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Thread: Standard macroeconomic model and implicit differentiation

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    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 $\displaystyle d^2Y/dI^2$.
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    Quote Originally Posted by Runty View Post
    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.
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