# Thread: Standard macroeconomic model and implicit differentiation

1. ## 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 $\displaystyle d^2Y/dI^2$.

2. Originally Posted by Runty
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 $\displaystyle d^2Y/dI^2$.
Taking the derivative with respect to I and using the chain rulegives

$\displaystyle \displaystyle \frac{dY}{dI}=\frac{df}{dY}\frac{dY}{dI}+1-\frac{dg}{dY}\frac{dY}{dI}$

$\displaystyle \displaystyle \frac{dY}{dI}=\frac{1}{\frac{dg}{dY}-\frac{df}{dY}}$

This should get you started.