# Mankiw Chapter 5 and Price-Specie Mechanism

I've recently been reading through Mankiw's Macroeconomics, 7th edition.

In Chapters 3 & 4 describe the Classical theory, which pretty much exactly lines up with the model that Keynes outlined in Book One of the General Theory (although Keynes didn't explicitly disaggregate the government sector).

##### Equilibrium in the market for goods and services: the supply and demand for the economy's output

$$\begin{array}{l} Y = C + I + G \\ \\ C = C(Y - T) \\ \\ Y = F(\bar{L},\bar{K}) = \bar{Y} \\ \\ \bar{Y} = C(\bar{Y} - \bar{T}) + I(r) + \bar{G} \\ \end{array}$$

##### Equilibrium in the financial markets: the supply and demand for loanable funds

$$\begin{array}{l} I = Y - C - G \\ \\ S = (Y - T - C) + (T - G) = I \\ \\ Y - C(Y - T) - G = I(r) \\ \\ \bar{Y} - C(\bar{Y}-\bar{T} - \bar{G}) = I(r) \\ \\ \bar{S} = I(r) \end{array}$$

In Chapter 5, he introduces an open economy:

Let $$r*$$ be the world interest rate.

$$\begin{array}{l} r = r* \\ \\ \end{array}$$

The supply side of the model is no different to how it is in chapter 3:

\begin{align} Y = F(\bar{L},\bar{K}) = \bar{Y} \\ \end{align}

The demand side of the equation now introduces net exports:

$$\begin{array}{l} Y = C + I + G + NX\\ \\ \bar{Y} = C(\bar{Y} - \bar{T}) + \bar{I}(r*) + \bar{G} + NX(\epsilon) \\ \end{array}$$

The primary channel that Mankiw uses to assess policy impacts is through the trade balance:

$$\begin{array}{l} NX = Y - C - I \\ NX = \bar{S} - I(r*) \end{array}$$

Policies that increase or decrease savings will increase or decrease net exports.

"The impact of any policy on the trade balance can be determined by examining its impact on saving and investment."

I'm trying to reconcile this in my mind with the pre-Keynes classical theory, and this means thinking about how the price-specie mechanism would fit into Mankiw's model. (Chapter 12 will introduce the IS-LM-BP model, or the Mundell-Fleming model, which is the modern day version of the price specie mechanism, but that's a later theoretical development).

If $$NX(\epsilon) \ge 0$$ then capital outflows will cause a increase in the money supply as gold flows in the opposite direction into the economy (under a Gold Standard).

$$\begin{array}{l} \uparrow{M}\bar{V} = \uparrow{P}\bar{Y} \\ \\ \Rightarrow \uparrow\epsilon = e.\frac{\uparrow{P}}{P*} \\ \end{array}$$ Given that $$NX$$ is a downward sloping function of $$\epsilon$$, this would cause $$\downarrow{NX(\epsilon)}$$. However, within Mankiw's system, where savings or investment lead net exports, this would require either a fall in savings or an increase in investment. In Mankiw's open economy model where $$r*$$ is fixed, it's not clear how this would mechanically happen.

If we assume a large enough economy whereby the export of savings affects the world interest rate, then $$\downarrow{r*}$$ leads to $$\uparrow{I(r*)}$$ which would close out our system, but this feels unsatisfying. Is there something I'm missing?

If you want to integrate the $$MV=PY$$ you actually do have to go to modified version of IS-LM model, since $$MV=PY$$ is the equilibrium on money market (in fact the New Keynesian LM curve, can be thought as a direct analogue to QTM equation you can see it when you solve both for price level; $$QTM: P= \frac{MV}{Y}$$, $$NKLM: P = \frac{M}{L(r,Y)}$$. When $$L(r,Y) = \frac{Y}{V(r)}$$ you get a special case that is QTM, although NKLM typically does not talk about V, but that is just terminological difference).
Hence, if you want to integrate the 'price specie mechanism' you can do that by making velocity function of interest rate $$V(r^*)$$, and then solve the system same way as you would solve the IS-LM model. Although, it will violate the classical assumption of $$V$$ being constant, that small alteration is necessary unless you want to do some modifications on IS side.
If you don't want to expand the model, then there is no non ad hoc way how to do it since these two models are not necessarily consistent with each other. You have to allow $$S$$ or $$Y$$, or $$V$$ to adjust (or $$r$$ like in the big country case), if you fix these you will have inconsistent model, and without explicitly going the extra mile to properly integrate the QTM you won't get any satisfying mechanism for adjustments that provides insight, things will simply change because there is nothing else that could change. If you integrate it using $$V(r)$$ you might get actually some satisfying mechanisms but you need to abandon the assumption that $$V$$ is fixed.