### ECON 200 – A consumer is facing the following utility

ECON 200 – Fall 2016 Problem set 3 Readings for the 4th week of class Chapters 8-9 of Hal Varian’s textbook, chapters 8-9 of Victor Lima’s lecture notes. Problem 1 A consumer is facing the following utility maximization problem: 1 2 maxx1 ,x2 x13 x23 s.t. p1 x1 + p2 x2 ? m x1 , x2 ? 0 a) Set up the Lagrangian for this problem and write down the first order conditions. c) What is the solution to the above problem when (p1 , p2 ) = (5, 4) and m = 30? d) Write down the Marshallian demands for goods 1 and 2. e) What happens to the Marshallian demand when prices and income double? What property of the Marshallian demand can you invoke to answer this question? f) What fraction of her income does the consumer spend on good 1? What fraction on good 2? g) Write down the indirect utility function associated of the above problem. What is the consumer’s indirect utility function associated with the solution to the above problem when (p1 , p2 ) = (5, 4) and m = 30? h) What is the partial derivative of the indirect utility function with respect to m? Interpret the sign of the partial derivative. i) What are the partial derivatives of the indirect utility function with respect to p1 and p2 ? Interpret the sign of these partial derivatives. l) Find the Marshallian demand for each good using Roy’s identity. m) Draw the income offer curve and Engel curves of the above problem. n) Draw the price offer curve and demand curves of the above problem with respect to good 1. Problem 2 A family (which we treat here as a single consumer, this is what economists call a unitary model of the household) has the following utility function: 1 u(xF , xM , xC ) = log(xF ) + log(xM ) + log(xC ) where xF is the consumption of the father, xM is the consumption of the mother and xC the consumption of the child. The family monthly income is $900, to be allocated to the consumption of its members. Assume that a unit of consumption for of the child has price pC = p¯ and that one unit of consumption of parents has price pF = pW = 2¯ p. a) Set up the utility maximization problem (UMP) of this household. b) Set up the corresponding lagrangian and write down the first order conditions. c) Solve for the optimal consumption of the three household members. d) What fraction of the household income is allocated to the child? How much does the child consume compared to the parents? Now, assume that the household has a target utility u, which it wants to achieve by minimizing expenditure E. e) Write down the expenditure minimization problem (EMP). f) Set up the corresponding lagrangian and write down the first order conditions. g) Solve for the optimal consumption of the three household members. h) Write down the Hicksian demand for this household. i) What is the expenditure function of this household? l) Write down the partial derivative of the expenditure function of this household with respect to u. Interpret your result, in particular with respect to the sign of this derivative. m) Write down the partial derivatives of the expenditure function of this household with respect to prices (prices are {pF , pM , pC }). Write down second partial derivatives of the expenditure function of this household with respect to prices. Interpret your result, in particular with respect to the sign of these derivatives. n) Show that the Hicksian demand is homogeneous of degree zero in prices and that the expenditure function is homogeneous of degree 1 in prices. o) What is the target utility u that leads the household to spend its entire income in the EMP? PRACTICE PROBLEM (not to be turned in) Assume that a consumer faces the 2 following UMP: maxx1 ,x2 ? x1 + x2 s.t. p1 x1 + p2 x2 ? m x1 , x2 ? 0 a) What class of preferences does the above utility function represent? Draw the indifference curves associated with these preferences. b) Can you make an example of goods that may be represented by these preferences? c) Set up the Lagrangian for this problem and write down the first order conditions. d) Find the Marshallian demands for goods 1 and 2. e) Graphically illustrate the solution to the UMP. f) What does the Lagrange multiplier mean in this context? g) Draw the income offer curve and Engel curves of the above problem. h) Show that this utility function is not homothetic. 3

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