1. Problem: Investigate the convexity of indifference curves and the behavior of the Marginal Rate of Substitution (MRS) for each utility function. A typical indifference curve is graphed as $U(x,y) = c$ for constant $c$. 2. For (a) $U(x,y) = 3x + y$: - Indifference curves: $3x + y = c \implies y = c - 3x$, which are straight lines. - MRS is $-\frac{\partial U/\partial x}{\partial U/\partial y} = -\frac{3}{1} = -3$, constant. - Since MRS is constant, the curve is linear and not convex. 3. For (b) $U(x,y) = \sqrt{x} \cdot y$: - Set $\sqrt{x} y = c \implies y = \frac{c}{\sqrt{x}} = c x^{-\frac{1}{2}}$. - MRS: $-\frac{\partial U/\partial x}{\partial U/\partial y} = -\frac{\frac{1}{2\sqrt{x}} y}{\sqrt{x}} = -\frac{y}{2x}$. - Substitute $y = \frac{c}{\sqrt{x}}$ gives $MRS = -\frac{c}{2 x^{3/2}}$, which decreases as $x$ increases. - Hence the curve is convex. 4. For (c) $U(x,y) = \sqrt{x} + y$: - Indifference curves: $\sqrt{x} + y = c \implies y = c - \sqrt{x}$. - MRS: $-\frac{\frac{1}{2 \sqrt{x}}}{1} = -\frac{1}{2 \sqrt{x}}$, which increases (in absolute value decreases) as $x$ increases. - The MRS declines as $x$ increases; curve is convex. 5. For (d) $U(x,y) = \sqrt{x^2 - y^2}$: - Indifference curves: $\sqrt{x^2 - y^2} = c \implies x^2 - y^2 = c^2 \implies y = \pm \sqrt{x^2 - c^2}$, hyperbolas. - MRS: $-\frac{\partial U/\partial x}{\partial U/\partial y} = -\frac{\frac{x}{\sqrt{x^2-y^2}}}{-\frac{y}{\sqrt{x^2-y^2}}} = \frac{x}{y}$. - As $x$ increases, $MRS$ increases if $y$ held constant, but on the curve $y$ changes with $x$. - The indifference curves are not strictly convex; they are hyperbolas and do not have declining MRS. 6. For (e) $U(x,y) = \frac{xy}{x + y}$: - Set $\frac{xy}{x + y} = c \implies xy = c(x + y) \implies xy - cx - cy = 0 \implies y(x - c) = c x \implies y = \frac{c x}{x - c}$ for $x \neq c$. - MRS computed from partials: $\partial U/\partial x = \frac{y(x+y) - xy}{(x+y)^2} = \frac{y^2}{(x+y)^2}$, $\partial U/\partial y = \frac{x(x+y) - xy}{(x+y)^2} = \frac{x^2}{(x+y)^2}$, - Hence $MRS = -\frac{y^2}{x^2} = -\left(\frac{y}{x}\right)^2$. - Along indifference curve, substituting $y = \frac{c x}{x - c}$, we see MRS changes nonlinearly. - MRS declines as $x$ increases (detailed derivative analysis confirms convexity). All functions except (a) and (d) have convex indifference curves with declining MRS.