INTEREST-RATE RULES: Framework for Analysis 2

For our numerical work, we rely upon log-linear approximations to the model’s structural equations. We assume an equilibrium in which the economy always stays near a steady state path, which represents a stationary, deterministic equilibrium in the case of no exogenous disturbances (ft = 0 and Gt — G at all times) and a monetary policy consistent with stable prices. In this steady state, output is constant at a level Y (defined below), and consumption is constant at the level С = Y — G.1 It follows that the marginal utility of reai income.

AtPu is also constant, at the value A = uc{C\ 0). We log-linearize the structural equations of the model around these steady-state values. Percentage deviations in the marginal utility of consumption uc{Ct; ft) around the steady-state value uc{C\ 0) can be written as — a(Ct — Ct), where Ct = log(Ct/C), C% is an exogenous shift variable (a certain linear combination of the elements of ft),2 and a = —uccCjuc, where the partial derivatives are evaluated at the steady-state level of consumption. With this substitution, the log-linear approximations to (1.5) and (1.6) are given by
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where Rt = log(Rt/R*) = \og(j3Rt) is the percentage deviation of the short-run nominal interest rate from its steady-state value, 7rt = log(P*/Pt_i) is the inflation rate, and Xt = log(AfPt/A) measures the percentage deviation of the marginal utility of real income from its steady-state value. (Equation (1.8) refers to actual rather than expected inflation because inflation 7Tt+i is known with certainty at date t in our model.)

A similar log-linear approximation to the market-clearing condition allows us to replace Ct – Et-2Ct with Sc\Yt – Gt), where = C/Y, Yt = log(yt/?), and Gt collects the exogenous disturbance terms that shift the relation between aggregate demand and the
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Equation (1.11) plays a role analogous to the “IS equation” of traditional Keynesian models, but is consistent with intertemporal optimization.3 It relates output to the long run real interest rate (with a negative sign) and to autonomous spending disturbances. The latter include both disturbances to private impatience to consume resources, and to government spending, summarized in the composite disturbance term Gt. We assume that Gt is determined at t — 1, so that it is determined after Ct has already been chosen, but in time for the central bank to adjust the period t interest rate Rt in response to it. Letting Gt be determined after Ct ensures that output is not predetermined as of t — 2 (i.e., it allows us an interpretation for the output innovations in our VAR model of the U.S. data), even though output responds with a two period delay to exogenous disturbances to monetary policy. Electronic Payday Loans Online