x eq will be supplied and sold at the market price. 2. Consumer and Producer Surplus When we assume that all consumers pay the equilibrium price, there are consumers who are paying less that they were willing to pay. If each consumer bought the commodity at the maximum price they were willing to pay, the total amount spent is represented by the area under the demand curve from zero to x eq. The difference between this amount and the amount spent at the equilibrium price is called the consumer surplus. It can be viewed as the consumers' gain from the trade. Consumer Surplus = area below the demand curve and above the line p = x eq = D( x)dx x eq (1) 0
When the commodity is sold at the equilibrium price, there is a gain to the producers also. Various producers were willing to supply the goods at the price given by the supply curve. However the supply price is always less than or equal to the equilibrium price. The difference between the amount received by the producers (i.e., x eq ) and the amount they would have received at the price level determined by the supply curve (the area under the supply curve from zero to x eq ) is called the producer surplus. Producer surplus = area above the supply curve and below the line p = x eq = x eq S( x)dx (2) 0 The sum of the consumer surplus and the producer surplus is called the total trade gain, and considered by economists to be a measure of the benefit to society of the transaction at the equilibrium point. 2.1 In the supply-demand figure shown (in Figure 1), shade in and label the areas that represent the consumer surplus and the producer surplus, respectively. Figure 1
2.2 For the linear model, calculate the consumer surplus and the producer surplus by means of a definite integral using the facts that D (x)= -0.5x +500 and S(x)=0.5x +20 in the formulas given in the earlier part of this section. Find the x eq and values from the tool. 2.3 Work through the following example: Find the consumer surplus and the producer surplus for an item whose supply and demand functions are given by: S( x) = 4x + 2 and D( x) = 20 x 2 for x thousands of units and prices in dollars/unit. Solution: First find the equilibrium point ( x eq, ) by equating S( x) and D( x) and solve for x (which will be x eq ). Start with: 3x + 2 = 20 x 2 The positive solution to this equation is x eq = (thousand units). The market price is = S( x eq ) = D( x eq ) = Set up the integrals for the consumer and producer surplus and the evaluate them. Show the intermediate steps. The consumer surplus = The producer surplus =
The total trade gain is dollars. 3. The Effect of Price Controls on Consumer and Producer Surplus Sometimes for the best motives, a government will interfere with the operation of the market process. For example price controls above the market can be imposed to insure the survival of marginal producers. The minimum wage and controls on cotton prices are two such examples. Also, prices can be kept artificially low. Rent control is an example of this policy. Select the box for price control to investigate these phenomena. 3.1 Use the price slider to select the controlled price p c to be higher than the market price. Notice that the producer surplus grows as the consumer surplus shrinks. Explain. 3.2 There is a new area on the screen that grows as the controlled price increases beyond the market price. This area (in yellow) represents the dead-weight loss (DWL) which is the difference between the total trade gain for the equilibrium price and quantity and the controlled price and quantity. The dead-weight loss is the loss in potential gains from all those transactions that never took place. On the graph (Figure 2) identify each of these areas.
Figure 2 3.3 Now look at the effect of rent controls when the price (the rent) is kept artificially low. When the rent cannot be greater than p c (for p c lower than ), the quantity of rental units supplied x c is less than x eq. The producer (landlord) surplus is the area above the supply curve and below y =. Notice that it appears to be reduced. This effect is balanced somewhat by the fact that those renters who have rental units have a greater consumer surplus. What does the dead-weight-loss represent in terms of renters and landlords? 4. The Effect of a Tax per Unit on Producer and Consumer Surplus Select the unit tax option at the top left of the tool. Suppose the government wishes to tax the producers a set amount for each item produced. This kind of tax is called a unit tax t. The unit tax adds a production cost equal to tx where x is the quantity of items produced. We can see from this that there is an additional marginal cost t (since
(t x) = t ). In the short run, the marginal cost curve is quite close to the supply curve (because dx in the short run, suppliers are willing to produce up to the point where the price they will receive for an additional item is equal to their marginal cost.) Consequently on the tool you can see that as you move the tax slider t, the supply curve moves up or down by the amount t. 4.1 Select the unit tax t = $60/item. Label the areas in the graph shown in Figure 3. Identify the areas that correspond to the consumer surplus, producer surplus, dead-weight loss and the amount of money accrued by the government by means of the tax. Figure 3 4.2 Experiment with various values of the tax t. Is it true that for positive t (that is, t is a tax, not a subsidy), the greater the unit tax t, the greater the dead-weight loss? Explain why tax increases might have this effect.
4.3 Now try negative values for the unit tax t. A negative unit tax is called a subsidy. The government is paying the producer for each item producer. Note that the big rectangle that contains the equilibrium point is the amount of money that the government pays out. Also see that part of this rectangle is filled in with yellow when the tax slider is released. That area represents the dead-weight loss (as before). What does the dead-weight loss mean in this context? 5. The Effect of a Sales Tax on Producer and Consumer Surplus A sales tax affects the demand curve as seen by the producers. Note that the price per item that the consumers see is p consumer = p producer + tp producer which lowers their demand according to their demand curve p = D(x) and gives a net price to the producer of p producer = p consumer 1 + t = D(x) 1 + t (3) In effect the producers see a lower demand curve. 5.1 Select the sales tax option on the tool and increase the tax t to verify this phenomenon for positive values of t. Select the option for the linear supply and demand curves. Note that the original equations are D(x)= -0.5x+500 and S(x)=0.5x+20. Determine the following slopes: Slope of the demand curve for 0% sales tax. (Recall that x is on thousands of units.) Slope of the demand curve (as seen by the producers) for a 50% sales tax. Recall that 50% =.5.
5.2 Now select the nonlinear case and set t =50%. Label the areas in Figure 4 that correspond to the producer and consumer surplus, the tax received by the government and the dead weight loss. Figure 4