Homework

Q1 (1)

SIM
• Households know the multiplier process and the parameters of the economy – perfect knowledge.
• Wealth is the equilibrium mechanism.
• Quick convergence rate
• Consumption function: Cd = α1*YD + α2*H-1
SIMEX
• Slow convergence rate – delayed expectations, it takes longer to approach to a steady state solution.
• Introduces imperfect knowledge/uncertainty into the model.
• Households must estimate the income they will receive and the amount of money they wish to hold.
• Households are assumed to have mistaken expectations – if realized income > expected income they will hold the difference in the form of larger cash balances.
• Role of money is the equilibrium mechanism. Money acts as a buffer and provides flexibility when expectations turn out to be incorrect.
• Consumption function: Cd = α1*YDe + α2*H-1

Despite expectations both models achieve the same stationary state.

(2)

Steady state: Y*=G/ Θ
If households’ realized income is higher than expected (W.Nse – W.Nd), households will hold the difference in larger than expected cash balances ΔHs – ΔHd. They will give up more money in the form of taxes (Td – Tse).

Expectations about income will remain unchanged before and after a shock. If actual income is higher than expected, the increase in wealth will be higher than anticipated which will make consumption grow.

In the model, if expected income is lower than realized income, the stock of wealth will grow until the consumption lost through mistaken expectations about income equals the additional consumption out of wealth. Conversely, if expected income is less than realized income, the stock of wealth will fall.

In the real world, people have mistaken expectations and there is a high level of uncertainty particularity in times of recession where people may consume less and save more. When the economy begins to recover, people may then start to consume more than they save. This demonstrates the flexibility people have with their income.


(3)

In period one there is no economic activity and none has existed. The government has injected no money into the economy and households have no income.

In period two the government spends $30, which initiates the economy and circulates within the system.
• Producers pay households 30units of cash
• Households pay 20% in taxes (6 units => YDe=24)
• Households consumption: C = α1.YDe + α2.H-1 (0.6)(24)+(0.4)(0) = 14.4

In period three the government has not injected any extra money into the economy, therefore government expenditure remains at $30.
• Households consumption: C = α1.YDe + α2.H-1 (0.6)(35.52)+(0.4)(21.12) = 30
• Expected income = realized income YDe = YD-1

In the infinite period
• GDP (Y) = G/ Θ = 30/0.2 = 150 Fiscal stance: the stationary level of income
• Taxes 150*0.2 = 30

Question 2.

It is possible to specify a version of SIM that replicates the ISLM model


The IS Curve – Saving & Investment (Goods Market)
Examining the IS curve with the SIM model in mind, we make the assumption that supply must always equal demand. Therefore, the below income equation will fit for supply and demand.

Y = C(Y-T) + I(Y, i) + G

An increase in the supply, Y, would indicate that there is and excess supply of goods. To return this equation to equilibrium, an opposite reaction would have to take place such as a fall in interest rates, i.
The fall in interest rates would be used to encourage a rise in consumption as high interest rates encourage investors to save money due to the high expense of loans or the advantages of investing in bonds.
This relationship is best displayed in the IS curve graph diagram. It can be seen in the graph that as the interest rate (i2) is increased, the output (Y2) decreases i.e. output is a decreasing function of the interest rate.

Given the above equation, and a set interest rate, we can establish that changes in consumption and taxes will shift the IS curve to the left or the right to remain in equilibrium. This is evidenced in the diagram for an increase in taxes.



Increase in Taxes: IS shift left
Decrease in Taxes: IS shift right
Increase in Consumption: IS shift right
Decrease in Consumption: IS shift left

Y: Income
C(Y-T): Consumer spending as a function of disposable income
I(Y, i): Investment as a function of real interest rate
G: Government spending


The LM Curve – Financial Markets
The below equation fits for money supply equals money demand in the SIM model.

M/P = YL (i)

An increase in Y would lead to an increase in the demand for money by households. To maintain equilibrium, it would be necessary for an increase in interest rates to bring the level of demand back to that of supply. This is best illustrated in the diagram of the LM curve where the interest rates are raised to i2 to maintain equilibrium.

Additionally, if there was an increase in the money supply, a cut of interest rates would be required. This would cause the LM curve to shift down.

Increase in Money Supply: Cut in interest rates & shift down of LM curve
Decrease in Money Supply: Increase in interest rates & shift up of LM curve.

Y: Real income
P: Price level
I: Interest rate

The IS-LM Model and some function in SIM

For the IS-LM model, we take the two graphs and combine them. The point at where they meet is the point at which both models achieve equilibrium. The basic theory behind this model is that, as a factor of either model changes, say the IS model for example, the IS model will move up or down along the axis of the LM model to maintain equilibrium.

Td = θ. W.Ns θ<1

From the previous diagram, we can see what happens as taxes are increased. Disposable income is affected which leads to and affect on output, thus the IS curve shifts to the left. As there is no tax function in the LM model, the model is not affected.

However, the change in taxes will have caused a change (decrease) in the interest
rates, as a result of the decrease in income (Y), so the IS model will move along the LM curve to equilibrium. Similar results would show for changes in money supply and consumption.

Stability

We can see from this example that a change in one factor has a direct affect on the IS model to which it is a function of and an indirect impact on the LM model. This shows how difficult it is to use just one policy to have the desired affect on the overall economy (and of course SIM).

Td :Tax demand
W.Nd :Wage rate times employment demand
Θ: proportion of change

References
Macroeconomics, Blanchard. Chapter 5
Monetary Economics, Godley & Lavoie. Chapter 3
Wikipedia
Egiwald
Roubini & Backus
MIT (.pdf)