Question

Show that the marginal utility of money income is homogeneous of degree -1 in (p, m)

Explain.

Answer 1

sol

let the utility function be u = xy, where x and y are two goods and budget constraint be p_xx + p_yy = m

so MU_X = y and MU_y = x

in equilibrium

y/x = p_x/p_y or p_xx = p_yy

putting in budget constraint

2p_xx = m or x = m/2p_x

similarly y = m/2p_y

putting x and y in utility function

u = m² /4p_xp_y

MU_m = 2m/4p_xp_y  

MU_m = m/2p_xp_y = f[m,p_x,p_y]

f[km,kp_x,kp_y] = km/2k²p_xp_y = k^-1f[m,p_x,p_y]

Thus marginal utility of money income is homogeneous of degre -1 in [p ,m]