Question

Sandy derives utility from consuming "all other goods," g, and clean air (measured by particulate matter removed per m3), a, as measured by the utility function U(g,a) =g0.6a0.4, such that the marginal utilities are: MUa = 0.4a–0.6g0.6 and MUg = 0.6g–0.4a0.4, whereas Sandy’s income is $100. The price of "all other goods" is $20 and the price of clean air (abatement) equals $10. Brian is the only other consumer in the market for clean air and demands 10 units of clean air. What is the market demand for clean air, a?

Answer 1

U(g,a) = g0.6a0.4  

MUg = 0.6g- 0.4a0.4

MUa = 0.4g0.6a0.6  

MRSg,a = MUg/MUa  

= 0.6g- 0.4a0.4 / 0.4g0.6a0.6  

= 3a/2g  

at optimal choice MRSg,a = Pg/Pa  

3a/2g = 20/10

3a/2g = 2

3a = 4g

g = 3a/4

Budget equation

Pgg + Paa = I

20g + 10a = 100

put g = 3a/4 in bidget equation  

20g + 10a = 100

20(3a/4) + 10a = 100

15a + 10a = 100

25a = 100

a = 100/25

a = 4

Demand for clean air by Sandy is 4 and demand for clean air by Brian is 10

Therefore total market demand for clean air is 4 + 10 = 14