Let p represent the price of a good and q the quantity produced. Define S(q) to
ID: 2850969 • Letter: L
Question
Let p represent the price of a good and q the quantity produced. Define S(q) to be the price a producer charges if they produce the quantity q, and let D(q) be the consumer demand for the good if q units are produced. A graph showing both supply and demand curves is shown below: It is common to write the intersection point of the two curves as (p_*, q_*). The area shaded in blue is called the consumer surplus, and the area shaded in red is called the producer surplus. Write both the consumer and producer surpluses as integrals involving the function S(q) and D(q).Explanation / Answer
The price per unit by p and the quantity supplied or demanded at that price by q. As is the convention in economics, we will always write p as a function of q.
Thus the supply curve will be denoted by the formula
p = S(q)
and represented by a graph where the x and y axes correspond to q and p values respectively.
Similarly,
we will use p = D(q) to denote the demand curve.
As we might expect, the supply function S is increasing – the higher the price, the more the producers will supply. The demand function D is decreasing – the higher the price, the less the consumers will buy.
total amount spent at equilibrium price = pe*qe.
now, divide the interval [0, qe] into n subintervals, each of length x = qe/n, with endpoints
x0 = 0, x1 = qe n , x2 = 2qe n , · · · , xn = nqe n = qe.
hence we get
(price per unit) × (number of units) = D(x1)x dollars
pe is approximately equal to
D(x1)x + D(x2)x + · · · + D(xn)x.
Total amount paid at maximum prices = Z qe 0 D(q) dq
hence consumer surplus we get throgh above relation is:
Consumer surplus = Z qe 0 D(q)dq peqe = Z qe 0 [D(q) pe] dq.
similarly producer surplus = peqe Z qe 0 S(q) dq = Z qe 0 [pe S(q)] dq.
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