U(x, y) = x2 + 12 yand her income is I = $10. The price of y is py = $1 per unit.(a) Obtain the marginal utilities of x and y respectively. Compute the marginal rate of substitution.Are Amanda’s indifference curves hammock-shaped? Explain

A consumer has an income of $10,000 to spend on two goods, X and Y, where the price of good X is $500 per unit, and the price of good Y is $1 per unit. The marginal rate of substitution is given by the formula Y/X. How many units of good Y does this consumer optimally consume?

U(x, y) = (3x + 2y)2 .The price of x is px = $10 per unit and his income is $200.(a) Obtain the equation of Johnathan’s indifference curve for the utility level U = 100. Drawthis indifference curve. (2 marks)(b) The price of y is py = $8 per unit. Obtain the marginal rate of substitution (MRS) and theequation of the budget line. Using a graph, find Johnathan’s optimal consumption bundle.In this graph, show the budget line, the optimal bundle, and the corresponding indifferencecurve. Make sure to label carefully all the curves. (3 marks)(c) Suppose that the price of y drops to py = $6 per unit (the price of x remains the same,at px = $10 per unit, and the income remains the same). Obtain the equation of the newbudget line. Using a new graph, find Johnathan’s optimal bundle with this new price. Inthis graph, show Johnathan’s new budget line, new optimal bundle, and the correspondingindifference curve. Make sure to label carefully all the curves. (3 marks)(d) Now suppose that the price of y is py = $8 per unit if Johnathan buys less than 10 units ofthis product, and py = $6 per unit if he buys 10 units of y or more (as an example, 20 unitsof y would cost $120). Assume that the price of x remains the same, at px = $10 per unit.Derive the equation of the budget line and draw it in a separate graph. (3 marks)(e) Using a new graph, find the optimal bundle(s) for the problem in part (d). In this new graph,show the budget line, the optimal bundle and corresponding indifference curve. Make sureto label carefully all the curves. (3 marks)

U(x, y) = 2xy.(a) Obtain the marginal rate of substitution. Obtain the equation of the indifference curves forU = 16 and U = 36 (solving for y as a function of x). Draw these indifference curves. Dothese indifference curves intersect the y-axis and/or the x-axis? Explain in your own wordswhy this is the case. (4 marks)(b) Are Julian’s preferences homothetic? Justify mathematically your answer. (2 marks)(c) At the current prices (which Julian knows, but you don’t), initially, Julian spends 30 dollarsin x and 20 dollars in y. Suppose that Julian’s income increases by 5 dollars (and pricesremain the same). Compute Julian’s expenditure in x and Julian’s expenditure in y afterthis increase in income. (2 marks)(d) There is a decrease in the price of x. Is the (total) price effect on Julian’s demand for xlarger, the same, or smaller than the substitution effect? Explain. (1 mark)

Alice has utility from hours of TV and hours of movies given by U (t, m) =5t + 4m.(a) Obtain the equation of the indifference curve for the utility level U = 100 (set min the vertical axis).(b) Draw three indifference curves for Alice, and label the levels of utility that theycorrespond to (you can choose the utility levels that you want). Set m in thevertical axis and t in the horizontal axis.(c) What is Alice’s marginal rate of substitution M RS? Hint: Adopt the conventionof treating t as x and m as y.(d) If Alice’s income is $50, and the price of movies is $2 per hour and the price ofTV is $1 per hour, how much of each will she consume?(e) What if the price of movies is $1 per hour and TV is $2 per hour?(f) Obtain the equation of the indifference curve for the utility level U = 20 (set min the vertical axis). Is this indifference curve steeper than the indifference curveyou derived in part (a)? Explain.3

Marginal rate of substitution(MRS) is the rate at which consumer is willing to trade one good for another. It must be true that:Group of answer choicesMRS is the slope of an indifference curve in reference to a particular bundle of goods.MRS is not the same along an indifference curve that is of usual shape.MRS is same along an indifference curve that is of usual shape.Both a) and b).