u(x₁, x₂) = ₁ + 122. Suppose that, because of a shortage of good 1, the government imposes a strict upper limit of ₁ on the quantity of good 1 that the consumer can consume. Assume throughout the following that w>p2. (a) Show that the consumer's preferences are strictly convex. Solution: Along any indifference curve x₁ + x₁x₂ = k, we have d²x₂ 1+x₂ = 2 da² > 0.
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- (c) If the preferences are concave, will the consumer ever consume both of the goods together?2. Sally consumes two goods, X and Y. Her utility function is given by the expression U = 3XY². The current market price for X is £10, while the market price for Y is £5. Sally's current in come is £500. (a) Sketch a set of two indifference curves for Sally in her consumption of X and Y. (b) Write the expression for Sally's budget constraint. Graph the budget constraint and determine its slope. (c) Determine the X,Y combination which maximises Sally's utility, given her budget constraint. (d) Calculate the impact on Sally's optimum market basket of an increase in the price of X to $15. What will happen to her utility as a result of the price increase?Let the following table represents the total utility of a given consumer, in the cardinal utility approach A) Calculate the MUx and MUy and fill the table in the 4th and 5th rows. B) If the two products (X&Y) are free goods how many of X and Y should the cons consumer take to maximize utility? C) What is the maximum utility of X and Y if they are free?. D) Assuming the consumer has any amount of money (enough budget) how many of X and Y should the consumer buy, to maximize utility? E) What is the total utility of X and Y? F) Let now price of X is 4 birr per unit and price of Y is 2 birr per unit and budget of the consumer for consumption of X and Y is 20 birr. Given budget constraint how many of X and Y should the consumer buy to maximize utility? G) What are the total utility of X and Y
- 4. The CES utility function is given by x°+y° U (x, y) (i). Show that the first-order conditions for a constrained utility maximum with this function require individuals to choose goods in the proportion = Pa 8–1 Py (ii). Show that the result in part (i) implies that individuals will allocate their funds equally between x and y for the case 8 = 0. (iii). How does the ratio depend on the PyY value of 8? Explain your results intuitively. (iv). Derive the indirect utility and expenditure functions for this case and check your results by describing the homogeneity properties of the functions you calculated.A consumer has the following utility function: Ulx, y) = xy -y, *21 where x and y represents the quantities consumed of goods X and Y. y 20 What will be the substitution and income effects for X and Yassuming that the consumer attempts to maintain the same level of utility achieved before price of Y increased (that is, when price of Y was $1)? SEx= +0.5 IEx = -0.5 SE, = -0.25 IE- = -0.25 SEx= +0.293 IE = -0.293 SEy = -0.414 IE, = +0.414 SEr= +0.25 IE SE, = -0.75 IE, = -0.75 = -0.25 SEx= +0.414 IEx = -0.414 SEy = -0.293 IE, = -0.207 Income = $3 Px= $1, Py= $24. Assume the prices of r and y are 1 and income is 10. For the following two utility functions, find the quantities of r and y consumed at the con- sumer's optimum. (Here, it may be helpful to solve by drawing out some indifference curves.) (a) U(r, y) = Min(x, 2y) (b) U(x, y) = 2x + y (c) Suppose the price of a rises to 3. What are the new optima in each case. What are the substitution effects?
- a good is normal, then an increase in the price of the good will lead to which of the following to be true for this good? (Assume that there are only two goods, the individual's preferences lead to well-behaved preferences with strictly convex indifference curves and an interior solution for all budgets). Let SE = substitution effect, IE = income effect) (a) The magnitude of the IE for this good must be larger than the magnitude of the SE (b) The magnitude of the SE for this good must be larger than the magnitude of the IE (c) The good could be a Giffen good d) The good must be an ordinary good ( (e) None of the aboveSuppose that the consumer maximizes his utilitysuch that the marginal utilities of goods X and Y are; MUx=80-16x MUy=60-2y Let Px=P4,Py=P2 and I=P80 Find the utility maximizing quantities X and Y2. Sally consumes two goods, X and Y. Her utility function is given by the expression U = 3XY². The current market price for X is £10, while the market price for Y is £5. Sally's current income is £500. (a) Sketch a set of two indifference curves for Sally in her consumption of X and Y. (b) Write the expression for Sally's budget constraint. Graph the budget constraint and determine its slope. (c) Determine the X, Y combination which maximises Sally's utility, given her budget constraint. (d) Calculate the impact on Sally's optimum market basket of an increase in the price of X to $15. What will happen to her utility as a result of the price increase?
- 1. Is it impossible for two indifference curves to intersect one another? why? 2. What does it mean that preferences are complete? Now would the real-life implica- tions of this assumption look like?Suppose a consumer has a monthly income of m = 100 which she spendson two commodities: french fries (x1) and beef jerky (x2). The price offrench fries is p1 = 2 and the price of beef jerky is p2 = 5.(a) Write down the consumer’s budget constraint (equation).(b) What is the maximal consumption of french fries (this is calledthe real income in french fries).(c) Find the maximal consumption of beef jerky (real income interms of beef jerky).2. What does it mean that preferences are complete? Now would the real-life implica- tions of this assumption look like?