ECON6008

Sample 2

Numerical Group Project

Question 1

It is demonstrated that with 1% shock to technology, consumption increased in the same proportion. This is because since

marginal cost, wt/at , has to remain constant. In the face of a 1% increase in productivity, wage also increases with the same percentage magnitude. However, since the utility function is in log-specification, the income effect and substitution effect cancels out each other, rendering the level of labour supply constant. (As shown in graph 3. With labour supply, N , constant, the 1% shock is passed on to  output by atNt   = yt , so output increased by the same magnitude. Lastly, because output equals to consumption in equilibrium, consumption also increases with the same magnitude, 1% (as shown in graph 1)

Graph 4 shows that inflation does not respond to technology shock. It is trivial that inflation does not deviate from its steady state value as there are no prices in the equations.

Also, in the presence of zero inflation, the real interest rate and the nominal interest rate would obviously be equal and therefore track each other in the impulse response function precisely. In response to the productivity shock, both interest rates deviate negative forty basis points, or 0.4 percentage points, per annum. They then   steadily return in the direction of the steady state to a level of approximately 5 basis points deviation below the steady state at period 20. The effects of the productivity shock first flow through to consumption. This then affects λ , which is the marginal  utility of consumption, inversely by the same amount (since ). Since consumption slowly return to the steady state level, future consumption is expected to be lower and therefore the marginal utility of consumption in the next period λt+1  would be greater than the current period λt. The intertemporal Euler equation demands that the present value marginal utilities across periods to be equal, and therefore interest rate would fall. But as contraction in consumption reduces with  time, the level of interest rate also decreases towards the steady state level.

Question 2, Part 1

With zero-inflation policy, the standard New Keynesian model is virtually identical to RBC since sticky prices has no significance in an economy with no price adjustment at all. So optimal price and relative price distortion is constant. With productivity increases by 1%. output also increases by 1% which in turn increases consumption by 1%. The shock, output and consumption then steadily decline until reaches a 0.2% deviation at 20 periods after the initial shock. Follow the same argument as Question 1, with zero inflation (shown in graph 4), real rate equals to nominal rate and the initial decline of 40 basis point (per annum) from steady state level is due to the fact the marginal utility in the current period is greater than the following period.  As consumption contraction becomes smaller with consumption approaches steady state level, interest rate also steadily increases towards steady state level (shown in graph 2)

Labour supply is also constant in this model due to a log-utility specification.

However notice that dynare output shows a fluctuation of 2x10^-7. This is assumed to be a computational result and does not affect our interpretation.

Question 2, Part 2

Compare to the RBC model in Question 1, due to the relative-price distortion, a 1% increase in productivity brings about a dampened increase in consumption, at close to 0.8% (shown in graph 1). The increase in consumption (and hence output) is lower than that of an flexible-price economy (RBC), hence opens up a negative output gap  that is responsible for the approximately 40% basis point per annum decline in inflation in period 1 (shown in graph 4). The Taylor-like rule prescribes that the nominal interest rate to be lowered in order to close the output gap and increase inflation, as shown in graph 2 with a close to 70 basis point per annum decrease in nominal interest rate. This is a large decrease in nominal rate compare to RBC model, but the real rate is less reduced than in RBC (35 basis point per annum compare to 40 in RBC) The productivity shock is also responsible for a persistent employment decline, as graph 3 demonstrate a 0.15% decrease from steady state in labour, compare to no fluctuation in labour in RBC. All variables in the graphs steadily move towards their respective steady state due to the accommodating central bank policy and the deterioration in the effects of the temporary technology shock.

Difference between this model and the previous models can be attributed to the fact that inflation is no longer stable and this allows relative-price distortion to affect the economy. With predetermined policy the optimal level of inflation is zero in the steady state as it minimises both relative price and average markup distortion, therefore the zero-inflation policy yields higher welfare for households. This is also supported by the fact that consumption is higher with zero-inflation policy than the Taylor-like rule. Further, τπ and τy  needs to be above a certain value in order for the model to be uniquely determined. Bullard and Mitra (2002) give us a necessary and  sufficient condition for the unique solution: K(τπ 一 1) + (1 一 β)τy  > 0 where K ≡ λ(σ + φ). In order to stabilize the system, the monetary author should act aggressively. However, larger τy would generate greater fluctuation in output gap and inflation, and hence will have larger welfare loss. Moreover, the smallest welfare loss could be achieved when monetary authority responds to changes in inflation only. As τπ increases, welfare loss would be lower. Therefore, by setting τy   = 0 and let τπ be great enough, the Taylor-like rule can mimic the allocation under optimal  Ramsey policy.

Question 3

With a 1% markup shock, the relative-distortion reduces output and therefore consumption by 0.1% (shown in graph 1). This again opens up a negative output gap of 0.13% compare to a flexible-price equilibrium (shown in graph 5). However, with the markup shock the firms now price higher and there creates a positive level of inflation, at 40 basis point per annum higher than steady state (shown in graph 4). While graph 2 shows that the central bank adopts a contractionary monetary policy, it is in fact in capable of closing both gaps at the same time. Since such policy would  reduce inflation but lowers output and further widening the output gap. The fact that all variables returns to their respective steady state and in a faster manner than previously can be attributed to a small and less persistent (at Pv   = 0.5) markup shock.