This is for equation reference only, for example, when you’re lost as hell during lectures! I will make real cheat sheets that explain what things are.
Calculating GDP
\[Y = C_a + I_a + G_a + NX_a\]where
- \(Y\) is GDP
- \(C_a\) is actual consumption
- \(I_a\) is actual investment, which is equal to gross investment less depreciation
- \(G_a\) is actual government expenditure
- \(NX_a = X_a - IM_a\) is actual net exports
- \(X_a\) is actual exports
- \(IM_a\) is actual imports
Simple Short-Run Model
Equilibrium GDP \(Y_e\) occurs when \(Y_e = AE\), the aggregate expenditure.
We have a few basic metrics:
- The autonomous spending \(A\) is \(AE\) when \(Y = 0\)
- \(\frac{\Delta Y}{\Delta AE} = z\) is the marginal propensity to spend
- \(\frac{\Delta A}{\Delta Y_e} = \frac{1}{1-z}\) is the simple multiplier
All our models will be linear, that is, \(AE = A + zY\). This should make it easy to identify \(A\) and \(z\). The term with no \(Y\)-dependency is called autonomous and with is called induced.
Assuming linearity also gives us the really nice equation \(Y_e = \frac{A}{1-z}\).
This also gives a relation for change in marginal propensity to spend: \(Y_{e_2} = \frac{1-z_1}{1-z_2}Y_{e_1}\).
Super Simple Model
In our simple model, we ignore government and trade, so that we can say
\(\begin{align} AE & = C + I \\ & = \underbrace{(a + I_0)}_{A} + \underbrace{(b)}_zY \end{align}\)
where
- \(C = a + bY\) is desired consumption
- \(a\) is autonomous consumption
- \(b\) is the marginal propensity to consume. The opposite (marginal propensity to save) is \(1-b\). The savings function is \(-A + (1-b)Y\).
- \(I = I_0\) is desired investment, which we set to a constant
Adding Government
We now add government expenditure.
\(\begin{align} AE & = C + I + G \\ & = \underbrace{(a + I_0 - bT_0 + G_0)}_{A} + \underbrace{b(1-t)}_{z}Y \end{align}\)
where the only changes from above are that
- \(C = a + bY_D\)
- \(Y_D = Y - T\) is disposable income
- \(T = T_0 + tY\) is net tax revenue
- \(T_0\) is autonomous tax that doesn’t depend on GDP, including transfers like CERB
- \(t\) is the net tax rate of all taxes net of subsidies
- \(T = T_0 + tY\) is net tax revenue
- \(Y_D = Y - T\) is disposable income
- \(G = G_0\) is desired government expenditure, which we set to a constant
Adding Trade
Adding the most annoying thing of all: the rest of the world.
\(\begin{align} AE & = C + I + G + NX \\ & = \underbrace{(a + I_0 - bT_0 + G_0 + X_0)}_{A} + \underbrace{(b(1-t) - m)}_{z}Y \end{align}\)
where the variables are the same except
- \(NX = X - IM\) is desired net exports
- \(X = X_0\) is desired exports, which we set to a constant
- \(IM = mY\) is desired imports
- \(m\) is the marginal propensity to import
Government Policy
The budget balance is \(T - G\) where \(T\) is net tax revenue and \(G\) is government expenditure.