Macro Summary

All models are wrong, some are useful. So much more true about Macroeconomics. Macro can’t be just a bunch of equations, it is highly contextual and talks about mainly one economy we have, the actual numbers, their magnitudes and the history which provides evidence are just as important for the beginners semi-empirical model.

This summary tries to anchor on algebra.

The tools with Fed are Monetary Base (or Money supply) which is more of a classical method, nominal interest rates (\(i\)), and measured value of inflation (\(\pi\)) because the real tool is \( \text{real interest }(r) = i - \pi.\) Modern policy primarily uses \(r\).

The goal of this summary is to call out empirical rules, intuitive linear models, and separate algebraic facts. Analysis can follow Algebra.

Classical Theory

Before 1930s Macro went with a simplistic theory that basically said if we doubled the money in the economy all prices and wages will adjust to 2x fairly quickly. This is called classical dichotomy, the use of money does not affect the real barter of goods and services, they are worth the same irrespective of the what money we use. Think iPhone cost in Yen vs USD, money is just a tool. This theory is called Quantity Theory of Money.

Real GDP = \(Y = C + I + G + NX\) = Consumption + Investment + Govt spending/investment(but mostly spending) + Net Exports is a simple identity but we will do some useful manipulations.

The output \(Y\) is a function of Technology \((A)\), Capital \((K)\), and Labor \((L)\) which is modeled by Cobb Douglas and it gives us useful insights and has some properties that match the economy.

\[ Y = A \, K^{0.3} \, L^{0.7} \] more generally replace \(0.3\) with \(\alpha\) and \(0.7\) with \(1-\alpha.\)
\[ \frac{\partial Y}{\partial K} = A \times 0.3 \times K^{-0.7} \times L^{0.7} = 0.3 \frac{Y}{K}, \quad \frac{\partial Y}{\partial L} = 0.7 \frac{Y}{L}. \]

Due to diminishing returns, marginal product is lower as we add more Labor (\(L\)) or Capital (\(K\)), matching the real-world observation that adding too many workers to the same machine yields less and less gain.

Quantity Theory of Money
\[ Y \times P = M \times V \] \(M\) is the money supply and \(V\) is the velocity (how many times a dollar changes hands) which is assumed constant in the long run. Using differential/log form, remember \(d(ab) = da + db\) for small changes. \[ dY + dP = dM + dV \] or \[ \pi = dP = dM + dV - dY. \] Per Quantity theory, if \(dV=0\) then \[ \pi = dM - dY. \] Historically, this does not always hold because \(dV\) is not zero and the effects showed up prominently in the 1970s.

Keynesian Theory

Classical theory over-simplified and assumed prices and wages quickly adjust to changing conditions, which is not always true. The Keynesian theory provides analysis for the Short Run.

The IS Curve

Keynes explained the Great Depression by noting that short-run changes are determined by Aggregate Demand, or Planned Expenditure \(Y_{pe}\): \[ Y_{pe} = C + I + G + NX. \] We model \(Y\) by modeling each term with a dependence on the real interest rate \(r\) and other parameters. Recall \(r = i - \pi\).

Consumption model:
\[ C = \bar{C} + mpc \cdot (Y - T) - c \, r. \] Here, \(\bar{C}\) is an exogenous constant, \(mpc\) is marginal propensity to consume, and \((Y-T)\) is disposable income. The term \(-c \, r\) captures that higher \(r\) reduces consumption.

Investment model:
\[ I = \bar{I} - d \, r_i, \] where \(r_i = r + \bar{f}\) is the real interest rate available for investment. If \(r_i\) is high, investment is lower.

Net exports model:
\[ NX = \overline{NX} - x \, r. \] Higher \(r\) leads to a stronger currency, making exports less competitive, hence reducing net exports.

In equilibrium, \(Y = Y_{pe}.\) Summing up the components and rearranging gives the IS curve: \[ Y = \frac{\bar{C} + \bar{I} - d \,\bar{f} + \bar{G} + \overline{NX} - mpc \,\bar{T}}{1 - mpc} \;-\; \frac{(c + d + x)}{1 - mpc} \; r. \] This is linear in \(r\). The slope is negative because higher \(r\) lowers \(Y\).

Monetary Policy Curve

The Fed’s monetary policy often follows the Taylor principle, adjusting the real interest rate \(r\) in response to inflation \(\pi\). A simple expression is: \[ r = \bar{r} + \lambda \,\pi. \] If \(\pi\) rises and the Fed does not increase \(i\) enough, then \(r = i - \pi\) falls, potentially causing an inflation spiral. Hence the Fed typically raises \(i\) to keep \(r\) from dropping.

Short Run Aggregate Supply Curve

Phillips Curve (short-run):
Empirically, when unemployment (\(U\)) is below the natural rate \(U_n\), wages (and hence inflation) tend to rise. A modern form: \[ \pi = \pi_e - \omega (U - U_n). \] Substituting Okun’s law \((U - U_n) = -0.5 \,(Y - Y_p)\), yields a short-run aggregate supply (SRAS): \[ \pi = \pi_e + \gamma \,(Y - Y_p), \] where \(\gamma\) is a positive constant.

The Aggregate Demand and Supply Model

In the short run, supply equals demand. The SRAS is: \[ \pi = \pi_e + \gamma (Y - Y_p). \] The aggregate demand curve comes from combining the IS curve with the monetary policy rule. Because \(r\) depends on \(\pi\), we can rewrite the IS curve in terms of \(\pi\). Solving simultaneously gives equilibrium \((Y^*, \pi^*)\).

Links

One reason the pure Quantity Theory can fail in the short run is price stickiness. However, during the pandemic, businesses were able to raise prices quickly (shrinkflation, greedflation, etc.), suggesting less “stickiness.” Some commentary suggesting a revival of the Quantity Theory:
Sep 2022 article about Quantity Theory
July 2023 article about Quantity Theory

In the short run, \(\pi = \pi_{\text{expected}} + \gamma (Y - Y_p)\). This WSJ article reports measurements of \(\pi\) with respect to expectations attributed to political views.

WSJ article discussing Phillips curve validity