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I am not sure I understand your question. $\lambda$ is a Lagrange multiplicator which has a value in the optimum. It is not a parameter and hence you cannot leave it in your solution. When solving the Lagrangian the optimal solution has the form $(x,\lambda)$ and in addition to your conditions $$x_1 = \frac{\alpha}{\lambda p_1} + b_1$$ $$x_2 = \frac{\beta}{\... 2 I am not sure whether the question asks about the function being convex, or simply the demand correspondence/function being a convex set. The latter is often asked and if that is the case here is an answer. This is a general answer to your question which may be more useful to you and future readers than a specific one. The following theorem holds in ... 1 For every x_i separately you get$$\frac {\gamma_i}{x_i-a_i} = \lambda p_i \tag{1}$$Re-arranging,$$\frac {\gamma_i}{\lambda} = p_ix_i - p_ia_i \tag{2}$$Sum over i$$\frac 1{\lambda}\sum\gamma_i = \sum p_ix_i - \sum p_ia_i \tag{3}$$Re-arrange to solve for the optimal \lambda taking into account the restrictions$$\sum p_ix_i = m ,\;\;\; \sum\...