## Wednesday, February 19, 2014

### Identities Can Be Deceiving

Many an economist may be heard complaining that accounting identities are not models. And about this, many an economist is correct. Unfortunately, this sad refrain bears repeating. Accounting identities are not models. In fact, they can be downright misleading. Take for example, the basic national accounting identity defining GDP: $$Y=C+I+G+X-M$$ Clearly, imports $M$ count against GDP. But does an increase in imports lower GDP? It looks like imports reduce GDP, but the equation does not tell us this. To see why this might be so, let us divide expenditures into domestic production and foreign imports. That is, $C=C_d+C_m$, $I=I_d+I_m$, $G=G_d+G_m$, $X=X_d+X_m$, and finally $M=C_m+I_m+G_m+X_m$. Then $$Y=C_d+I_d+G_d+X_d$$ It appears that imports do not enter into GDP at all. Yet it is no less correct to write $$Y=\left(C-C_m\right)+\left(I-I_m\right)+\left(G-G_m\right)+\left(X-X_m\right)$$ which again suggests that imports reduce GDP one-for-one. Which equation is correct? They all are. They all provide exactly the same information, yet invite the reader to different interpretations. Equation (2) invites the reader to believe that $C_d$ is independent of $C_m$ (which may or may not be true.) Equation (3) invites the reader to believe that $C$ is independent of $C_m$ (which also may or may not be true.)

Rather, we require a model to tell us how the various parts move. For example, we might say $G$ and $X$ are fixed, but an additional dollar of $M$ increases $C$ by \$1.50 and reduces investment by \$1.00. The accounting identity would then tell us GDP falls by 50 cents for every dollar of additional imports. Is it true? The result depends on the model, and the model need not be reasonable. Suppose instead that an additional dollar of imports leads to a million dollars of additional consumption. Garbage in, garbage out. But the identity must hold.

As we will see in a future post, identities get more deceptive when used in combination with other identities.