# Does duality hold for u(x, y) = x^2 + y^2? (Corner solution)

I've been told that "if preferences are strongly monotonic, duality holds."

In the case of utility u(x,y) = x^2 + y^2, we will get a corner solution depending on the relative price of x and y. Suppose p_x > p_y => x* = 0.

Then MUx = 0 at x*, so preferences are not strongly monotonic around the optimum. Hence, duality may not hold (Hicksian demand ≠ Marshallian demand). After plugging in the indirect utility function into Hicksian demand, I found that indeed duality does not hold here.

Does duality hold? If yes, please show why mathematically.

• This utility function is strongly monotonic. Commented Apr 26 at 15:44
• Not globally though? Marginal utility is equal to zero for x or y equal to zero, which is exactly the interesting case? Although this may not be the formal definition of strong monotonicity, I feel like it interferes with duality here. Do you think duality holds? Commented Apr 26 at 15:54
• You could actually provide your calculations. Your utility function is strictly increasing everywhere. A strictly positive derivative everywhere is a sufficient but not necessary condition for a function to be strictly increasing. Commented Apr 26 at 16:26

The Marshallian demand correspondence is given by: $$x(p_x, p_y,m) = \begin{cases} \{m/p_x\} &\text{ if } p_x < p_y\\ \{0, m/p_x\} &\text{ if } p_x = p_y\\ \{0\} &\text{ if } p_x > p_y \end{cases}.$$

The indirect utility function is given by: $$v(p_x, p_y, m) = \left(\frac{m}{\min\{p_x, p_y\}}\right)^2$$

Now consider the cost minimization problem: $$e(p_x, p_y, \bar u) = \min p_x x + p_y y \text{ s.t. } x^2 + y^2 \ge \bar u.$$ If $$p_x < p_y$$ it will be best to set $$x = \sqrt{\bar u}$$ and if $$p_y < p_x$$ it will be best to set $$y = \sqrt{\bar u}$$. So we get that the Hicksian demand correspondence is given by: $$h_x(p_x, p_y, \bar u) = \begin{cases}\{\sqrt{\bar u}\} & \text{ if } p_x < p_y,\\ \{0, \sqrt{\bar u}\} \text{ if } p_x = p_y,\\ \{0\} &\text{ if } p_x > p_y\end{cases}.$$ The expenditure function is given by: $$e(p_x, p_y, \bar u) = \min\{p_x, p_y\}\sqrt{\bar u}.$$

Now, if $$p_x < p_y$$ then: $$h_x(p_x, p_y, v(p_x, p_y, m)) = \left\{\sqrt{v(p_x, p_y, m)}\right\} = \left\{\frac{m}{\min\{p_x, p_y\}}\right\} = \left\{\frac{m}{p_x}\right\} = x(p_x, p_y, m).$$ If $$p_x > p_y$$ then: $$h_x(p_x, p_y, v(p_x, p_y, m)) = \{0\} = x(p_x, p_y, m)$$ If $$p_x = p_y$$ we have: $$h_x(p_x, p_y, v(p_x,p_y,m)) = \left\{0, \sqrt{v(p_x, p_y, m)}\right\} = \left\{0, \frac{m}{p_x}\right\} = x(p_x, p_y, m).$$ This shows that strong duality does hold in this case.

The conceptual difficulty for this example is that if $$p_x = p_y$$, then demand is not single-valued. As such, we have to work with demand correspondences rather than demand functions.

• Wow thank you for this thorough answer! I completely missed that method for deriving Hicksian demand! Commented Apr 27 at 12:58

I'm posting here, because I don't have enough reputation points to just comment. So, if I understand this correctly, monotonic means that preferences, do not change, right? Like, I'm familiar with the concept of monotonic transformation, but what do you mean by duality? Also, is there a scenario, where preferences are not monotonic?

I don't mean to be rude. I genuinely want to know. Also, pls don't feel like you need to write long paragraphs to explain it (although that would be great), if you could recommend a video or an article I would really appreciate it too.