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4 An intertemporal household problem with power utility In the previous set of practice problems, we considered an intertemporal problem of a household with preferences represented by the logarithmic utility function U = In C1 + 8 In C2 subject to the intertemporal budget constraint C2 = W. 1 + R (2) Question 4.1 What do parameters S and W represent? What does the intertemporal budget constraint represent? Now consider a different preference specification, which is a generalization of the loga- rithmic utility function: Cl-y Cl-Y U = + B 2 (3 1 -y COThis is the socalled power utility function, and the parameter 6/ E [0, 00) is the coefcient of risk aversion (and captures the curvature of the utility function). Question 4.2 Consider a household that maximizes utility (3) subject to the intertemporal budget constraint (2). Proceed exactly as in the case of logarithmic utility and derive the Euler equation that represents the relationship between consumption 01 and 0'2. I Question 4.3 The Euler equation you should have obtained is 0f7=13(1+R)02_7. Argue that this Euler equation is indeed a generalization of the Euler equation for the logarithmic utility and that logarithmic utility can indeed be viewed as a special case of power utility, despite the seemingly very different functional form of the expression for U. For which parameter 7 do we obtain the case of logarithmic preferences? I Question 4.4 Using the Euler equation for power utility, show that when 02 is fixed, interest rates and consumption in period 1 are inversely related