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by: Miss Kristy Veum


Marketplace > Texas A&M University > Economcs > ECON 636 > MACROECONOMIC THEORY I
Miss Kristy Veum
Texas A&M
GPA 3.8

L. Auernheimer

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L. Auernheimer
Class Notes
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This 6 page Class Notes was uploaded by Miss Kristy Veum on Wednesday October 21, 2015. The Class Notes belongs to ECON 636 at Texas A&M University taught by L. Auernheimer in Fall. Since its upload, it has received 17 views. For similar materials see /class/225825/econ-636-texas-a-m-university in Economcs at Texas A&M University.




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Date Created: 10/21/15
Econ 636 Macroeconomic Theory I Fall 2008 LAuernheimer Handout 5 The RCK Model with Endogenous Labor The RCK model with variable labor is at the core of Real Business Cycles RBC theory There is a good treatment of the model in Romer in discrete time and with a particular utility function In order to preserve some continuity in our analysis what follows is a compact description of the model The only difference between the analysis and the standard RCK framework is that here rather than assuming a fixed supply of labor for any labor force in turn equal to population every individual agent chooses the work effort Given a xed maximum number of work effort you can interpret this as hours per day the option is either work which earns a wage and leisure which provides utility In particular normalizing the maximum work effort to unity the level of labor per individual per unit of time is then given by l 0 l Leisure It is convenient in this case to split the economy into a household sector and the sector of business rms as we did at some point in the standard RCK model Households The single individual s utility function is UCJ Ugt0Ult0 U50 UHlt0U 0 where as before 0 is the ow of labor per individual Since it is leisure what provides utility then from the de nition in l labor will provide quotdisutilityquot Notice that the second derivative with respect to labor is positive contrary to what is in the transparency I distributed and I used in class the mistake came because I was thinking of the marginal disutility of labor We also assume separability between consumption and labor in the utility function Individuals care about their descendants so that their utility after t 0 is magni ed by the size of their quotfamilyquot Assuming that population grows at the rate n and maintaining the same notation we have been using the functional to be maximized at any initial time t 0 is 2 Uct0t aw dt 0 Households hold real assets 1 which pay a real interest rate provide work and receive wages denoted as w and consume Their budget constraint then can be written as 239 dadtatrwt0tatnct The value of assets evaluated at their shadow price i is Vt A a so that the quotmodi ed Hamiltonianquot to be maximized is H Uct0te quot k 1 dt dt Differentiating and equating to zero yields Uccte mt X 0 60 U 0te mt kw 0 60 krn 0 6a dt From the rst of these conditions 3 UCt UMUD W so that we can write a quotsupply of laborquot function 0 0cw QClt0 0Wgt0 Notice the simple economic interpretation of 3 showing that the marginal disutility of labor divided by its price the wage rate needs to be equal to the marginal utility of consumption Finally the Euler optimization condition is 4 Business Firms As in the standard RCK model rms do not own stoks They rent capital for which they pay a net return equal to the real interest rate hire labor and maximize pro ts which since rms are perfectly competitive end up being zero The constant returns to scale production function is Y A F K 0 N whereA gt 0 is a xed coef cient and N is equal to the labor force in turn identically equal to population In per capita terms y Af k 0 From a simple pro t maximization procedure see the case of the standard RCK model with households and rms we derive the optimality conditions Afkk08r 5 Akow which simply mean that firms will rent capital and hire labor up to the point at which their marginal product is equal to their rental or wage The Aggregate Economy A couple of simple substitutions allows to Write U 11 Ucltcltrgtgt Ak l and IZtckA 50 1230 12Agt0 Then in the aggregate the system can be expressed in terms of consumption and capital per capita U AfkkllckA5p AfkllckA 7M5 nc Figure 1 is a graphical representation of the system IQ1t 0 J 7 1km o r Figure 1 Appendix The slope of the dc dt Olocas in the RCK model with Variable Labor The locus is given by equating the expression U c Afkk0ckA a p as to zero ie by the equation 1 Afkk0CkA5P Ie along the dc dt 0 line the marginal cost of capital will be constant Since the production function is CRS this means that the k ratio will be constant and so will also be the marginal product of labor ie the wage rate The individual39s equilibrium condition concerning the laborleisure choice is given by the utility function U c 0 2 UcgtoaUmlt0alaltwilt05Uci 0 and the condition U 0 t 3 Uccr D W where w is constant along 1 Differentiating 3 yields 4 UH0td0 wUmdc But since the k ratio is constant along 1 so that replacing this value in 4 yields 5 ie a negative slope Look at two particular cases First consider the utility function see Handout 7 6 U lnc 01 gt06gt0 Computing the corresponding rst and second derivatives for the equilibrium condition 3 and performing a couple of substitutions yields the result E E dk c ie the elasticity of the dc dt 0 locus is constant and given by the coefficient 0 of the utility term for labor 6 Second consider an quotall logquot utility function such as U 11110 l alnl 0 lgtagt0 In this case equilibrium condition 3 becomes and the particular expression for 5 reduces after some substitutions to dc wa 0 dk 1 54 ie a negative and constant term so that l is a negatively sloped straight line Additionally if you like very general expressions you can verify that substituting equilibrium condition 3 into expression 5 yie s m UMUN lt0 dk c UmU90 where you should recognize in the rhs the ratio of the temporal elasticities of substitutions of labor ie the negative of leisure and consumption


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