In macroeconomics, the simple "consumption function" relates national expenditure on consumption goods, CONSUMP \(_{t}=\) aggregate consumption, in period \(t\) to national income, \(I N C O M E_{t}=G N P_{t}\). Specify the consumption function \(\operatorname{CONSUMP}_{t}=\beta_{1}+\beta_{2} I N C O M E_{t}+e_{t}\). Suppose that \(I N V_{t}\) is aggregate investment. In the simplest model, the income identity is \(I N C O M E_{t}=\operatorname{CONSUMP}_{t}+I N V_{t}\).

a. Substitute the income identity into the consumption function and solve for consumption in terms of investment.

b. Find the covariance between \(I N C O M E_{t}\) and the random error \(e_{t}\).

c. Find the covariance between \(I N V_{t}\) and \(I N C O M E_{t}\).

d. Suppose \(I N V_{t}\) is uncorrelated with the random error \(e_{t}\). Does it satisfy the conditions for an IV?