This is a utility maximzation problemmaximize x a + y b subject to p 1 x + p 2 y = w (utility maximization problem)Anyone has any idea, there are no restrictions on a and b, as far as i can see it. many thanks!!!

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This is a utility maximzation problemmaximize       x    a    +      y    b   subject to       p    1    x  +      p    2    y  =  w (utility maximization problem)Anyone has any idea, there are no restrictions on   a and   b, as far as i can see it. many thanks!!!

Alice Harmon · Accepted Answer

When consumers maximize utility   u  (  x  ,  y  )  =      x    a    +      y    b   with respect to a budget constraint       p    1    x  +      p    2    y  =  w, the indifference curve is tangent to the budget line, therefore:  M  R      S          x      y        =      p    1        /        p    2  with   M  R      S          x      y        =            u      x              x      y       and                              u          x                                    =                              ∂            u            (            x            ,            y            )                                ∂            x                          =        a                  x                      a            −            1                                                        u          y                                    =                              ∂            u            (            x            ,            y            )                                ∂            y                          =        b                  y                      b            −            1                              that is            p      1              p      2        =            a              x                  a          −          1                            b              y                  b          −          1                    So the optimal             x      ^       and             y      ^       are solution of                    w                            =                  p          1                                      x            ^                          +                  p          2                                      (                                          p                2                                            p                1                                                    a              b                                                                        x                  ^                                                            a                −                1                                      )                                              1                              b                −                1                                                                        w                            =                  p          1                                      (                                          p                1                                            p                2                                                    b              a                                                                        y                  ^                                                            b                −                1                                      )                                              1                              a                −                1                                                    +                  p          2                                      y            ^

Olive Guzman · Accepted Answer

You could use Lagrange multipliers but I think that the problem could be easy if you extract &quot;y&quot; from the constraint and plug it in the objective function which becomes only dependent on &quot;x&quot;. Now, serach for the maximum.

This is a utility maximzation problem maximize <msup> x a </msup> + <msup>

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