### Thompson’s Reformulation of Macroeconomic Theory, Part III: Solving the FF-LM Model

In my two previous installments on Earl Thompson’s reformulation of macroeconomic theory (here and here), I have described the paradigm shift from the Keynesian model to Thompson’s reformulation — the explicit modeling of the second factor of production needed to account for a declining marginal product of labor, and the substitution of a factor-market equilibrium condition for equality between savings and investment to solve the model. I have also explained how the Hicksian concept of temporary equilibrium could be used to reconcile market clearing with involuntary Keynesian unemployment by way of incorrect expectations of future wages by workers occasioned by incorrect expectations of the current (unobservable) price level.

In this installment I provide details of how Thompson solved his macroeconomic model in terms of equilibrium in two factor markets instead of equality between savings and investment. The model consists of four markets: a market for output (C – a capital/consumption good), labor (L), capital services (K), and money (M). Each market has its own price: the price of output is P; the price of labor services is W; the price of capital services is R; the price of money, which serves as numeraire, is unity. Walras’s Law allows exclusion of one of these markets, and in the neoclassical spirit of the model, the excluded market is the one for output, i.e., the market characterized by the Keynesian expenditure functions. The model is solved by setting three excess demand functions equal to zero: the excess demand for capital services, XK, the excess demand for labor services, XL, and the excess demand for money, XM. The excess demands all depend on W, P, and R, so the solution determines an equilibrium wage rate, an equilibrium rental rate for capital services, and an equilibrium price level for output.

In contrast, the standard Keynesian model includes a bond market instead of a market for capital services. The excluded market is the bond market, with equilibrium determined by setting the excess demands for labor services, for output, and for money equal to zero. The market for output is analyzed in terms of the Keynesian expenditure functions for household consumption and business investment, reflected in the savings-equals-investment equilibrium condition.

Thompson’s model is solved by applying the simple logic of the neoclassical theory of production, without reliance on the Keynesian speculations about household and business spending functions. Given perfect competition, and an aggregate production function, F(K, L), with the standard positive first derivatives and negative second derivatives, the excess demand for capital services can be represented by the condition that the rental rate for capital equal the value of the marginal product of capital (MPK) given the fixed endowment of capital, K*, inherited from the last period, i.e.,

R = P times MPK.

The excess demand for labor can similarly be represented by the condition that the reservation wage at which workers are willing to accept employment equals the value of the marginal product of labor given the inherited stock of capital K*. As I explained in the previous installment, this condition allows for the possibility of Keynesian involuntary unemployment when wage expectations by workers are overly optimistic.

The market rate of interest, r, satisfies the following version of the Fisher equation:

r = R/P + (Pe – P)/P), where Pe is the expected price level in the next period.

Because K* is assumed to be fully employed with a positive marginal product, a given value of P determines a unique corresponding equilibrium value of L, the supply of labor services being upward-sloping, but relatively elastic with respect to the nominal wage for given wage expectations by workers. That value of L in turn determines an equilibrium value of R for the given value of P. If we assume that inflation expectations are constant (i.e., that Pe varies in proportion to P), then a given value of P must correspond to a unique value of r. Because simultaneous equilibrium in the markets for capital services and labor services can be represented by unique combinations of P and r, a factor-market equilibrium condition can be represented by a locus of points labeled the FF curve in Figure 1 below.

The FF curve must be upward-sloping, because a linear homogenous production function of two scarce factors (i.e., doubling inputs always doubles output) displaying diminishing marginal products in both factors implies that the factors are complementary (i.e., adding more of one factor increases the marginal productivity of the other factor). Because an increase in P increases employment, the marginal product of capital increases, owing to complementarity between the factors, implying that R must increase by more than P. An increase in the price level, P, is therefore associated with an increase in the market interest rate r.

Beyond the positive slope of the FF curve, Thompson makes a further argument about the position of the FF curve, trying to establish that the FF curve must intersect the horizontal (P) axis at a positive price level as the nominal interest rate goes to 0. The point of establishing that the FF curve intersects the horizontal axis at a positive value of r is to set up a further argument about the stability of the model’s equilibrium. I find that argument problematic. But discussion of stability issues are better left for a future post.

