A consumer has a Cobb-Douglas utility function: $ U(x_1,x_2)= x_1^\alpha x_2^\beta$, where $x_1,x_2$ denote the consumption of the two goods and $\alpha,\beta$ are positive constants. The prices of the goods are $p_1,p_2$ and the consumer's income is $m$.
So the question asks to express the consumer's problem as a constrained maximisation problem and then to find the first-order conditions and find the demand functions.
So I have to maximise $x_1^\alpha x_2^\beta$ subject to $p_1x_1+p_2x_2=m$.
This leads to first-order conditions as $\alpha x_1^{\alpha-1} x_2^\beta = \lambda p_1$ and $\beta x_1^\alpha x_2^{\beta - 1} = \lambda p_2$. No problems here.
The answer for the demand functions are: $x_1 = \frac{\alpha m}{(\alpha + \beta)p_1} , x_2 = \frac{\beta m}{(\alpha + \beta)p_2}$
Further to this I am asked to give an economic interpretation of the parameters $\alpha$ and $\beta$, I was slightly lost here too.
Could anyone shed some light on how to arrive at the demand functions? I found myself doing lots of algebraic calculations which didn't seem to get me anywhere.
Thanks