Macroeconomics Lecture NotesTOPICS 2. The Solow Growth model 2-5 2. The Ramsey Growth model 6-11 3. Real BusinessCycle Theory 12-20 4. Traditional Keynesian Theories of Fluctuations 21-23 5. Microeconomic Foundations of Incomplete Nominal Adjustments 24- 39 6. Consumption 40-44 7. Investment 45-508. Inflation and Monetary policy 51-901Growth1.1The Solow Growth ModelAt any point in time, the economy has some amount of labor (L), capital (K) and knowledge (A) to produce output (Y). The production function can be written asY (t) = F(K(t),A(t)L(t),(1)where t denotes time. The production function is assumed to have constantreturns to scale in capital and effective labor, (F(ck,cAL) = cF(K,AL) where c 0). Hence, we can write1 ALF(K,AL) = F?K AL,1? .(2)Defining k = K/AL, y = Y/AL and f(k) = F(K,1), we can rewrite equation (2) as:y = F(k)(3)Here, the output per effective labor depends on the capital per effective labor. The ”intensive” or ”per capita” form of the production function f(k)satisfies f(0) = 0,f0(k) 0,f00(k) 0 and k is rising. If k kthen (k) 0, n (1 )g 0(12)Note that determines the households willingness to shift consumption be-tween two periods (1/ can be defined as the elasticity of intertemporal sub- stitution between consumption at any two points in time.) As goes to 0, marginal utility falls more slowly as consumption rises so that the household is more willing to allow consumption to vary overtime. The last assumption, n (1 )g 0, ensures that lifetime utility does not diverge.2.1.1The behavior households and firmsHouseholds budget constraint: The representative household take r and w as given. Since the household has L/H members, its labor income at t is W(t)L(t)/H and consumption expenditures are C(t)L(t)/H. Since interest rates over time can change, we write R(t) as (=Rt =0r()d). One unit of good invested at t = 0 yields eR(t)units of the good at time t (showing the effects of continuous compounding). The households consumption is C(t)L(t)/H and its initial wealth is 1/H of wealth at time 0 or K(0)/H The households budget constraint is therefore:Zt=0eR(t)CtLt Hdt K0 H+Zt=0eR(t)WtLt Hdt.(13)The constrain implies that the household cannot spend more than its initial wealth. In other words, one cannot issue debt and roll it forever. The issuermust pay offthe debt. Households maximization problem: The representative household wants to maximize its lifetime utility subject tobudget constraint. Define c(t) as the effective labor consumption, consump- tion per worker, C(t), can be written as A(t)c(t). Hence, the households objective function can be written in the intensive form asU = BZt=0etc(t)1 1 dt(14)7where B A(0)1L(0)/H and n (1 )g, where 0. We can also write the household budget condition in the intensive formZt=0eR(t)c(t)e(n+g)tdt k(0) +Zt=0eR(t)w(t)e(n+g)tdt(15)Household Behavior: The household chooses the path of c(t) to maximize lifetime utility (14) subject to budget constraint (15).To solve the problem, we set up the LagrangianL=BZt=0etc(t)1 1 dt(16)+? k(0) +Zt=0eR(t)w(t)e(n+g)tdt Zt=0eR(t)c(t)e(n+g)tdt?The household chooses c at each point in time. Hence maximizing the lagrangian we getBetc(t)= eR(t)e(n+g)t(17)which characterizes the household behavior. Taking the log and the time derivative of both sides we obtain c(t) c(t)=r(t) g (18)which implies that consumption per worker rises as long as real return r is greater than the discount factor . This equation is also known as the euler equation and describes how consumption behaves over time. To interpret (20), recall that consumption per worker is C(t) = c(t)A(t), hence we can writeC(t) C(t)=r(t) .(19)Equation (19) implies that consumption per worker is rising if real return exceeds households discount rate.82.1.2The Dynamics of the EconomyRecalling that r(t) = f0(k(t), we can write 20 as c(t) c(t)=f0(k(t) g (20)and the dynamics for capital stock can be written ask(t) = f(k(t) c(t) (n + g)k(t)(21)Combining the two equations in a phase diagram we can compute the modi-fied golden rule kwhich happens to be smaller than the golden-rule level of k given by the peak of thek(t) schedule (see figure 2.3 in your book). Welfare: Since markets are competitive and there are no externalities,the first welfare theorem from microeconomics hold. The decentralized equi- librium produces highest possible utility among allocations that treat all households in the same way.2.2Properties of the balanced growth pathThe properties of the economy at equilibrium is identical to that of the Solow economy on the balanced growth path. Capital, output and consumptionper unit of effective labor are constant, and at that point savings rate is also constant! Total capital stock and output and consumption grow at rate n+g. Hence, central implications of the Solow growth model does not hinge on constant saving rate.The only difference i