Suppose that a consumer has a Cobb-Douglas utility function:
U(a,b) = a^? * b^(1-?)
where a and b are consumption of environmental goods and consumer goods, respectively (assume that 0 < ? < 1). Suppose further that the prices of the two goods are pa and pb, and that total income is M.
(i) Form the lagrangian and derive the first-order conditions.
(ii) Solve for the optimal demands for a and b as a function of ?, pa, pb, and M. Explain intuitively the effect of these four parameters on the consumer's optimal choices.
(iii) Plot the optimal demands for a and b as of pb. Explain why the functions are shaped as they are. How does the share of income spent on each good change with pb? If consumers' preferences defined by a Cobb-Douglas utility function, what does your result suggest about their expenditure patterns?