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notes_3 - Consumers Theory February 8 2009 1 Problem Set 1...

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Consumers Theory February 8, 2009 1 Problem Set 1: Solution Exercise 2.6 The data are P Q 4 30 4 . 5 27 5 24 It’s pretty easy to see that these points lie above a line with slope equal to -6: Δ Q Δ P = 27 - 30 4 . 5 - 4 = - 6 = 24 - 27 5 - 4 . 5 In order to identify a line you need just 2 parameters: the slope and the vertical intercept. So, if the equation of a generic demand is Q = a - bP and we already know that b = - 6, then we can find a by just plugging in any one of the three points 30 = a - 6(4) = a = 54 The direct demand is Q = 54 - 6 P , and the inverse demand is P = 9 - 1 / 6 Q Now, in order to find the elasticity we can use the formula for linear de- mands D = - b P Q so P Q 4 30 - 4 / 5 4 . 5 27 - 1 5 24 - 5 / 4 1
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Exercise 2.7 The existence of a shortage implies that the official price, P 0 is below the equilibrium one. At this price, therefore, the quantity de- manded is greater than the quantity supplied. Black market tickets have a price above the official one: the scalping price cleans the market. Increasing penalties for scalping has the effect of increasing costs for black market traders: as a consequence their supply function shrinks. Their new prices are higher than before, thus reducing the amount of customers inter- ested in the transaction. Exercise 2.16 The economic system is represented by the following equa- tions Q d = 90 - 2 P - 2 T Q s = - 9 + 5 P - 2 . 5 R Equilibrium is found at the intersection of demand and supply: Q d = Q s . Therefore Q * = 46 P * = 12 For the price elasticities, use the formulas = slope P * Q * . The slope of the demand function is -2 (the coefficient in front of P ), and the slope of the supply is +5. Therefore the elasticities are d = - 2 12 46 = - 12 23 s = 5 12 46 = 30 23 The cross price elasticity of gold balls with respect to the price of titanium is Q,T = Δ Q Δ T T Q = ∂Q ∂T T Q now, the first term of the expression is the partial derivative of Q with respect to T : -2 (since T enters linearly it is just the coefficient in front of T in the demand function). Therefore Q,T = - 2 10 46 = - 10 23 Exercise 2.26 1. January: the economic system is defined by Q s = 30 P - 30 Q d = 120 - 2 P 2
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Q P 6 120 D Jan D Mar 7 P^D Jan = 6 - Q/20 P^d Mar = 7 - Q/20 P^s Jan = 1 + Q/30 1 2 P^s Feb = 2 + Q/30 A B C 48 60 Equating demand and supply we get P Jan = 3 Q Jan = 60 2. February: the economic system is defined by Q s = 30 P - 60 Equating demand and supply we get P Feb = 18 5 Q Feb = 48 3. March: the economic system is defined by Q d = 140 - 20 P Equating demand and supply we get P Mar = 4 Q Mar = 60 2 Utility Functions Utility functions are representation of the level of satisfaction that a con- sumer achieves by purchasing certain quantities of several items. They 3
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are functions (possibly multivariate) that take as arguments quantities con- sumed of different items. Let’s consider the case of an economy with just 2 goods that we label x, y . The the utility for each individual is given by the function U ( x, y ).
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