Question

By practicing, Brian can play more notes in tune, GB, on an upcoming music exam. GB is a function of the # of hours he spends on scales, S, and the number of hours he spends on arpeggios, A. Specifically, GB = 2.5S0.4A 0.6 . The tone production function of his roommate Cesar is GC = 2.5S0.75A 0.25

. a. What is Brian’s marginal productivity from scales?

b. What is Cesar’s?

c. What is Brian’s marginal rate of technical substitution between the two types of practice?

d. What is Cesar’s?

Answer 1

a. Brian's marginal productivity from scales, MUs = ∂(GB)/∂S = 0.4(2.5)S^0.4-1A^0.6 = S^-0.6A^0.6
So, MUs = S^-0.6A^0.6

b. Cesar's marginal productivity from scales, MUs = ∂(GC)/∂S = 0.75(2.5)S^0.75-1A^0.25 = 1.875S^-0.25A^0.25
So, MUs = (1.875)S^-0.25A^0.25

c. Brian's marginal productivity from arpeggios, MUa = ∂(GB)/∂A = 0.6(2.5)S^0.4A^0.6-1 = 1.5S^0.4A^-0.4
MRTS_SA = MUs/MUa = S^-0.6A^0.6/1.5S^0.4A^-0.4 = A^0.6+0.4/1.5S^0.6+0.4 = A/1.5S
So, MRTS = A/1.5S

d. Cesar marginal productivity from arpeggios, MUa = ∂(GC)/∂A = 0.25(2.5)S^0.75A^0.25-1 = 0.625S^0.75A^-0.75
MRTS_SA = MUs/MUa = 1.875S^-0.25A^0.25/0.625S^0.75A^-0.75 = 3A^0.25+0.75/S^0.75+0.25 = 3A/S
So, MRTS = 3A/S