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Consider the equation system $$ \left\{ \begin{array}{rcl} g(x,y) & = & 1, \\ g(\alpha x, \beta y) & = & 1, \end{array} \right. $$ where $\alpha,\beta > 0$, $a \neq \beta$ are parameters, function $g$ is concave and continuous: $$ g(\lambda x' + (1-\lambda) x'', \lambda y' + (1-\lambda) y'') \geq \lambda g(x',y') + (1-\lambda)g(x'',y'') \;\;\; \forall \lambda \in [0,1]$$ and positively homogeneous of order 1: $$g(\lambda x, \lambda y) = \lambda g(x,y) \; \forall \lambda > 0$$ Is it true that if this system have a solution then it have the unique solution? If it isn't true are there some handy conditions of such uniqueness?

Consider an example: $g(x,y) = ax+by$. The system $$ \left\{ \begin{array}{ccccl} ax &+ &by & = & 1, \\ (a\alpha) x & + & (b\beta) y &= &1, \end{array} \right. $$ have the unique solution if $\alpha \neq \beta$, $\alpha, \beta, a, b > 0$.

Another example of convenient function $g(x,y)$ is given by the so-called CES-function: $g(x,y) = c ( ax^{-\rho} + (1-a)y^{-\rho} )^{-\frac{1}{\rho}}$, where $c>0$, $\rho > 0$, $a \in (0,1)$, $x,y \geq 0$.

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What's an example of such a function? (Are you assuming that $g$ is continuous? Your using the word algebraic seems to indicate so, and that it is in fact a polynomial...) –  Mariano Suárez-Alvarez Apr 2 '12 at 18:12
    
For example, linear function $g(x,y) = ax + by$ or $g(x,y) = ( ax^{-\rho} + (1-a)y^{-\rho})^{-\frac{1}{\rho}}$, $\rho > 0$. –  Nimza Apr 2 '12 at 18:17
    
Ok. You might then remove the word algebraic from the title, as an «algebraic system of equations» usually means a polynomiak system of equations. Adding the examples to the question itself would also be helpful. –  Mariano Suárez-Alvarez Apr 2 '12 at 18:18
    
Your second example is only convex in each quadrant, no? –  Mariano Suárez-Alvarez Apr 2 '12 at 18:25
    
CES-function is considered only in first quadrant and it is concave when $\rho \geq -1$ and convex when $\rho \leq -1$. –  Nimza Apr 2 '12 at 18:44

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