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5. (30 points) A consumer has the utility function over goods r and y, u(x, y) = 12ry'. Let the price of good r be given by pr, let the price of good y be given by py, and let income be given by M. (a) What is the slope of the consumer's indifference curve at the consump- tion bundle. (1,1) ? (b) Derive the consumer's generalized demand function for goods r and y. (c) If we have px = 2, py = 2, and M = 24, compute the utility marimizing consumption bundle.Question 1) There are two goods, X and Y, with prices p, and py respectively. a) Show the Slutsky Equation for the own-price effect for good X. Interpret it (concisely). (b) Show the Slutsky Equation for the cross-price effect for good X. Interpet it (concisely). (c) Now assume that px falls from p, to pi. Using the Slutsky Equation show how X and Y can be gross complements and net substitutes simultaneously. Show this graphically, with X on the horizontal axis. (Question 2) A consumer has the following expenditure function: E(PrPyll) = Px * Py (Px + Py) (a) State concisely exactly what an expenditure function is. (b) Derive the compensated demand for good X. (c) Derive the ordinary demand for good Y. (d) Graph the relationship between a compensated demand and an ordinary demand curve for a normal good. (Question 3) Whitney is a huge fan of the Boston Celtics and has the following utility of wealth function: U(W) = In(W) where Wis Whitney's wealth. He starts with an initial wealth Wo. He can take an advance on his paycheck (assume at no cost) and use the money to bet on the Celtics. He thinks that the probability that the Celtics will win the next game is p. The bookie offers the following bet: you can bet $.x on the Celtics and if they win, you get your $x back, and if they lose you forfeit your $.x. (There is no cost to placing the bet.) Show how Whitney chooses the level of x to maximize his expected utility. (Question 4) A firm faces the constant-returns-to-scale and constant elasticity-of-substitution (CRS and CES) production function: q(K, L) = [KP + LP]UP) where p > 0 Show the following, making sure you show your derivation: (a) MPx = (# ) "+) and MP, = (4 ) ") at the optimal choices of L and K. (b) RTS = ( )(1-)Question 2: Use the 2 factor,-2 goods-2 countries Heckscher-Ohlin model. (i) Describe the production possibility frontier (PPF) for that model. Illustrate, why this model implies that the opportunity costs of a good increase if the production of that good increases. What assumptions are behind this conclusion? (ii) Illustrate the autarky and the free trade equilibrium for a small country that is capital abundant. State the conditions that characterize the competitive equilibrium (both for the autarky and the free trade equilibrium) and give an interpretation of these conditions. (iii) Why does this country gain from trade? FoodQuestion 3: Use in the standard trade model with two goods, cloth and food: (i) Describe the production possibility frontier (PPF) assumed in that model and explain why this model implies that the opportunity costs of a good increase if the production of that good increases. What assumptions are behind this conclusion? (ii) Illustrate the autarky and the free trade equilibrium for a country that has a comparative advantage in the production of cloth. State the conditions that characterize the competitive equilibrium (both for the autarky and the free trade equilibrium) and give an interpretation of these conditions. (iii) Why does this country gain from trade?Problem 4 (15 pts) You are given data on the following variables in an economy: Government spending 300 Planned investment 200 Net exports 50 Autonomous taxes 250 Income tax rate 0.1 Marginal propensity to consume 0.5 a) [2 pts] Consumption (C) is 600 when income (Y) is equal to 1500. Solve for autonomous consumption. b) [4 pts] Solve for the equilibrium level of output in the following two scenarios: i) there is an income tax t=0.1, ii) there is no income tax in the economy. Denote these two variables by Y"w and Y" wo respectively. c) [2 pts] In the economy with an income tax of 10%, what is the budget balance of the government? d) [3 pts] Solve for the change in net exports that would bring the equilibrium output level in the economy with the income tax to the level of Y"we that you found in part b. Specify both the magnitude of the change and whether it is an increase or a decrease. What would be the new level of net exports after this change?Problem 3 (10 pts) You have the following information on 3 countries: Soccerland, Handeggland and Neverland Soccerland Handeggland Neverland GDP in the beginning 5000 1000 2000 of 2013 Growth rate per year 5% 8% 10% a) [1 pt] How long will it take until Soccerland's GDP increases by 75%? b) [2 pts] How long will it take until Soccerland and Handeggland have the same GDP? c) [2 pts] Soccerland's population is not happy that, eventually, Handeggland is going to have higher GDP than their country. They feel that they are a much better country, so they are going to work harder to ensure that Handeggland will never catch up with Soccerland. If Soccerland new growth rate is constant every year, what is the minimum growth rate that ensures that Soccerland will always have a higher GDP than Handeggland? d) [5 pts] Neverland's ambition is to host the World Cup. At the beginning of 2013, its GDP was 2,000 and it was growing at 10% a year. Moreover, it will keep this pace until it hosts the World Cup. In order to host such a big event, Neverland's GDP must be at least 6,000. After the moment it reaches that GDP level, it will host the World Cup in the beginning of the next year available for the event (remember that the World Cup takes place every four years, in the years: 2014, 2018, 2022, etc.) During the year that Neverland hosts the World Cup, their GDP will grow at 75%. Then, everything returns to normal, and their GDP will keep growing at 10% a year forever onwards. i) (3 pts) When will Neverland's GDP reach 20,000