# Troubleshooting Utility Maximization with the Lagrange Method

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?

• 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. – Ubiquitous Apr 5 '17 at 7:05
• Is B a function or a constant? What are the constraints? – Klas Lindbäck Apr 5 '17 at 11:58
• 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 – Klas Lindbäck Apr 5 '17 at 12:01
• 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. – Klas Lindbäck Apr 5 '17 at 12:05
• B is a constant! x>0, y>0, B>0, a>0 – Elizabeth Apr 6 '17 at 8:41

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 $w$.