To this end, there are at least two methods:
The most mechanical way is to construct Lagrange function to solve it, and the speed is not slow if you are skilled. It's just a little more process. For example, with budget lines as constraints,
① construct laplace function, l =1/3 * lnq1+2/3 * lnq2+λ (m-p1q/-p2q2),
(2) Then find the partial derivative and eliminate λ,
③ Q2P2/Q 1P 1=2 can be obtained.
④ Then through the budget line, Marshall demand function can be obtained, namely: Q 1=m/(3*P 1), Q2=(2*m)/(3*P2).
There is another ingenious way. This utility function is actually the [linear affine] of the C-D production function, so some parameters obtained by this utility function, such as various elasticity and various optimization functions, are in the same form. Knowing this, we can recall that the utility function in C-D form has several very common properties, one of which is the ratio of its power to the expenditure of two commodities, that is, (1/3)/(2/3) = p1q1/p2q2, which directly leads to the method 650.
# If you are familiar with the properties, you can also do this: Continue (Method 2) and recall the C-D utility function, such as U = A * Q 1 (A 1) * Q2 (A2)? The form of (Marshall) demand curve is Qi=ai*m/Pi, so the result can also be obtained directly.
The second question is simple. If you directly substitute the result, you get Q 1=Q2= 100.