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

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.

• In the part 2 I think you may need to look in cases when the price is different, because the good $z$ is like a substitute of the combination of $x & y$ so if the price of $z$ is much higher than the prices for the combination $x & y$ the consumer will consume only $x & y$ solving the demand as a standard coub douglas, while if the price of $z$ is much lower than the prices for the combination $x & y$ the consumer will consume only $z$. Nov 16 '21 at 18:17

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.
• @Kinno: You can usually recover $z$ from the budget constraint after solving for $x$ and $y$. You should probably start from your condition $3$, and use the fact/assumption that $p_1,p_2,p_3>0$ to deduce that $\lambda>0$, which in turn will give you strict equality in condition $7$, and go from there. Nov 17 '21 at 13:39
• @Kinno: Also, the MU-to-Price Ratio gives you a relation between $x$ and $y$, e.g. $x=2yp_2/p_1$. Plug this into either $1$ or $2$ for $x$, and you'll get $y$ as an expression of prices and $\lambda$. Nov 17 '21 at 13:48