How can I derive Hicksian demand, when from the FOC I only get $\frac{p_x}{p_y} = \frac13$.

The problem is to minimize

$$p_x \cdot x + p_y \cdot y \qquad\text{s.t.}\qquad x + 3y > U$$

If I plug the price relation into the budget constraint $I =p_x \cdot x + p_y \cdot y$ I get the income in the demand function, but then this is Marshallian demand. Plugging the relation in expenditure function, obtained from the indirect utility function also doesn't lead to the Hicksian demand (that I obtained via Shephard's lemma and equals $h_x = U + x + 3y$).

How can I derive Hicksian demand, when from the FOC I only get $\frac{p_x}{p_y} = \frac13$ without the usual x & y. So they cannot be derived directly from FOC, but if I plug the price relation into the budget constraint $I =p_x \cdot x + p_y \cdot y$ I get the income in the demand function, so this is Marshallian demand. Plugging the relation in expenditure function, obtained from the indirect utility function also doesn't lead to the Hicksian demand (that I obtained via Shephard's lemma and equals $h_x = U + x + 3y$).

The problem is to minimize

$$p_x \cdot x + p_y \cdot y \qquad\text{s.t.}\qquad x + 3y ≥ U$$

How can I derive Hicksian demand, when I get from FOC only px/py=1/3.

The problem is to minimize pxx + pyy s.t. x+3y>U

If I plug the price relation into the budget constraint I=pxx+pyy I get the income in the demand function, but then this is Marshallian demand. Plugging the relation in expenditure function, obtained from the indirect utility function also doesn't lead to the Hicksian demand(, that I obtained via Shephard's lemma and equals hx=U+x+3y).

How can I derive Hicksian demand, when from the FOC I only get $\frac{p_x}{p_y} = \frac13$.

The problem is to minimize

$$p_x \cdot x + p_y \cdot y \qquad\text{s.t.}\qquad x + 3y > U$$

If I plug the price relation into the budget constraint $I =p_x \cdot x + p_y \cdot y$ I get the income in the demand function, but then this is Marshallian demand. Plugging the relation in expenditure function, obtained from the indirect utility function also doesn't lead to the Hicksian demand  (that I obtained via Shephard's lemma and equals $h_x = U + x + 3y$).