Approximate quadratic-linear optimization problem

1. Approximate quadratic-linear optimization problem Based on Pierpaolo Benigno and Michael Woodford

2. The Quadratic Approximation to the Utility Function • Consider the problem

3. The first-order condition

4. The second-order approximation to the utility function

5. The second-order approximation to the constraint

6. Substitute the second-order approximation to the constraint into the linear term of the second-order approximation to the utility function, using the FOC, yields a quadratic objective function

7. The approximate optimization problem Subject to:

8. Which is supposed to be(?) a first order approximation of

9. A Linear-Quadratic Approximate Problem • Begin by computing a Taylor-series approximation to the welfare measure, expanding around the steady state. As a second-order (logarithmic) approximation, BW get:

10. The Quadratic Approximation to the Utility Function • Consider the problem

11. The first-order condition

12. The second-order approximation to the utility function

13. The second-order approximation to the constraint

14. Approximate optimization • Substitute the second-order approximation to the constraint into the linear term of the second-order approximation to the linear term of the second-order approximation of the utility function, using the first-order conditions, yields a quadraticobjective function. • The approximate optimization is to maximize the quadratic objective function, subject to the first-order approximation of the constraint.The first-order condition is equal to the first order approximation of the FOC of the original problem.

15. The Micro-based Neo-Keynesian Quadratic-linear problem Based on Pierpaolo Benigno and Michael Woodford

16. The Micro-based Quadratic Loss Function

17. Welfare measure expressed as a function of equilibrium production Demand of differentiated product is a function of relative prices

18. The Deterministic (distorted) Steady State Maximize with respect to Subject to constraints on

19. BW show that an alternative way of dealing with this problem is to use the a second-order approximation to the aggregate supply relation to eliminate the linear terms in the quadratic welfare function.

20. A Linear-Quadratic Approximate Problem • Begin by computing a Taylor-series approximation to the welfare measure, expanding around the steady state. As a second-order (logarithmic) approximation, BW get:

21. There is a non-zero linear term in the approximate welfare measure, unless • As in the case of no price distortions in the steady state (subsidies to producers that negate the monopolistic power). This means that we cannot expect to evaluate this expression to the second order using only the approximate solution for the path of aggregate output that is accurate only to the first order. Thus we cannot determine optimal policy, even up to first order, using this approximate objective together with the approximations to the structural equations that are accurate only to first order.

22. Welfare measure expressed as a function of equilibrium production Demand of differentiated product is a function of relative prices

23. The Micro-based Quadratic Loss Function of Benigno and Woodford

24. There is a non-zero linear term in the approximate welfare measure, unless • As in the case of no price distortions in the steady state (subsidies to producers that negate the monopolistic power). This means that we cannot expect to evaluate this expression to the second order using only the approximate solution for the path of aggregate output that is accurate only to the first order. Thus we cannot determine optimal policy, even up to first order, using this approximate objective together with the approximations to the structural equations that are accurate only to first order.

25. The Deterministic (distorted) Steady State Maximize with respect to Subject to constraints on

26. BW show that an alternative way of dealing with this problem is to use the a second-order approximation to the aggregate supply relation to eliminate the linear terms in the quadratic welfare function.

27. MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING

28. MICROFOUNDED CAGAN-SARGENT PRICE LEVEL DETERMINATION UNDER MONETARY TARGETING FLEX-PRICE, COMPLETE-MARKETS MODEL

29. Complete Markets Value of portfolio with payoff D = price kernel

30. Interest coefficient for riskless asset Riskless Portfolio

31. Budget Constraint Where T is the transfer payments based on the seignorage profits of the central bank, distributed in a lump sum to the representative consumer

32. No Ponzi Games: For all states in t+1 For all t, to prevent infinite c The equivalent terminal condition

33. Lagrangian

34. Transversality condition: Flow budget constraint:

35. Market Equilibrium Market solution for the transfers T

36. Monetary Targeting: BC chooses a path for M Fiscal policy assumed to be: Equilibrium is S.t. Euler-intertemporal condition condition FOC-itratemporal condition TVC Constraint For given

37. We study equilibrium around a zero-shock steady state:

38. Derive the LM Curve From the FOC: At the steady state:

39. Separable utility : Define: The “hat” variables are proportional deviations from the steady state variables.

40. Similar to Cagan’s semi-elasticity of money demand

41. We log-linearize around zero inflation define Log-linearize the Euler Equation and transform it to a Fisher equation: Elasticity of intertemporal substitution g is the “twist” in MRS between m and c

42. Add the identity We look for solution given exogenous shocks

43. Solution of the system This is a linear first-order stochastic difference equation ,where, Exogenous disturbance (composite of all shocks):

44. given There exists a forward solution: From which we can get a unique equilibrium value for the price level: This is similar to the Cagan-Sargent-wallace formula for the price level, but with the exception that the Lucas Critique is taken care of and it allows welfare analysis.

45. I. Interest Rate Targeting based on exogenous shocks Choose the path for i; specify fiscal policy which targets D: Total end of period public sector liabilities. Monetary policy affects the breakdown of D between M and B: No multi-period bonds Beginning of period value of outsranding bonds End of period, one-period risk-less bonds

46. Steady state (around ) fix

47. PRICE LEVEL IS INDETERMINATE: Real balances are unique Future expected inflation is unique Is unique But, neither Can uniquely be determined!

48. To see the indeterminancy, let “*” denote solution value: v is a shock, uncorrelated with (sunspot), the new triple is also a solution, thus: Price level is indeterminate under the interest rule!

49. II. Wicksellian Rules: interest rate is a function of endogenous variables (feedback rule) V=control error of CB Fiscal Policy Exogenous Endogenous

50. Steady State: Log-linearize: