Question

Find the consumer and producer surpluses by using the demand and supply functions, where p is the price (in dollars) and x is the number of units (in millions). Demand Function Supply Function p = 410 − x p = 160 + x
consumer surplus $ millions
producer surplus $ millions

Answer 1

Answer:

$7812.5million

$7812.5million

Step-by-step explanation:

From the question, the demand quantity D(x) is 410-x and the supply quantity S(x)=160+x. We determine the equilibrium quantity by equating the Demand quantity and the Supply quantity i.e

D(x)=S(x).

$410-x=160+x\\x=125$

Hence the equilibrium quantity is 125.

Next we determine the equilibrium price. This can be obtain by just substituting the equilibrium quantity into either the demand quantity or the supply quantity. I prefer using the Demand quantity.

Equilibrium price=$410-125=285$.

Next we write the expression for the Consumer Surplus

$CS=\int\limits^q_0 {D(x)} \, dx -pq$

where p and q are the equilibrium price and equilibrium quantity respectively.

By substituting values we have

$CS=\int\limits^q_0 {(410-x)} \, dx (285)(125)$

$CS=/410x-\frac{x^{2} }{2} /^{125}_{0}$

By carrying out simple arithmetic we arrive at

$CS=(51250-7812.5)-(285)(125)\\CS= 7812.5million$.

To determine the producer surplus, we use the expression below

$PS=pq-\int\limits^q_0 {S(x)} \,dx$

Hence if we substitute values we arrive at

$PS=(285)(125)-\int\limits^q_0 {(160+x)} \,dx\\PS=(285)(125)-[(160x-\frac{x^{2} }{2})]^{125}_{0}$.

By simply simplification we arrive at

$PS=(285)(125)-(20000+7812.5)\\PS=35625-27812.5\\PS= 7812.5million$