0
$\begingroup$

I'm trying to solve a utility maximization problem through the Lagrange method. The utility function is something like $u(x,y)=x+B(y-a)$.

However, I'm running into problems as x and y do not come up in the First Order Conditions. I was wondering if anyone had an intuitive explanation as for why, and how to get around it?

$\endgroup$
  • 5
    $\begingroup$ A suggestion: chose some arbitrary values for $B$ and $a$ and draw the indifference curves (i.e. find some bundles that all gives the consumer the same utility). Then draw a budget line and look where the optimal bundle is. This should help you see why the Lagrangian method isn't working. $\endgroup$ – Ubiquitous Apr 5 '17 at 7:05
  • 1
    $\begingroup$ Is B a function or a constant? What are the constraints? $\endgroup$ – Klas Lindbäck Apr 5 '17 at 11:58
  • $\begingroup$ When x and y don't appear in the differential then it is easy to find lambda. See this example: en.wikipedia.org/wiki/Lagrange_multiplier#Example_1 $\endgroup$ – Klas Lindbäck Apr 5 '17 at 12:01
  • $\begingroup$ If x or y don't appear in the differential of the utility function then the partial derivatives are constant which means u(x,y) forms a plane. $\endgroup$ – Klas Lindbäck Apr 5 '17 at 12:05
  • $\begingroup$ B is a constant! x>0, y>0, B>0, a>0 $\endgroup$ – Elizabeth Apr 6 '17 at 8:41
1
$\begingroup$

You needn't use the Lagrange here. And in general, this will be true for additively separable, linear utility functions.

Note that the marginal rate of substitution between any two goods $i,j$ is a utility function of the form $u(x_1,x_2,...,x_n)=a_1x_1+...+a_nx_n$ will be constant at $MRS_{i,j}=\frac{a_i}{a_j}$

A consumer will buy none of a good $x_i$ whenever $\frac{u_i}{p_i}$ is below the maximum such ratio of the $n$ goods. A consumer will exhaust her budget on whichever good has the maximum such ratio. And if all such ratios are equal, the consumer is indifferent between any consumption bundle that exhuats wealth $w$.

For your case of two goods, indifference curves are linear. For more than two goods, they are hyperplanes.

| improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ This assumes B is a constant and not a function. $\endgroup$ – 123 Apr 5 '17 at 21:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.