The utility maximation problem answers how to allocate the total income between salad (X1) and candles (X2) to maximise total utility. The function is U(X1, X2) = ln(X1) + X2 The prices of salad P1 and candles P2, and Gwyneth's income in m, we want to maximise U(X1, X2) = ln(X1) + X2, subject to the constraints, the budget constraint, P1 × X1 + P2 x X2 = m First, we derive the lagrangian, L(1) L(X1, X2,1) = In(X1) + λ(m − P1X1 - P2X2) Now we differentiate L with respect to X1,X2 and 1 and set to 0 JL 1 - P1λ = 0 ǝx1 X1 ᎧᏞ = 1 P2λ = 0 ax2 JL = m P1X1 - P2X2 = 0 αλ Simplifying gives us, 1 X1 = (1) Ρ1λ 1 = λP2 Now we solve the system of equations for X1, X2 and λ, to obtain the demand functions for salad X1 and candles X2 in terms of prices (P1, P2) and income (m) We can substitute 1 from (2) into (1) for X1 in terms of P1 and P2 to get X1 = 27 P2 P1 Then, Form - P1X1 - P2X2 = 0, we have m = P1 + P2X2 m = P2+ P2X2 m = P2(1 + X2) m m = 1 + x2 hence, we get X2 = - 1 P2 P2 (a) Gwyneth is conducting research into consumer behaviour. She has decided to use the utility function u(x1,x2) = Inx₁ + x2, where x₁ is her consumption of salad and x2 is her consumption of candles. Let prices of x1 and X2 be denoted by p₁ and p2. Gwyneth earns an income of m. Set up the utility maximisation problem. Use a Lagrangian to derive the demands for salad and candles. Illustrate the income offer curve and engel curves for salad. (b) Do you think the utility function in (a) is sensible? Explain.