Question

I've seen this problem solved with various answers. Still having difficulty understanding the process, please explain.

(20 points) Arnold and Betty have identical marginal benefit formulas for a public good. The MBs are given by : MB, = a(Y+98)-BQ* MB, = a(Y, +q)-BQ* where, Q* = 9, +93 a and B are constant positive numbers, a<l. YA, YB, are incomes of Arnold and Betty. 9A, QB are purchases of the public good by Arnold and Betty. The marginal cost of a unit of the public good is constant: MC-C. a. As in lecture, determine the best response functions for Arnold and Betty. Suppose now that a=.5, B=5, YA=Y=200, and MC=10. b. What is the strategic equilibrium quantity of the public good that gets produced? c. What is the socially optimal quantity of the public good that gets produced? d. Claim: as the slope of the MB curve gets steeper, the difference between the strategic equilibrium and the socially optimal quantity increases. Agree or disagree. How do the formulas help you answer the question? In words briefly explain the economics of this result.

Answer 1

Solution:

a)
Arnold would maximize

π1 = MBA*qA – CqA = α(YA + qB)qA – β(qA+qB)qA – Cqa

FOC; d π1/dqA = 0 i.e.

α(YA + qB) - 2 βqA – βqB – C = 0 or

2βqA = α(YA + qB) – βqB – C or

Therefore Best response function for Arnold is given as:

qA = (1/2β)*[ α(YA + qB) – βqB – C] …………… (1)

Because of symmetry, the best response function for Betty can be written as;

qB = (1/2β)*[ α(YB + qA) – βqA – C] …………… (2)

b)

α=.5, β=5, YA=YB=200, and MC=10

Substituting these values in (1) and (2)

qA = (1/2β)*[ α(YA + qB) – βqB – C] = [1/(2*5)]*[0.5(200+qB)-5*qB-10] = (1/10)*[100+0.5qB – 5qB -10] = 0.1*[90-4.5qB] Or

qA = 0.1*[90-4.5qB] …………… (1)

qB = (1/2β)*[ α(YB + qA) – βqA – C] = [1/(2*5)]*[0.5(200+qA)-5*qA-10] = (1/10)*[100+0.5qA – 5qA -10] = 0.1*[90-4.5qA] Or

qB = 0.1*[90-4.5qA] …………… (2)”

Solving (1)” and (2)”

qA = 0.1*{90-4.5*[0.1*(90-4.5qA)]} or

qA = 9 - 0.01*4.5*(90-4.5qA) or

qA = 9-4.05-0.2025qA or

0.7975qA = 4.95, implies qA = 6.206897

And qB = 0.1*[90-4.5qA] = 0.1*(90-4.5*6.206897) = 6.2068964

So, strategic quantity of the public goods (Q) = qA + qB = 12.414

c)

For social optimal quantity, we derve marginal social benefit (MSB)

MSB = MBA + MBB = α(YA + qB) – βQ + α(YB + qA) – βQ

MSB = α(YA+YB+Q*)-2βQ* …………… (3)

Substituting α=.5, β=5, YA=YB=200

MSB = 0.5(400+Q*)-10Q = 200-9.5Q

At social optimal point, MSB = MC or

200-9.5Q = 10, implies social optimal quantity (Q) = 20

d)

Steepness of MB curve is denoted by ‘β’ i.e. increase in ‘β’ would make MB curve steeper.

We disagree with the statement that as MB becomes steeper i.e. as β increases, the difference between the strategic equilibrium and the socially optimal quantity increases.

From equation (1) and (2), we note that increase in β would cause strategic quantities to increase and from equation (3) we note that increase in β would result in social optimal quantity to fall. This implies that the increase in β i.e. as MB curve becomes steeper, the difference between the strategic equilibrium and the socially optimal quantity will fall.

This happens because β denote the disvalue of public goods to the each agent. So when value of β increases, social optimal quantity decreases and hence the difference between the strategic equilibrium and the socially optimal quantity decreases.