Question

Q1. John derives utility from consuming good X and Y.
His utility function is given by U(X,Y) = LnX + 10Y.
The price of good Y is 1 and the price of good X is Px and his income is M.
(a) Derive John’s demand for X and Y.
(b) Suppose the price of X fell from $2 to $1, calculate the substitution effect and
income effect for the change in X.

Need help in part b

Answer 1

$$Solution$$

$U(X,Y)=LnX+10Y\\ Derive\ the\ first\ order\ conditions\ for\ X\ and\ Y and\ then\ solve\ for\ Y.\\ MU_X=\frac{\Delta\ U }{\Delta\ X}=Ln\\ MU_Y=\frac{\Delta\ U}{\Delta\ Y}=10\\ Marginal\ rate\ of\ substitution(MRS)\\ =\frac{MU_Y}{MU_Y}=\frac{Ln}{10}\\ Ln=10P_x\\ M=P_XQ_X+Q_Y\\ =\frac{ln}{10}QX+Q_Y\\ Q_Y=\frac{Ln}{10}Q_X$