9 Appendix

The appendix contains a list of equations and variables, and a handy algebraic result.

9.1 List of equations and variables

This is just a short list of variables that are used in the text. If I’ve forgotten any, let me know!

Linear aggregate consumption function $$C = a + b(Y-T)$$ Linear aggregate savings function: $$S = - a + (1-b)(Y-T)$$ MPC: $$b$$

MPS: $$(1-b)$$

Linear investment function: $$I = d - hr$$

In partial equilibrim:

Investment multiplier: $$\frac{\Delta Y}{\Delta I} = \frac{1}{1-b}$$

Government spending multiplier: $$\frac{\Delta Y}{\Delta G} = \frac{1}{1-b}$$ Tax multiplier: $$\frac{\Delta Y}{\Delta T} = \frac{-b}{1-b}$$ Central Bank Policy Rule (Taylor Rule’’) for nominal rates:

$$i^{Fed} = r^* + \pi^* + \alpha (Y - Y^{N}) + \beta (\pi - \pi^*)$$ Real interest rate (MP curve): $$r = i^{Fed} - \pi + \epsilon = r^* + \alpha (Y - Y^{N}) + (\beta - 1) (\pi - \pi^*) + \epsilon$$ AD curve equation (where IS = MP ): $$Y = \frac{1}{1 - b + h \alpha} \left[a + d - b T + G - h(r^* + \pi^*) + h \alpha Y^{N} - h \times (\beta - 1) \times (\pi - \pi^*) - h \epsilon \right]$$

Greek letters:

$\Delta$ - capital delta – generically, the change in a variable

$\alpha$ - lower case alpha – the weight on the output gap in the Taylor Rule

$\beta$ - lower case beta – weight on inflation gap in the Taylor rule.

$\epsilon$ - lower case epsilon – interest rate “shock” (deviation from the Taylor rule)

9.2 The “write-it-as-changes” trick

With this model, we’ll mainly work with systems of linear equations. If you need a refresher, you should check out sections 3 and 5 of the math review document.

This isn’t really a trick, but a fact about algebra:

Suppose that

$$X = q + m Y + n Z$$

And use $\Delta$ to indicate a change in a variable, e.g. $\Delta X$ is a change in $X$. Suppose $q$ is a constant. That means if $X$ is changing, it is because of some change in either $Y$ or $Z$. This implies

$$X + \Delta X = q + m (Y + \Delta Y) + n (Z + \Delta Z)$$

Notice that we can then write:

$$X + \Delta X - X= q + m (Y + \Delta Y) + n (Z + \Delta Z) - (q + m Y + n Z)$$

and hence

$$\Delta X = m \Delta Y + n \Delta Z$$

This is handy if we want to calculate how much $X$ changes with $Z$ changes, all else equal: just use the “changes” version of the expression with $\Delta Y = 0$.

If you’ve taken calculus, this may seem slightly familiar. We can think of our linear model as describing the linearized, approximate behavior of a more complicated model (e.g., using the tangent lines around some value to approximate the behavior of the potentially more complicated system). The way we calculate “changes” is by taking total differentials.