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Dynamic Macroeconomics 2018-19 Universidad Carlos III de Madrid Problem Set 1 Discussion in class: February 8

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Numerical exercise: the CES utility function

A class of utility functions that is very important is the class of constantelasticity-of-substitution (CES) or power utility functions:1 ( 1−γ c if γ ∈ [0, 1) o γ > 1, 1−γ u(c) = ln c if γ = 1. In this exercise we study the properties of this function numerically. Use a spreadsheet program such as OpenOffice or Excel or another maths/statistics program to answer the following questions. 1. In a table, calculate the utility from consuming the quantities c = 0.1, 0.2, . . . 2.9, 3.0 for each of the parameter values γ = 0, 0.3, 0.7, 1, 2, 4, 10. 2. With the computer, graph the function u for each of the values γ = 0, 0.3, 1, 2. 3. Briefly comment (about 3-5 sentences) on the properties of the function and how they depend on γ . 1

In the following lectures, we will see why it is called this way and why the logarithmic function covers the case γ = 1.

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Constrained utility maximization

Consider a household facing the following utility-maximization problem: max {φ ln c + (1 − φ) ln (1 − l)} c,l

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s.t. c = I and I = l 2 , where c and l denote, respectively, the level of consumption and the amount 1 of time spent working and 0 < φ < 1 is a constant.2 Also, I = l 2 represents an elementary production function for this household. By using both the substitution and the Lagrangian method of optimization, calculate the consumption and labor supply.

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The business cycle in Spain

Download the file Spanish Cycle (1970-2011) from Aula Global. This file gives you quarterly business-cycle data from the Instituto Nacional de Estad´ıstica.3 The block on the left-hand side gives demand data at constant prices of 1986 (so we do not have to worry about the effects of inflation). The file is in Spanish, but you will be told later in this problem exactly which series are the ones we are interested in. The block on the right-hand side of the sheet gives the component relating to the trend τt of the logarithms of the respective data series.4 Use a spreadsheet (such as Excel ) or statistics program to answer the following questions; familiarize yourself with how to copy the formulas from one cell to another if you do not already know how to do so.

3.1

Trend and deviations

Calculate the deviations y˜t , c˜t , ˜it and g˜t of yt = ln Yt (Y : gross domestic product, GDP), ct = ln Ct (C: domestic private consumption), it = ln It 2

1 is the amount of available time, so (1 − l) is the amount of leisure time. The series were created by combining two series from INE’s website (section Econom´ıa, subsection Cuentas econ´omicas). 4 The trend was computed using the Hodrick-Prescott filter with λ = 1600. 3

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(I: gross fixed capital formation) and gt = ln Gt (G: public consumption) (y) from their respective trends τt (for example: y˜t = yt − τt ).5 1. Make a graph of the log gross domestic product yt and its trend τt against time, and another of the deviations of the trend y˜t against time. Identify at least two exceptional periods in the (economic) history of Spain. 2. What is the economic interpretation of the deviations y˜t etc. that you have just calculated? 3. Make a graph of the series y˜t , c˜t , ˜it and g˜t against time. Comment.

3.2

Volatility and covariation

1. Calculate the standard deviations of: y˜t , c˜t , i˜t and g˜t . (a) What is the interpretation of these standard deviations you have just calculated? (b) Are they similar to the patterns in volatilities in the U.S. (that we saw in class)? Comment. 2. Calculate the coefficients of correlation of c˜t , i˜t and g˜t with y˜t . (a) What is the interpretation of the correlations you have just calculated? (b) Are they similar to the patterns of correlation in the U.S. (that we saw in class)? Comment.

3.3

Free choice

Focus on whichever pattern or peculiarity in the data you consider interesting (for example: changes in volatility over time, extreme events,. . . ). 1. Describe the pattern/peculiarity in words and justify your argument with some statistics obtained from the data. 5

In the spreadsheet, these will be listed, respectively, as ’producto interior bruto’, ’consumo privado interior’, ’formaci´on bruta del capital fijo’, and ’consumo p´ ublico’. The rows are for a˜ no=year, and trim=quarter

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2. Try and find economic/political/historical reasons for what we are seeing in the pattern/peculiartiy that you are describing.

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