Let Y denote total expenditure on consumer durables, nondurables, and services and E_d, E_n, and E_s are the expenditures on the three categories. As defined, Y = E_d + E_n + E_s. Now, consider the expenditure system

E_d = α_d + β_dY + γ_ddP_d + γ_dnP_n + γ_ds P_s + ε_d,
E_n = α_n + β_nY + γ_ndP_d + γ_nnP_n + γ_ns Ps + ε_n,
E_s = α_s + β_sY + γ_sdP_d + γ_snP_n + γ_ss P_s + ε_s.

Prove that if all equations are estimated by ordinary least squares, then the sum of the expenditure coefficients will be 1 and the four other column sums in the preceding model will be zero.