The maximization problem is:Maximize u ( x 1 , x 2 ) = min [ a 1 x 1 , a 2 x 2 ] s.t. p 1 x 1 + p 2 x 2 ≤ w, in which x i , p i is the amount and price of good i, w is the total budget available.What I have been told to deal with this min[.,.] function is to solve it graphically. It's very easy to see on the graph that the maximization happens when a 1 x 1 = a 2 x 2 . But I wonder if there is a way to solve this algebraically? I'm stumped at the first step, which is to derive ∂ u ( x i ) ∂ x i .

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The maximization problem is:Maximize   u  (      x    1    ,      x    2    )  =  min  [      a    1        x    1    ,      a    2        x    2    ]       s.t.          p    1        x    1    +      p    2        x    2    ≤  w, in which       x    i    ,      p    i   is the amount and price of good   i,   w is the total budget available.What I have been told to deal with this min[.,.] function is to solve it graphically. It&#039;s very easy to see on the graph that the maximization happens when       a    1        x    1    =      a    2        x    2  . But I wonder if there is a way to solve this algebraically? I&#039;m stumped at the first step, which is to derive                     ∂        u        (                  x          i                )                    ∂                  x          i                    .

Ellie Meyers · Accepted Answer

Since the utility function has the Leontief form, then the two goods are perfect complements. Therefore the consumer will always choose the kink point where       a    1        x    1    =      a    2        x    2  , i.e. the maxima   (      x    1    ∗    ,      x    2    ∗    ) satises       a    1        x    1    ∗    =      a    2        x    2    ∗  .Also since the consumer will spend all his/her income, we can have two variables (x∗1,x∗2) and two equalities:      {                                        a            1                                x            1            ∗                                              =                      a            2                                x            2            ∗                                                            p            1                                x            1            ∗                    +                      p            2                                x            2            ∗                                              =          w                        Therefore by solving the two equations, we can nd the demand function for each good:      {                                        x            1            ∗                                              =                                                    a                2                            w                                                      a                2                                            p                1                            +                              a                1                                            p                2                                                                                      x            2            ∗                                              =                                                    a                1                            w                                                      a                2                                            p                1                            +                              a                1                                            p                2                                                            Note that we cannot equate the MRS with the slope of the budget line here, because the MRS is not deﬁned at the point where       a    1        x    1    =      a    2        x    2  .

The maximization problem is: Maximize u ( x 1 </msub> , x

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