10. (LO2) The marginal propensity to consume is given by MPC=0.75+(4)/(sqrt (Y)) where Y is national income. If consumption is 20 when Y=0 find the consumption when Y=25. e) 34 a) 28 b) 24 c) 36 d) 78.75

To find the consumption when \( Y = 25 \), we need to integrate the marginal propensity to consume (MPC) function with respect to \( Y \) to obtain the consumption function.<br /><br />Given:<br />\[ MPC = 0.75 + \frac{4}{\sqrt{Y}} \]<br /><br />The consumption function \( C(Y) \) is obtained by integrating the MPC function:<br />\[ C(Y) = \int \left( 0.75 + \frac{4}{\sqrt{Y}} \right) dY \]<br /><br />Let's integrate term by term:<br />\[ C(Y) = \int 0.75 \, dY + \int \frac{4}{\sqrt{Y}} \, dY \]<br /><br />The first integral is straightforward:<br />\[ \int 0.75 \, dY = 0.75Y \]<br /><br />For the second integral, we use the power rule for integration:<br />\[ \int \frac{4}{\sqrt{Y}} \, dY = \int 4Y^{-1/2} \, dY = 4 \int Y^{-1/2} \, dY = 4 \cdot 2Y^{1/2} = 8\sqrt{Y} \]<br /><br />So, the consumption function is:<br />\[ C(Y) = 0.75Y + 8\sqrt{Y} + C_0 \]<br /><br />We are given that consumption is 20 when \( Y = 0 \):<br />\[ C(0) = 0.75(0) + 8\sqrt{0} + C_0 = 20 \]<br />\[ C_0 = 20 \]<br /><br />Thus, the consumption function is:<br />\[ C(Y) = 0.75Y + 8\sqrt{Y} + 20 \]<br /><br />Now, we need to find the consumption when \( Y = 25 \):<br />\[ C(25) = 0.75(25) + 8\sqrt{25} + 20 \]<br />\[ C(25) = 18.75 + 8(5) + 20 \]<br />\[ C(25) = 18.75 + 40 + 20 \]<br />\[ C(25) = 78.75 \]<br /><br />Therefore, the correct answer is:<br />d) 78.75