10- Calculate the consumer surplus and producer surplus for the following demand an supply equations: Demand: P=100-2Q Supply: P=20+3Q Assume the market is in equilibrium. (10 points)

To calculate the consumer surplus we first need to find the equilibrium price and quantity. The equilibrium occurs where the demand and supply equations are equal, i.e., where $P_D = P_S$.<br /><br />Setting the demand equation equal to the supply equation:<br />$100 - 2Q = 20 + 3Q$<br /><br />Solving for $Q$:<br />$100 - 20 = 3Q + 2Q$<br />$80 = 5Q$<br />$Q = 16$<br /><br />Now, substitute $Q = 16$ into either the demand or supply equation to find the equilibrium price $P$:<br />$P = 100 - 2(16) = 100 - 32 = 68$<br /><br />The equilibrium price is $P = 68 quantity is $Q = 16$.<br /><br />Consumer surplus is the area between the demand curve and the equilibrium price level, up to the equilibrium quantity. It can be calculated as:<br />$\text{Consumer Surplus} = \frac{1}{2} \times (\text{Base}) \times (\text{Height})$<br /><br />The base is the equilibrium quantity ($Q = 16$), and the height is the difference between the maximum price consumers are willing to pay (when $Q = 0$) and the equilibrium price:<br />$\text{Height} = 100 - 68 = 32$<br /><br />So, the consumer surplus is:<br />$\text{Consumer Surplus} = \frac{1}{2} \times 16 \times 32 = 256$<br /><br />Producer surplus is the area between the supply curve and level, up to the equilibrium quantity. It can be calculated as:<br />$\text{Producer Surplus} = \frac{1}{2} \times (\text{Base}) \times (\text{Height})$<br /><br />The base is the equilibrium quantity ($Q = 16$), and the height is the difference between the equilibrium price and the minimum price producers are willing to accept (when $Q = 0$):<br />$\text{Height} = 68 - 20 = 48$<br /><br />So, the producer surplus is:<br />$\text{Producer Surplus} = \frac{1}{2} \times 16 \times 48 = 384$<br /><br />Therefore, the consumer surplus is $256$, and the producer surplus is $384$.