Notes on Macroeconomic TheorySteve WilliamsonDept. of EconomicsUniversity of IowaIowa City, IA 52242August 1999Chapter 1Simple Representative AgentModelsThis chapter deals with the most simple kind of macroeconomic model,which abstracts from all issues of heterogeneity and distribution amongeconomic agents. Here, we study an economy consisting of a represen-tative ¯rm and a representative consumer. As we will show, this isequivalent, under some circumstances, to studying an economy withmany identical ¯rms and many identical consumers. Here, as in all themodels we will study, economic agents optimize, i.e. they maximizesome objective subject to the constraints they face. The preferences ofconsumers, the technology available to ¯rms, and the endowments ofresources available to consumers and ¯rms, combined with optimizingbehavior and some notion of equilibrium, allow us to use the model tomake predictions. Here, the equilibrium concept we will use is competi-tive equilibrium, i.e. all economic agents are assumed to be price-takers.1.1 A Static Model1.1.1 Preferences, endowments, and technologyThere is one period and N consumers, who each have preferences givenby the utility function u(c; `); where c is consumption and ` is leisure.Here, u(¢; ¢) is strictly increasing in each argument, strictly concave, and12 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELStwice di®erentiable. Also, assume that limc!0u1(c; `) = 1; ` > 0; andlim`!0u2(c; `) = 1; c > 0: Here, ui(c; `) is the partial derivative withrespect to argument i of u(c; `): Each consumer is endowed with oneunit of time, which can be allocated between work and leisure. Eachconsumer also ownsk0Nunits of capital, which can be rented to ¯rms.There are M ¯rms, which each have a technology for producingconsumption goods according toy = zf(k; n);where y is output, k is the capital input, n is the labor input, and z isa parameter representing total factor productivity. Here, the functionf(¢; ¢) is strictly increasing in both arguments, strictly quasiconcave,twice di®erentiable, and homogeneous of degree one. That is, produc-tion is constant returns to scale, so that¸y = zf(¸k; ¸n); (1.1)for ¸ > 0: Also, assume that limk!0f1(k; n) = 1; limk!1f1(k; n) = 0;limn!0f2(k; n) = 1; and limn!1f2(k; n) = 0:1.1.2 OptimizationIn a competitive equilibrium, we can at most determine all relativeprices, so the price of one good can arbitrarily be set to 1 with no loss ofgenerality. We call this good the numeraire. We will follow conventionhere by treating the consumption good as the numeraire. There aremarkets in three objects, consumption, leisure, and the rental servicesof capital. The price of leisure in units of consumption is w; and therental rate on capital (again, in units of consumption) is r:Consumer's ProblemEach consumer treats w as being ¯xed, and maximizes utility subjectto his/her constraints. That is, each solvesmaxc;`;ksu(c; `)1.1. A STATIC MODEL 3subject toc · w(1 ¡`) + rks(1.2)0 · ks·k0N(1.3)0 · ` · 1 (1.4)c ¸ 0 (1.5)Here, ksis the quantity of capital that the consumer rents to ¯rms, (1.2)is the budget constraint, (1.3) states that the quantity of capital rentedmust be positive and cannot exceed what the consumer is endowedwith, (1.4) is a similar condition for leisure, and (1.5) is a nonnegativityconstraint on consumption.Now, given that utility is increasing in consumption (more is pre-ferred to less), we must have ks=k0N; and (1.2) will hold with equality.Our restrictions on the utility function assure that the nonnegativityconstraints on consumption and leisure will not be binding, and in equi-librium we will never have ` = 1; as then nothing would be produced,so we can safely ignore this case. The optimization problem for the con-sumer is therefore much simpli¯ed, and we can write down the followingLagrangian for the problem.L = u(c; `) + ¹(w + rk0N¡ w` ¡c);where ¹ is a Lagrange multiplier. Our restrictions on the utility func-tion assure that there is a unique optimum which is characterized bythe following ¯rst-order conditions.@L@c= u1¡¹ = 0@L@`= u2¡¹w = 0@L@¹= w + rk0N¡w` ¡c = 0Here, uiis the partial derivative of u(¢; ¢) with respect to argument i:The above ¯rst-order conditions can be used to solve out for ¹ and cto obtainwu1(w + rk0N¡ w`; `) ¡ u2(w + rk0N¡ w`; `) = 0; (1.6)4 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELSwhich solves for the desired quantity of leisure, `; in terms of w; r; andk0N: Equation (1.6) can be rewritten asu2u1= w;i.e. the marginal rate of substitution of leisure for consumption equalsthe wage rate. Diagrammatically, in Figure 1.1, the consumer's budgetconstraint is ABD, and he/she maximizes utility at E, where the budgetconstraint, which has slope ¡w; is tangent to the highest indi®erencecurve, where an indi®erence curve has slope ¡u2u1:Firm's ProblemEach ¯rm chooses inputs of labor and capital to maximize pro¯ts, treat-ing w and r as being ¯xed. That is, a ¯rm solvesmaxk;n[zf(k; n) ¡rk ¡ wn];and the ¯rst-order conditions for an optimum are the marginal productconditionszf1= r; (1.7)zf2= w; (1.8)where fidenotes the partial derivative of f(¢; ¢) with respect to argu-ment i: Now, given that the function f(¢; ¢) is homogeneous of degreeone, Euler's law holds. That is, di®erentiating (1.1) with respect to ¸;and setting ¸ = 1; we getzf(k; n) = zf1k + zf2n: (1.9)Equations (1.7), (1.8), and (1.9) then imply that maximized pro¯tsequal zero. This has two important consequences. The ¯rst is that wedo not need to be concerned with how the ¯rm's pro¯ts are distributed(through shares owned by consumers, for example). Secondly, supposek¤and n¤are optimal choices for the factor inputs, then we must havezf(k; n) ¡rk ¡ wn = 0 (1.10)1.1. A STATIC MODEL 5Figure 1.1:6 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELSfor k = k¤and n = n¤: But, since (1.10) also holds for k = ¸k¤andn = ¸n¤for any ¸ > 0; due to the constant returns to scale assumption,the optimal scale of operation of the ¯rm is indeterminate. It thereforemakes no di®erence for our analysis to simply consider the case M = 1(a single, representative ¯rm), as the number of ¯rms will be irrelevantfor determining the competitive equilibrium.1.1.3 Competitive EquilibriumA competitive equilibrium is a set of quantities, c; `; n; k; and pricesw and r; which satisfy the following properties.1. Each consumer chooses c and ` optimally given w and r:2. The representative ¯rm chooses n and k