Solve for the Walrasian demand, Utility of three variables, and Convexity of Preferences?

Question

I am given $U(x,y,z) = x^\frac{2}{3}y^\frac{1}{3} + z$. I am asked to solve the following:

(i) Prove the convexity of these preferences (convex, strictly convex or neither?)
(ii) Solve for the Walrasian Demand?

For part 1, I calculated the determinant of the bordered hessian matrix and got $\frac{320}{81}*\frac{1}{x^\frac{2}{3}y^\frac{4}{3}}$. Here I concluded that if x and y are greater than zero (This was not given to me I assumed this), then determinant is greater than zero so $U$ must be quasi-concave and hence preferences are convex. Is this correct?

For part 2, I considered three cases.

Case 1: when $z = 0$ and $x,y > 0$. This was just the Walrasian for the standard Cobb Douglas we are left with.

Case 2: when $x$ or $y = 0$ and $z > 0$. All wealth is spent on z as well.

Case 3: $x,y,z > 0$. This I was unable to compute and did not know how to proceed.

Note: Budget constraint is standard $P_1X+P_2Y+P_3Z = W$

For Case 3: I set up Lagrangian and used Kuhn Tucker Conditions:

1. $\frac{2}{3}$*$(\frac{y}{x})^{\frac{1}{3}} - \lambda P_1 \le 0$
2. $\frac{1}{3}$*$(\frac{x}{y})^{\frac{2}{3}} - \lambda P_2 \le 0$
3. $1 - \lambda P_3 \le 0$
4. $x[\frac{2}{3}$*$(\frac{y}{x})^{\frac{1}{3}} - \lambda P_1] = 0$
5. $y[\frac{1}{3}$*$(\frac{x}{y})^{\frac{2}{3}} - \lambda P_2] = 0$
6. $z[1 - \lambda P_3] = 0$
7. $W - P_1X - P_2Y - P_3Z \ge 0 $
8. $\lambda [W - P_1X - P_2Y - P_3Z] = 0$

Imposing $x, y, z > 0$ and Walras's law, I know that I can equate 1, 2, 3, and 7 to zero. Essentially I end up with an equation which says that at optimal Marginal Utility to Price Ratio of each good is same.

After simplifying by equating $\lambda$, I get this from 1, 2:

$\frac{x}{P_2} = \frac{2y}{P_1}$.
My problem is that I can't solve for $z$ as I can't get my budget constraint all in terms of one variable.

Answer 1

For part (i), in complete rigor, you should also check the determinants of all the leading principal minors of the bordered Hessian and make sure that they have alternating signs. Your final conclusion looks correct though.

For part (ii), recall that Walrasian demand is the solution to utility maximization subject to budget constraint. So you should setup a Lagrangian, derive the Kuhn-Tucker conditions, and then solve for $x,y,z$ as functions of the prices and income. These will be the Walrasian demands.

Answer 2

$u(x,y,z)=x^{\frac{2}{3}}y^{\frac{1}{3}}+z$ is the sum of two concave functions: $u_1(x,y,z)=x^{\frac{2}{3}}y^{\frac{1}{3}}$ and $u_2(x,y,z)=z$, and therefore, it is concave. Since $u$ is concave, it is also quasi-concave. Preferences represented by $u$ are, therefore, convex. However, these preferences are not strictly convex. To show this, consider $(x_1,y_1,z_1)=(2,2,0)$ and $(x_2,y_2,z_2)=(0,0,2)$, and $\lambda = \frac{1}{2}$. Clearly, $u(\lambda (x_1,y_1,z_1) + (1-\lambda)(x_2,y_2,z_2))=u(1,1,1)=2=u(2,2,0)=u(0,0,2)=\min(u(x_1,y_1,z_1),u(x_2,y_2,z_2))$

Demand can be obtained by solving this problem: \begin{eqnarray*} \max_{x,y,z} & \ x^{\frac{2}{3}}y^{\frac{1}{3}}+z \\ \text{s.t. } & p_Xx+p_Yy+p_Zz\leq M \\ \text{and } & x\geq 0, \ y\geq 0, \ z \geq 0\end{eqnarray*} where $p_X>0, p_Y>0, p_Z>0, M\geq 0$

and we get the demand as

$(x^d,y^d,z^d)(p_X,p_Y,p_Z,M)\in\begin{cases} \left\{\left(\dfrac{2M}{3p_X},\dfrac{M}{3p_Y},0\right)\right\} &\text{if } \frac{3p_X^{\frac{2}{3}}p_Y^{\frac{1}{3}}}{2^{\frac{2}{3}}}<p_Z \\ \left\{\left(0,0,\dfrac{M}{p_Z}\right)\right\} & \text{if } \frac{3p_X^{\frac{2}{3}}p_Y^{\frac{1}{3}}}{2^{\frac{2}{3}}}>p_Z \\ \left\{\left(\dfrac{2\alpha M}{3p_X},\dfrac{\alpha M}{3p_Y},\dfrac{(1-\alpha)M}{p_Z}\right)|0\leq\alpha\leq 1\right\}&\text{if } \frac{3p_X^{\frac{2}{3}}p_Y^{\frac{1}{3}}}{2^{\frac{2}{3}}}=p_Z\end{cases}$

Note: Approach I have used to find the demand here is same as the one I have used here: https://economics.stackexchange.com/a/51562/11824