Generally good qualitative points, although the implicit assumptions in your math are way too strong. In particular:
In the first period you have this ratio as 1, and in the second period it equals 2. So something has changed in our preferences between the periods. If you wanted to hold preferences constant, then given the price changes you pose, brass consumption should be 10x apple consumption not 5x.
This is not true in general; you’re assuming constant elasticity of substitution, which is a very strong assumption. In general, it’s entirely possible for the preferences/utility function to stay the same, but the elasticity of substitution to change as the amount of goods consumed changes (which is what I had in mind when writing the example).
This carries through to your example utility function. The Cobb-Douglas form you use implicitly assumes constant elasticity of substitution. Indeed, it is the only form of utility function (up to isomorphism) with constant elasticity of substitution; any other (non-equivalent) utility form whatsoever would not have that issue.
Fair enough—my point was not to come up with the exact quantitative growth rate, but to show some of the assumptions that the original post glossed over.
Generally good qualitative points, although the implicit assumptions in your math are way too strong. In particular:
This is not true in general; you’re assuming constant elasticity of substitution, which is a very strong assumption. In general, it’s entirely possible for the preferences/utility function to stay the same, but the elasticity of substitution to change as the amount of goods consumed changes (which is what I had in mind when writing the example).
This carries through to your example utility function. The Cobb-Douglas form you use implicitly assumes constant elasticity of substitution. Indeed, it is the only form of utility function (up to isomorphism) with constant elasticity of substitution; any other (non-equivalent) utility form whatsoever would not have that issue.
Fair enough—my point was not to come up with the exact quantitative growth rate, but to show some of the assumptions that the original post glossed over.