eco204_summer_2009_HW_2

eco204_summer_2009_HW_2 - University of Toronto, Department...

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Unformatted text preview: University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain ECO 204 Summer 2009 S. Ajaz Hussain HW 2 Please help improve the course by sending me an email about typos or suggestions for improvements Question 1 Suppose an individual's indifference curves are "thick: What kind of preferences does this individual not have? Question 2 In this question, you will apply microeconomics to a macroeconomics setting. From ECO 100, recall that when money supply increases by (say) x%, then: In a classical economy, all pecuniary variables (prices and income) also increase by x%. You might have seen the equation for the quantity theory of money: MV = PY where M is money, V is velocity of money, P is the overall price level and Y is GDP. Thus, with V and Y constant, as M increases, P must also increase by the same percentage. In a Keynesian economy, some pecuniary variables are "sticky". For example, prices will increase by x% but income written in contracts which take time to change may not. 1 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain The question is, how will demand change in a classical vs. Keynesian setting. To explore this, consider two consumers Ajax and Barbara. Ajax has the utility function U = Q1 Q2 and Barbara has the utility function U = min(Q1 , Q2). (a) Solve for Ajax's optimal choices. (b) Solve for Barbara's optimal choices. (c) Suppose Ajax and Barbara live in a Classical (Macro)economy. If the money supply increases by 10% and therefore all pecuniary variables also increase by 10%, how will this impact their demand for goods 1 and 2? (d) Suppose Ajax and Barbara live in a Keynesian (Macro)economy. If the money supply increases by 10% and all prices (but not income) also increase by 10%, how will this impact their demand for goods 1 and 2? Question 3 In this question you will practice the envelope theorem in a UMP. Consider a consumer with utility function: U = Q1 Q2. (a) What is the impact of ceteris paribus higher prices and income on optimal utility? For example, if ceteris paribus, P1 , will the consumer be "happier" or "sadder"? Hint: apply the envelope theorem on the Lagrangian of the UMP. (b) What is the impact of ceteris paribus higher and on optimal utility? For example, if ceteris paribus, , will the consumer be "happier" or "sadder"? Hint: apply the envelope theorem on the Lagrangian of the UMP. Hint: apply a monotonic transformation to the utility function. Question 4 In this question you will investigate the complements utility function and its properties. G Indart has utility function U = min(2Q1 , 4Q2) (a) What are and in U = min(Q1, Q2)? (b) Suppose GIndart purchases goods 1 and 2 for (uniform) prices P1 and P2. Assume GIndart is a price taker. Derive an expression for GIndart's demand of and expenditure on goods 1. (c) What is GIndart's price elasticity of good 1? (d) What is GIndart's income elasticity of good 1? 2 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain Question 5 Suppose your company produces good 1 (Q1). Suppose all your customers have the utility function U = min(Q1, Q2). One of your managers wants to raise sales of good 1: he proposes an ad campaign designed to raise the parameter . (a) What does it mean to raise for an advertising campaign? (b) Derive the demand function for good 1. Does raising raise sales of good 1? Question 6 You've seen the CobbDouglas, complements and perfect substitutes utility functions repeatedly. For these cases, you should look at whether demands for good 1 and 2 depend on prices and income. For your convenience here is a summary: If U = Q1 Q2 then Q1 = {/( + )} Y/P1 and Q2 = {/( + )} Y/P2. Observe how demands depend on Y and own but not cross price. Moreover demands are non linear. If U = min(Q1, Q2) then Q1 = Y/(P1 + P2) and Q2 = Y/(P1 + P2). Observe how demand depend on Y and all prices. Moreover demands are nonlinear. If U = Q1 + Q2 then if P1 > P2, Q1 = 0 and Q2 = Y/P2 while if P1 < P2, Q1 = Y/P1 and Q2 = 0 and if P1 = P2, Q1 = [0, Y/P1 ] and Q2 = [0, Y/P2 ]. Moreover demands are nonlinear. Now you'll see a utility function widely used in public economics in which demand for one good if it is consumed is independent of income. This is the quasi linear utility function and takes the form: U = f(Q1) + Q2 Where f(x) is some function. Suppose f(x) = log x. Derive the optimal demands for goods 1 and 2. Question 6 In ECO 100 you've always seen a linear demand curve; in ECO 204, you've only thus far seen nonlinear demand curves. This begs the obvious question: are there any preferences, and a utility function that represents these, which generate a linear demand curve? Actually there is, but it takes a slightly interesting form. Suppose a consumer has preferences over good 1 (Q1) and "all other goods" (Q2). In economics especially macro it is common to measure the price of a good in terms of another, the so called relative price. Suppose P2 = 1 so that P1 is the price of good 1 in terms of good 2. If a consumer has utility function: 3 University of Toronto, Department of Economics, ECO 204 20082009 S. Ajaz Hussain U = Q2 + (Q1 )2/2 Show that the demand function for good 1 is linear where: Q1 = P1 4 ...
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