Next-nano inc. is about to release a software upgrade that includes some new functionalities. The market study reveals that the demand to beq = D(p) = 1000sqrt(200-2p) . The willingness to order such an upgrade is given by the supply curve q = S(p) = 1000sqrt(10p-400).
b) Determine the consumers’ surplus using integrals at the equilibrium price (p = 50, q = 10000)
c) Determine the producers’ surplus using integrals at the equilibrium price (p = 50, q = 10000)
b) To calculate the consumers' surplus at the equilibrium price, we need to find the integral of the demand curve between the maximum reservation price (when the quantity demanded is zero) and the equilibrium price.
First, let's find the maximum reservation price:
D(p) = 1000sqrt(200-2p) = 0
200-2p = 0
2p = 200
p = 100
Now, let's find the integral of the demand curve between p = 100 and p = 50:
Consumers' Surplus = ∫[D(p) - 10000]dp from p = 50 to p = 100
= ∫[1000sqrt(200-2p) - 10000]dp from p = 50 to p = 100
To find the integral, we can use substitution. Let u = 200-2p, so -2dp = du, and dp = -du/2:
Consumers' Surplus = (-1/2)∫[1000sqrt(u) - 10000]du from u = 0 to u = 100
Now we just need to find the antiderivative:
F(u) = (-1/2)[(2/3)1000u^(3/2) - 10000u]
Now we can find the definite integral:
Consumers' Surplus = F(100) - F(0) = (-1/2)[(2/3)1000(100)^(3/2) - 10000(100)] - 0
= (-1/2)[(20000) - (1000000)]
= (1/2)(980000)
Consumers' Surplus ≈ 490,000
c) To calculate the producers' surplus at the equilibrium price, we need to find the integral of the supply curve between the equilibrium price and the minimum production price (when the quantity supplied is zero).
First, let's find the minimum production price:
S(p) = 1000sqrt(10p-400) = 0
10p - 400 = 0
10p = 400
p = 40
Now, let's find the integral of the supply curve between p = 40 and p = 50:
Producers' Surplus = ∫[10000 - S(p)]dp from p = 40 to p = 50
= ∫[10000 - 1000sqrt(10p-400)]dp from p = 40 to p = 50
To find the integral, we can use substitution. Let v = 10p - 400, so 10dp = dv, and dp = dv/10:
Producers' Surplus = (1/10)∫[10000 - 1000sqrt(v)]dv from v = 0 to v = 100
Now we need to find the antiderivative:
G(v) = (1/10)[10000v - (2/3)1000v^(3/2)]
Now we can find the definite integral:
Producers' Surplus = G(100) - G(0) = (1/10)[10000(100) - (2/3)1000(100)^(3/2)] - 0
= (1/10)[(1000000) - (20000)]
= (1/10)(980000)
Producers' Surplus ≈ 98,000