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 nominal interest - inflation = real interest rate (r). 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.
dY/dK = A * 0.3 * (1/K^0.7) * L^0.7 = A * 0.3 *( K^0.3/K) * L^0.7 = 0.3 Y / K
Likewise
dY/dL = 0.7 * Y/ L

Due to inverse relationship, marginal production is lower as we add more Labor (L) or Capital (K). Which matches with real world property, adding 10 people on the same lawn mover is not going to improve productivity much. Quantity Theory of Money Y * P = M * V , M is the money and V is the number of times a dollar circulates which is a constant in the long run. Remember d(ab) = da + db for small changes. dY + dP = dM + dV Or Inflation (pi) = dP = dM + dV -dY per Quantity theory dV = 0 so pi = dM - dY This does not work and dV is not zero, and the effects showed up prominently in 1970

Keynesian Theory

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

The IS Curve

Keynes explained Great Depression by 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 primarily real interest rate (r) and other params. Because we can control r by measuring inflation (pi) and setting nominal rate (i). i = pi + r
Since our model is primarily relating to r, we split the terms into components not related to r (exogenous) and a function of r. Our analysis will consider impacts of exogenous components as shifts in the curve. This is basically changing the constant b in the standard line equation (y=mx+b). Exogenous terms are marked with _bar
C = C_bar + mpc * (Y - T) - cr
mpc = marginal propensity to consume,
(Y-T) = disposable income available to private spenders,
c = a constant
r = real interest rate, cr models the effect or real interest rate on consumption. We will use similar model for other terms.
Intuition if r is high it reduces consumption

I = I_bar - dr_i
r_i = real interest rate available for investment, due to financial friction f_bar.
r_i = r + f_bar
Intuition if r_i is high it reduces investment.
I = I_bar -d(r + f_bar)
NX = NX_bar - xr
Intuition: higher r will increase USD exchange rate because it offers higher interest than before. Higher exchange rate lower net export because our goods are more expensive than before!
Keynes observes equilibrium occurs when Y = Y_pe
By adding up the models and moving the Y term from C to the left we get the IS curve equation. It is called IS curve because it essentially captures the real interest determined by Investment , Saving equilibrium.

The IS curve:
Y = [ C_bar + I_bar - d*f_bar + G_bar + NX_bar - mpc*T_bar ]/(1 - mpc) - (c+d+x)/(1-mpc) * r
Think of it as
Y = some_constant - some_slope*r
Key points compare to line equation (y=mx+b),
the constant b is made up of several exogenous components which are important as well, notably G_bar which is a controllable factor. C_bar, mpc, and I_bar are affected by overall sentiment. The line equation tells us how the line shifts with changing constant.
The slope m = -(c+d+x)/(1-mpc) is negative, refer to the modeling, you can see the parameters are +ve. So it is a downward sloping curve.
Macro is highly contextual and it is important to be aware that r is usually below 10%, do not extrapolate and always double check conclusions because all models are wrong. But we will see that quite a bit of analysis can be done with the simple models.
IS equation is an equilibrium curve points away from the curve shift to the line.

Looking at a closed economy, i.e. NX=0 we can see that I and S=Y-T-C (savings) are both functions of r (the real interest rate). This shows that real interest rate is a function of Investment and Savings in an economy at least in the long run but with prices and wages taking time to catch up in the short run Fed can manipulate r by measuring the current inflation.

Further IS curve above is combined with the Monetary Policy r = r_bar + lambda*pi, how Fed thinks about setting r. This will convert IS curve to be a function of pi. So in practice we use pi (inflation) vs Y (output).

Monetary Policy Curve

Fed’s monetary policy is to increase real interest rate (r) (indirectly by measuring inflation (pi) and setting nominal interest (i), i = r + pi) because if pi rises and the Fed does not act inflation spirals up.
The spiraling process is explained by Taylor principle
pi increase -> r decreases (r = i - pi) -> Y increases (IS curve) -> pi increases (if Y>Y_p prices rise, Y increase results in cost increase, demand for labor increases costs) -> r decrease -> Y increase -> pi increase.
This causes inflation spiral, and high inflation is detrimental to economy. Fed will act to keep r stable by rising i.
This is expressed as r = r_bar + lambda * pi
Fed can raise r_bar as well for other reasons as well. These are autonomous r_bar changes. Change with pi (lambda * pi) is automatic change, literally it can be left as automatic change as we measure inflation.

Short Run Aggregate Supply Curve

Phillips law 1958, (UK 1861-1957) and modern law:
Empirical observation 1958. U -ve correlation to wages, later developed to substitute wages to inflation (pi)
Shortage of workers -> higher wages, due to costs -> higher prices -> higher inflation
Friedman and Phelps (1967,1968) (Nobel) point out workers and firms care of real wages! Wages and inflation will rise 1:1 with expected inflation also introduced natural Unemployment (U_n)
pi = pi_e - omega*(U - U_n)
Since it is only true in Short Run, we need analysis for shifts in the Long Run
If U < U_n it increases pi , it leads to higher pi_e and the curve shifts up and higher pi. So inflation and expected inflation feed on each other.
Inflation spiral stops when U = U_n (why does this settle?)
Okun’s law:
strong correlation observed between Y and U: (U - U_n) = -0.5*(Y - Y_p)
Aggregate Supply Curve
Short Run Aggregate Supply (SRAS) curve
Combine Philips + Okun
pi = pi_e + omega*(0.5*(Y - Y_p))
pi = pi_e + gamma*(Y - Y_p) ; gamma = omega*0.5 (actually 0.5 comes from correlation)
However this is true only for Short Run, Long Run Aggregate Supply, LRAS = AK^alpha L^(1-alpha) the Cobb Douglas function. It is not impacted by pi but it grows steadily over time due to factors.
Just like Phillips curve SRAS has inflation spiral, remember increase in pi causes increase in pi_e. This stops when output gap (Y-Y_p) gets to zero.

The Aggregate Demand and Supply Model

Short Run
Supply = Demand (planned expenditure) so Y_s = Y_pe
Y_s from SRAS = Y_pe from IS curve
pi = pi_e + gamma*(Y_s - Y_p) ; Y_p is Y potential modeled by Cobb Douglas
Y_pe = some_constant - some_slope *r
but r = r_bar + lambda*pi from monetary policy
So Y_pe = some_constant - some_slope*pi (now since we clubbed r_bar into a constant, we primarily dealing with automatic changes to r by Fed)
Since Y_pe = Y_s we set Y_pe = Y_s = Y* at eq and substitute for pi into
Y = some_constant - some_slope*(pi_e + gamma*(Y - Y_p))
We can set pi_e = pi_-1 (previous inflation), though not always true, and solve for pi. We get Y_p by fitting Cobb-Douglas to history.
Now we can solve for the equilibrium Y* and get pi* as well.

Links

One of the reasons Quantity Theory does not apply is due to the stickiness effect of prices. However during pandemic inflation became a common man's term and was widely used, generating several variations such as shrinkflation, greedflation. Social media and the internet also propagated information and memes in particular quite fast working against stickiness. During the post pandemic period, businesses could easily raise prices without any negative sentiment from their customers, this is quite unusual and works against stickiness. Here are articles that speak about revival of Quantity Theory of Money.
Sep 2022 article about Qunatity Theory
July 2023 article about Quantity Theory

In the short run inflation pi = pi_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