A consumer's demands x, y for two different goods are chosen to maximize the utility function U(x,y)=x+y(x0,y0) subject to the budget constraint px+qy=m (where p,q,m>0 ). (a) Show that the utility function is concave. (4 marks) (b) Write out the Lagrangian for the constrained maximization problem. (1 mark) (c) Write out the first-order conditions for a constrained maximum at (x,y). (2 marks) (d) Find the utility-maximizing demands for both goods, as well as the Lagrange multiplier , all as functions of the three variables (p,q,m). Simplify all expressions as much as possible. (6 marks) (e) Find the maximized utility U(p,q,m). Simplify the expression as much as possible. (4 marks) (f) Show that mU is the Lagrangian multiplier . (2 marks) A consumer's demands x, y for two different goods are chosen to maximize the utility function U(x,y)=x+y(x0,y0) subject to the budget constraint px+qy=m (where p,q,m>0 ). (a) Show that the utility function is concave. (4 marks) (b) Write out the Lagrangian for the constrained maximization problem. (1 mark) (c) Write out the first-order conditions for a constrained maximum at (x,y). (2 marks) (d) Find the utility-maximizing demands for both goods, as well as the Lagrange multiplier , all as functions of the three variables (p,q,m). Simplify all expressions as much as possible. (6 marks) (e) Find the maximized utility U(p,q,m). Simplify the expression as much as possible. (4 marks) (f) Show that mU is the Lagrangian multiplier . (2 marks)