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

Hailee Stout

Hailee Stout

Answered question

2022-05-10

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 .

Answer & Explanation

Ellie Meyers

Ellie Meyers

Beginner2022-05-11Added 15 answers

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 .

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