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Math Help - 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) Y=C+I+\overline{X}-M

    (ii) C=f(Y)

    (iii) M=g(Y)

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

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

    (b) Find an expression for 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) Y=C+I+\overline{X}-M

    (ii) C=f(Y)

    (iii) M=g(Y)

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

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

    (b) Find an expression for d^2Y/dI^2.
    Taking the derivative with respect to I and using the chain rulegives

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

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

    This should get you started.
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