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Objective function of Lagrangian can be set up either with $+\lambda$ or $-\lambda$, depending on how you solve the budget constraint. Actually, for the solution it does not matter if $\lambda$ has negative or positive sign in the equation. You can clearly see it from the formula if you expand the second term: $$ \mathcal{L}(x,y, \lambda) = U(x,y) + \lambda(...


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$$u(x, y) = 2$$ is a simple example of a continuous utility function with thick indifference curve. \begin{eqnarray*} u(x, y)= \begin{cases} x & x < 1 \\ 1 & 1 \leq x < 2 \\ x - 1 & x \geq 2 \end{cases} \end{eqnarray*} is another continuous utility function with thick indifference curve for $u=1$.


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