The mathematical equations characterizing the real business cycle model are given below:
Nt =Nd(wt,At,Kt) Nt =Ns(wt,t) Ct = Cd(Yt Gt, Yt+1 Gt+1, rt)
It =Id(rt,At+1,qt,Kt) Yt =AtF(Kt,Nt) Yt = Ct + It + Gt
Nt is labor, wt is the real wage, At and At+1 are productivity, Kt is capital stock, t is an exogenous labor supply shock, Ct is consumption expenditure, Yt and Yt+1 are output, Gt and Gt+1 are government expenditures, rt is the real interest rate, It is investment, and qt is an exogenous investment shock.
5. The mathematical equations characterizing the real business cycle model are given below: N4 = N(wt, At, Kt) N4 = N* (wt,0t) C4 = Cd (Y4 - Gt, Yt+1 - Gt+1,rt) It = I(rt, At+1, qt, Kt) Y4 = AF(Kt, N+) Y4 = C++ It +Gt Nt is labor, Wt is the real wage, At and At+1 are productivity, Kt is capital stock, 0t is an exogenous labor supply shock, Ct is consumption expenditure, Yt and Yt+1 are output, Gt and Gt+1 are government expenditures, rt is the real interest rate, It is investment, and qt is an exogenous investment shock. (a) Briefly explain what each equation is and where it comes from. If relevant, discuss the signs (positive or negative) of the partial derivatives of the functions. In cases where there is ambiguity concerning the sign of the partial derivative, state where that is and what assumption allows us to sign the partial derivative. (b) What are the exogenous variables of the model? (c) What are the endogenous variables of the model? (d) Write down the definition of the Ys curve. Which of the above equations are summarized by the Y' curve? Graphically derive the Ys curve. Label the figures. (e) Write down the definition of the IS curve. Which of the above equations are summarized by the IS curve? Graphically derive the IS curve. Label the figures. (f) Suppose that there is an exogenous increase in At. Graphically show how this affects the equilibrium. Label the figures. In words, discuss how Ct and It ought to react to the increase in At. (Note: Consider a similar question where you allow other exogenous variables to change.) 5. The mathematical equations characterizing the real business cycle model are given below: N4 = N(wt, At, Kt) N4 = N* (wt,0t) C4 = Cd (Y4 - Gt, Yt+1 - Gt+1,rt) It = I(rt, At+1, qt, Kt) Y4 = AF(Kt, N+) Y4 = C++ It +Gt Nt is labor, Wt is the real wage, At and At+1 are productivity, Kt is capital stock, 0t is an exogenous labor supply shock, Ct is consumption expenditure, Yt and Yt+1 are output, Gt and Gt+1 are government expenditures, rt is the real interest rate, It is investment, and qt is an exogenous investment shock. (a) Briefly explain what each equation is and where it comes from. If relevant, discuss the signs (positive or negative) of the partial derivatives of the functions. In cases where there is ambiguity concerning the sign of the partial derivative, state where that is and what assumption allows us to sign the partial derivative. (b) What are the exogenous variables of the model? (c) What are the endogenous variables of the model? (d) Write down the definition of the Ys curve. Which of the above equations are summarized by the Y' curve? Graphically derive the Ys curve. Label the figures. (e) Write down the definition of the IS curve. Which of the above equations are summarized by the IS curve? Graphically derive the IS curve. Label the figures. (f) Suppose that there is an exogenous increase in At. Graphically show how this affects the equilibrium. Label the figures. In words, discuss how Ct and It ought to react to the increase in At. (Note: Consider a similar question where you allow other exogenous variables to change.)
