Lucy consumes only scoops of ice-cream (x) and cones (y). Moreover, she insists on consuming these two goods in the combination of 1 scoop of ice-cream and 1 cone. If there are more scoops of ice-cream than cones, she throws the extra ice-cream away. If there are more cones than scoops of ice-cream, she throws the extra cones away. Her utility function for ice-cream and cones is given by U(x, y) = min {x, y} . (The function min {x, y} returns the smaller number between x and y. In other words, if x < y, min {x, y} = x; if x > y, min {x, y} = y; and if x = y, min {x, y} = x = y.) (a) Draw a couple of indifference curves for Lucy. Put ice-cream on the horizontal axis. (Hint: Draw the 45◦ line. Choose a point on the line. If you increase the amount of x but not y from that point, how would Lucy’s utility change? If you increase the amount of y but not x from that point, how would Lucy’s utility change?) (b) Suppose each scoop of ice-cream costs $2, and each cone costs $1. Lucy has an income of $6. Draw her budget line in the diagram you have drawn in part (a). (c) Using the diagram you have drawn, find Lucy’s utility maximising consumption bundle. Lucy’s sister, Mary, also insists on a particular combination of ice-cream and cones. Unlike Lucy, however, Mary must have 2 scoops of ice-cream with every cone. Mary’s utility function is given by U(x, y) = min  x 2 , y . (d) In a separate diagram, draw a couple indifference curves for Mary. (Hint: Instead of drawing the 45◦ line, draw the line y = x/2. Then follow the procedures as in part (a).) (e) Mary faces the same prices as Lucy, but has an income of $10 instead. Using the diagram you have drawn in part (d), find Mary’s utility maximising consumption bundle

Lucy consumes only scoops of ice-cream (x) and cones (y). Moreover, she insists on consuming these two goods in the combination of α1 scoops of ice-cream and α2 cones, where α1 and α2 are real numbers greater than 1. If there are more scoops of ice-cream than cones, she throws the extra ice-cream away. If there are more cones than scoops of ice-cream, she throws the extra cones away. (a) (4 marks) Draw a couple of Lucy’s indifference curves when: (i) α1 = α2; (ii) α1 > α2; and (iii) α1 < α2. Put ice-cream on the horizontal axis. (b) (8 marks) Suppose each scoop of ice-cream costs α2, and each cone costs α1. Lucy has an income of (α1 2 + α2 2 ) dollars. Obtain Lucy’s utility maximising consumption bundle for general α1, α2 (i.e., without making any of the assumptions about α1 and α2 in (a)).

Lucy’s sister, Mary, also insists on a particular combination of ice-cream and cones. Unlike Lucy, however, Mary must consume these two goods in the com￾bination of α2 scoops of ice-cream and α1 cones. Mary is envious of Lucy and wants to achieve the same utility level that Lucy gets when she is consuming her optimal bundle in (b). What is the minimum income that Lucy’s and Mary’s parents must give to Mary so that she can achieve the utility level of Lucy at the current prices? Is this income greater or lower than the income of Lucy in (b)

Mary (consumer 1) and Lucy (consumer 2) are the only two consumers in the economy.Each of them consumes only two goods, fish (good x) and chips (good y), which they also own.Consumer 1’s utility function is given byU 1(x, y) = x2y3She has 25 units of x and 5 units of y. Consumer 2’s utility function is given byU 2(x, y) = ln x + ln y.She has 6 units of x and 20 units of y. Let p be the price of x and normalise the price of y to 1.(a) Draw the Edgeworth Box of this economy, marking clearly the endowment point. For eachconsumer, sketch the indifference curve passing through the endowment point.(b) Calculate the marginal rate of substitution for each consumer at the endowment point.(c) Who is going to sell x and buy y? Why?(d) Find the demand of consumer 1 for x—denote it as x1—and the demand of consumer 1for y—denote it as y1.(e) Find the demand of consumer 2 for x—denote it as x2—and the demand of consumer 2for y—denote it as y2.(f) Find the equilibrium price of good x.(g) Find the equilibrium consumption bundle of each consumer.(h) Mark your answer to part (g) in the Edgeworth Box you have drawn for part (a). Drawthe budget line under the equilibrium price. For each consumer, draw the indifferencecurve passing through the equilibrium consumption bundle.1

Suppose there are only two goods in the economy: Yo-yos and Marbles. Bob only cares about his consumption of Yo-yos; the more Yo-yos he has the happier he is, regardless of his consumption of Marbles. In contrast, Lucy only cares about his consumption of Marbles; the more Marbles she has the happier she is, regardless of her consumption of Yo-yos. (a) Draw Bob’s indifference curves, putting Yo-yos in the x-axis and Marbles in the y-axis. (b) Draw Lucy’s indifference curves, putting Yo-yos in the x-axis and Marbles in the y-axis. (c) Do Bob’s and Lucy’s preferences comply with the property of diminishing marginal rate of substitution? (d) Are Bob’s and Lucy’s preferences transitive?

Using the MU theory to derive the marginal utility for ice cream which is a normal good