Suppose that preferences can be represented by the following utility function U(x, y) = xyb A...

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Suppose that preferences can be represented by the following utility function U(x, y) = xªyb A consumer maximises utility subject to a budget constraint M = Pxx +Pyy Where p, is the price of good x, p, is the price of good y and M is the budget available. a. Derive an expression for the marginal utility of x. Under what condition is the marginal utility diminishing. (5 marks) b. Derive an expression for the marginal utility of y. Under what condition is the marginal utility diminishing. (5 marks) (5 marks) c. Define non-satiation. Is this satisfied in the given utility function? d. Using your answers for parts a and b, find the slope of an indifference curve. (5 marks) e. Solve the budget constraint for x and use this to eliminate x in the utility function. Why are we able to make this substitution? (5 marks) f. Take the natural log of the function that you derived in part e and differentiate with respect to y. Solve this derivative for the optimal demand for good y and using this solve for the optimal demand for good x. Why are we able to take the natural logarithm? (5 marks) g. Take the second derivative with respect to y and check that the solution is a maximum. (5 marks) h. Fill in the following table and create a series of graphs showing the relationship between income (x axis) and utility (y axis). Comment on your findings. (15 marks) Suppose that preferences can be represented by the following utility function U(x, y) = xªyb A consumer maximises utility subject to a budget constraint M = Pxx +Pyy Where p, is the price of good x, p, is the price of good y and M is the budget available. a. Derive an expression for the marginal utility of x. Under what condition is the marginal utility diminishing. (5 marks) b. Derive an expression for the marginal utility of y. Under what condition is the marginal utility diminishing. (5 marks) (5 marks) c. Define non-satiation. Is this satisfied in the given utility function? d. Using your answers for parts a and b, find the slope of an indifference curve. (5 marks) e. Solve the budget constraint for x and use this to eliminate x in the utility function. Why are we able to make this substitution? (5 marks) f. Take the natural log of the function that you derived in part e and differentiate with respect to y. Solve this derivative for the optimal demand for good y and using this solve for the optimal demand for good x. Why are we able to take the natural logarithm? (5 marks) g. Take the second derivative with respect to y and check that the solution is a maximum. (5 marks) h. Fill in the following table and create a series of graphs showing the relationship between income (x axis) and utility (y axis). Comment on your findings. (15 marks)

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a The marginal utility of x is the partial derivative of the utility function with respect to x MUx Ux axa1yb The marginal utility of x is diminishing if a 1 meaning that the additional utility obtain... View the full answer