Consider a consumer who uses his or her income Y to purchase ca apples at the price of pa per apple and cb bananas at the price of pb per banana, subject to the budget constraint Y ≥ paca + pbcb. Suppose that the consumer’s preferences over apples and bananas are described by the utility function α ln(ca) + (1 − α) ln(cb) where ln(c) denotes the natural logarithm of c and α, satisfying 0 < α < 1, determines how much the consumer likes apples relative to bananas. Note that the choice of the function u(c) = ln(c) to help describe how much utility the consumer gets from each good implies that u’ (c) = 1/c, an expression that will help in solving for consumer’s optimal choices. Set up the Lagrangian for this constrained optimization problem: choose ca and cb to maximize the utility function subject to the budget constraint. Then, using the first-order conditions together with the budget constraint, see if you can obtain equations that show how the consumer’s optimal choices c∗ a and c∗ b depend on his or her income Y as well as the prices pa and pb and the parameter α from the utility function. Note that to do this, you will also have to find an equation that shows how the value λ∗ of the Lagrange multiplier associated with the solution to the consumer’s problem depends on Y and/or pa, pb, and α. Using these equations, do you notice any relationship between the amounts pac∗ a and pbc∗ b that the consumer optimally spends on apples and bananas and the the parameter α from the utility function?
