Full Question 1: 1. Consider a consumer with the utility function u(x, x) = ln(x) + x and the budget constraint px + Px = m. Suppose good 1 represents coffee and good 2 represents everything else consumed. a. b. C. d. Derive the consumer's demand functions for x and x2 using constrained optimization. Show all working. [20 marks] What do the resultant demand functions suggest about the consumer optimal choices? [5 marks] Calculate the optimal demand for both goods if income, m = $30, price of coffee, p = $2, and price of everything else, p2 = $6. [2 marks] Consider a case where a quantity tax on coffee is introduced at t = $1. Explain how this affects the consumer's optimal choice. [5 marks]