Monday, February 25, 2008

Homework 3

Question 1

1.1 Explain the differences between SIM and SIMEX when both models are in their steady state


The SIM model assumes that consumers have perfect foresight with regard to money and that consumers’ disposable income is constant and actual. The consumption function for the model is Cd = α1.YD + α2.H-1

The SIMEX model on the other hand considers the fact that consumers do not know precisely what their disposable income is going to be, thus assumes that households make some estimate of the income they will receive within the given period. The consumption function is adjusted to reflect expected disposable income as opposed to actual known income; Cd = α1.YDe + α2.H-1, where YDe denotes an expected value.

Given fixed expectations and perfect foresight both models eventually converge to the steady state level of national income. Convergence in the SIMEX model takes longer however, due to the fact that people act on wrong expectations.

1.2 Second, what does it mean for the stability of the model when the presence of mistakes allows household’s incomes to suffer? Can you draw any general conclusions about the real world from this model?

The fact that households can incorrectly estimate disposable income does not affect the stability of the model. When people act on wrong expectations, for example underestimating disposable income, saving is higher than expected and hence the stock of wealth grows faster than in the perfect foresight state. Consumption will eventually reach the same steady state value that it would have reached in a perfect foresight model, and the model will solve for the same steady state level of national income.

From this one can conclude that people will often underestimate their disposable income and that they will revise their behavioural consumption patterns as their wealth stocks deplete and their expectations about future income are adjusted.


1.3 Solve SIMEX for the following values for 3 periods: G=30, α1=0.6, α2=0.6, θ=0.2. Follow the format of table 3.6 on page 81 of GL in presenting your results.


Calculations are as follows:

Period 1: This period assumes no economic activity, thus reflected by the zero values.

Period 2: Expected disposable income, YDe, is derived by calculating the individual’s tax liability, θ*G = 0.2 * 30 = 6, and then subtracting this figure from government expenditure, 30 – 6 = 24. This figure is then used to solve for consumption, Cd = α1.YDe + α2.H-1 = 0.6*24 + 0.4*0 = 14.4. The figure for income is solved by adding government expenditure and consumption, 30+14.4=44.4. The tax rate is 20% of income, 44.4*0.2=8.88. These figures are then plugged into the remaining calculations to solve for ΔHs, ΔHh, H, ΔHd and Hd.

Period 3: Expected disposable income, YDe, is the same figure for period 2’s actual disposable income, 35.52. All other figures are derived as per period 2’s calculations. One difference however, is that there is now a value for H-1 which is equal to period 2’s H figure.

Period ∞: Y* is calculated using the fiscal stance formula, i.e. G/θ. This gives a value of 150.

Question 2
2.1 Is it possible to specify a version of SIM that replicates the ISLM model?

Yes it is possible to specify a version of the SIM that replicates the ISLM model. This is because their consumption functions are similar.

The ISLM model’s consumption is as follows: C= α0 + α1YD, where the intercept α0 represents autonomous consumer expenditure and the coefficient α1 indicates the consumer’s MPC.

The SIM model’s consumption function is Cd = α1.YD+ α2.H-1, where α1 represents the propensity to consume out of regular (present income) and α2 represents the propensity to consume out of past wealth.

2.2 Write one down and comment on the stability of this model

One of the main properties of the stationary state, i.e. SIM equilibrium, is that consumption must equal disposable income. In other words the average propensity to consume must equal one. This implies that YD=C, which can be replicated in the ISLM model by assuming an MPC of 1.

Equilibrium Shocks to Interest Rate
Show a change in interest rate on bonds from 0.07 to 0.1. What effect does this have on the u-vh space?

The second diagram illustrates the change in the value of the capital output ratio that has occurred as a result of increasing the interest rate on bonds. The diagram illustrates an initial increased dip in the curve but then proceeds to increase at a steady rate to finish at a much higher value than before the interest rate shock.

Before shock

After shock

Similarly the value of the household decreases slightly but then increases at a rapid pace with the final value at 11, in comparison to the pre shock value of less than 4.5. This shock has had a very positive impact on the value of the household.

Before shock


After shock


Reset the model. Show a change in the value of α from 0.3 to 0.7

The reaction to the value of the capital output ratio to the change in alpha is initially good as it jumps to 1. However, it then falls and steadies out just above 0.9 which is less than the value prior to the shock.

Before shock


After shock


The after shock diagram illustrates how value of the household experiences a massive decline and is deemed almost worthless after the change in alpha.

Before shock



After shock