Corresponding to the FF curve, it is straightforward to derive another curve, closely analogous to the Keynesian LM curve, with which to complete a graphical solution of the model. The two LM curves are not the same, Thompson’s LM curve being constructed in terms of the nominal interest rate and the price level rather than in terms of nominal interest rate and nominal income, as is the Keynesian LM curve. The switch in axes allows Thompson to construct two versions of his LM curve. In the conventional case, a fixed nominal quantity of non-interest-bearing money being determined exogenously by the monetary authority, increasing price levels imply a corresponding increase in the nominal demand for money. Thus, with a fixed nominal quantity of money, as the price level rises the nominal interest rate must rise to reduce the quantity of money demanded to match the nominal quantity exogenously determined. This version of the LM curve is shown in Figure 2.

A second version of the LM curve can be constructed corresponding to Thompson’s characterization of the classical model of a competitively supplied interest-bearing money supply convertible into commodities at a fixed exchange rate (i.e., a gold standard except that with only one output money is convertible into output in general not one of many commodities). The quantity of money competitively supplied by the banking system would equal the quantity of money demanded at the price level determined by convertibility between money and output. Because money in the classical model pays competitive interest, changes in the nominal rate of interest do not affect the quantity of money demanded. Thus, the LM curve in the classical case is a vertical line corresponding to the price level determined by the convertibility of money into output. The classical LM curve is shown in Figure 3.

The full solution of the model (in the conventional case) is represented graphically by the intersection of the FF curve with the LM curve in Figure 4.

Note that by applying Walras’s Law, one could draw a CC curve representing equilibrium in the market for commodities (an analogue to the Keynesian IS curve) in the space between the FF and the LM curves and intersecting the two curves precisely at their point of intersection. Thus, Thompson’s reformulation supports Nick Rowe’s conjecture that the IS curve, contrary to the usual derivation, is really upward-sloping.

#### 10 Responses to “Thompson’s Reformulation of Macroeconomic Theory, Part III: Solving the FF-LM Model”

1. August 14, 2012 at 8:01 pm

David: let me just check to see if my intuition matches yours:

Capital is always fully employed in this model, because there is no opportunity cost to using the services of existing capital goods, so the supply curve of capital goods services (at a given time) is vertical.

But labour services has an opportunity cost, so the labour supply curve slopes up. And if workers misperceive the current price level, unexpectedly tight monetary policy will cause employment of labour to fall, employment of capital to stay the same, the MPK to fall, so the real interest rate falls too. Hence the IS curve slopes up, not down as it does in the standard ISLM.

The interest paid on demand deposits is always (say) 2 percentage points below the interest paid on capital, because there are costs of providing banking services equal to that same (say) 2 percentage points, and it’s a competitive banking system. So the LM curve is vertical, because the opportunity cost of holding demand deposits rather than capital is a fixed (say) 2%, and is independent of the rate of interest.

OK, it’s making sense so far.

How does the central bank shift the LM curve right and left? (Maybe I should re-read your earlier posts).

Walras’ Law makes me very queasy in a monetary exchange economy, but that’s an argument for another day. Maybe it doesn’t actually matter here anyway, since we always have market clearing?

I really don’t like that Lucas/Rapping/Friedman labour supply function. But that’s another story too.

2. August 15, 2012 at 12:03 pm

Nick, You have admirably summarized the basic logic of the model.

Remember there are two cases relevant to the LM curve. In the competitive (classical) case that you just described, the central bank is out of a job. Money is convertible into output at a fixed conversion rate, so the postion of the LM curve depends on just one thing: the choice of the conversion rate. There’s a problem with that only when there are two outputs and money is convertible into just one of the outputs so that a change in the relative price of the outputs can affect the price level. In the conventional (Keynesian) case, however, the central bank determines exogenously the nominal quantity of money and the nominal quantity of money determines the postion of the upward-sloping LM curve.

Yes, there is always market-clearing, but the labor market may be inefficient when price-level and wage expectations are incorrect. I agree that the labor supply function is not all that we would like. Armen Alchian always raved about Donald Gordon’s paper “A Neoclassical Theory of Keynesian Unemployment” in which firms keep wages rigid as a kind of implicit contract with workers, who consider any wage-cutting opportunistic by the employer. Employers don’t cut wages because if they did, new workers would stop coming to work for them because the employer would have a reputation for taking unfair advantage of them. What do you think of that one?

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