Standard macroeconomic model and implicit differentiation

A standard macroeconomic model for income determination in an open economy is

**(i)** $\displaystyle Y=C+I+\overline{X}-M$

**(ii)** $\displaystyle C=f(Y)$

**(iii)** $\displaystyle M=g(Y)$

where $\displaystyle 0<f'(Y)<1$. Here $\displaystyle \overline{X}$ is an exogenous constant that denotes exports, whereas $\displaystyle M$ denotes the volume of imports. The function $\displaystyle g$ in **(iii)** is called an import function. By inserting **(ii)** and **(iii)** into **(i)**, we obtain an equation that defines $\displaystyle Y$ as a function of exogenous investment $\displaystyle I$.

**(a)** Find an expression for $\displaystyle dY/dI$ by implicit differentiation. What is the likely sign of $\displaystyle g'(Y)$? Discuss the sign of $\displaystyle dY/dI$.

**(b)** Find an expression for $\displaystyle d^2Y/dI^2$.