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.
B
a function or a constant? What are the constraints? $\endgroup$ – Klas Lindbäck Apr 5 '17 at 11:58x
andy
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 12:01x
ory
don't appear in the differential of the utility function then the partial derivatives are constant which meansu(x,y)
forms a plane. $\endgroup$ – Klas Lindbäck Apr 5 '17 at 12:05