You Have Collected Data for a Cross-Section of Countries in Two

You have collected data for a cross-section of countries in two time periods, 1960 and 1997, say. Your task is to find the determinants for the Wealth of a Nation (per capita income)and you believe that there are three major determinants: investment in physical capital in both time periods (X_1,T and X_1,0), investment in human capital or education (X_2,T and X_2,0), and per capita income in the initial period
(Y₀). You run the following regression:
ln(Y_T)= β₀ + β₁X_1,T + β₂X_1,0 + β₃X_2,T + β₄X_1,0 + ln(Y₀)+ u_T
One of your peers suggests that instead, you should run the growth rate in per capita income over the two periods on the change in physical and human capital. For those results to be a parsimonious presentation of your initial regression, what three restrictions would have to hold? How would you test for these? The same person also points out to you that the intercept vanishes in equations where the data is differenced. Is that correct?