Favorite Resurrection: Barro and Grossman

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Last week’s post called attention to a few of the multiple abandoned-model resurrections enabled by the GEM Project’s microfounding of MWR. My favorite is the recue from oblivion of the fixed-wage general-equilibrium (FWGE) model class that burst on to the macro scene in the 1970s

The principal contribution to the FWGE literature is Barro-Grossman (1971, 1976), in which they posit nominal wage rigidity in order to investigate the relationship between aggregate demand and involuntary job loss that results from the interdependence of rationing in the labor and goods markets. Such analysis must be general; partial-equilibrium labor-market analysis is inherently inadequate to the task. Barro-Grossman identify several regimes, the applicability of each depending on which markets are experiencing excess demand or supply. GEM Project modeling is most compatible with their Keynesian regime, in which the labor market exhibits excess supply.

Resurrected analysis.  The generalization of exchange powerfully enriches the original Barro-Grossman (1971) FWGE model. Adding rational workplace behavior to their single (marketplace) venue microfounds their crucial assumption of wage rigidity. The enhanced model variables are:

  • XJ and XK are the quantities of commodities produced in the large-establishment (LE) and small-establishment (SE) venues respectively;
  • HJ and HK are the quantities of labor services;
  • MJ and MK are the initial stock of nominal money balances held by households supplying labor to the LE and SE venues respectively;
  • mJ and mK are the increments to household money balances (in commodity units);
  • ΠJR and ΠKR are profits (in commodity units);
  • PJ and PK are money prices of real commodities produced;
  • P denotes the price index; and
  • WJR and WKR are wage rates (in commodity units) consistent with the WMS.

Labor is the only variable input in both venues’ production. Commodities (X) are consumable and the only form of current production; there is no investment. Money is the store of value, the medium of exchange, and unit of account; its quantity is exogenous and constant. Barro-Grossman further simplify their analysis by positing that households receive profits according to a predetermined distribution function. Firms maximize profits; households maximize utility.

In the small-establishment venue (SEV), market-price-taking firms understand profit maximi-zation to be constrained only by the production function. Firms can hire all the labor they desire and sell all the output they produce, implying the specification of real profits such that ΠKR=XKSWKRHKD, subject to the production function, XK=XK(HK), demonstrating positive, diminishing marginal productivity). The superscripts S and D denote, respectively, supply and demand quantities. Profit maximization implies XKS=XK(HKD) and HKD=HK(WKR), such that δXKHK=WKR

In the large-establishment venue (LEV), the familiar market-clearing conditions break down. Posit that current production (XJ) is equivalent to demand-determined sales and that XJ<XJS, where XJS becomes notional supply. The profit-maximization problem, as described by Patinkin (1956), is then restricted to selecting the minimum labor hours needed (HJD’) to produce XJ. Profit maximization implies that: HJD’=FJ-1(XJ), for δFJHJWJR. It follows from the constraint XJ<XJS that HJD’<HJD, with HJD’ approaching HJD as XJ approaches XJS. An effective-demand shock ΔHJD’<0 forces involuntary job loss. It also follows that the effective demand for LEV labor services (HJD’) can vary despite unchanged WJR. The market breakdown is Keynesian (i.e., involuntary job loss resulting from combining inadequate demand and wage rigidity) rather than classical (i.e., persistent unemployment wholly caused by excessively high real wages).

In the enriched Barro-Grossman model, the workplace venue produces a dominant labor-pricing optimizing decision-rule equilibrium, which constrains the marketplace venue’s subordinate decision-rule equilibrium. The two sectors are linked by Harris-Todaro labor transfer. (Chapter 3) If the acquisition of labor-market information is, more intuitively, permitted to follow a lagged learning process (especially useful in allowing job losers to engage in time-intensive price-discovery, calibrating their previous wage rents and better informing WƦ), joblessness consistent with continuous equilibrium is made even more congruent with the available evidence.

Consumption analysis.  Barro-Grossman use their model to support Clower’s interpretation of the Keynesian consumption function. In that insightful contribution, the relation between income and consumer spending is a manifestation of Walrasian disequilibrium in the labor market. Worker income, now representing the constrained effective demand for current output resulting from the excess supply of labor, critically influences rational household consumption-saving choice.

LEV job rationing has been demonstrated to be consistent with optimizing workplace exchange anchored by continuous general equilibrium. Households cannot sell all the labor services that they prefer (HJ<HJS) and, as a result, receive labor income WJRHJ<WJRHJS. From the employees’ perspective, hours at work are exogenously given, pushing households off their market supply curve and making effective supply inelastic.

The LEV household optimization problem is reduced to the rational disposition of constrained income WJRHJ+ΠJ, optimizing UJ(HJ,XJD’,(M/PJ+mJD’) subject to WJRHJ+Πj=XJD’+mJD’. Effective consumption demand is understood to be XJD’, while effective demand for more real balances is mJD. Maximizing UJ(∙) implies that XJD’=XJD’(WJRHJ+ΠJ,(M/P)J) and mJD’=mJD’(WjRHJ+ΠJ,(M/P)J). It follows that household consumption and saving ultimately depend on labor income and wealth. From Barro-Grossman (1971, p.88): “The important property of [the commodity and real-balances effective demands] is that they do have the form of the usual Keynesian consumption and saving functions. Labor income enters the consumption and saving functions as it represents the constraints upon the demand for current output imposed by the excess supply of labor.” Patinkin’s analysis is motivated by profit maximization subject to an output constraint, while Clower’s model maximizes utility subject to an employment constraint.

Rationed Jth-venue jobs also forces SEV employees to accept zero hours in high-wage jobs for which they are qualified, well below the quantity of higher-productivity work they prefer. Their optimization problem is similarly reduced to the rational disposition of constrained income, which becomes the central determinant of their consumption decision. Putting the pieces together, GEM-enriched FWGE modeling provides a coherent continuous-equilibrium interpretation of Keynesian consumption, the centerpiece of stabilization-relevant effective demand. It is a big deal.

Parting word. Barro-Grossman (1971, 1976) made one of the truly important contributions to stabilization-relevant macroeconomics. Around the time their book came out, however, they totally recanted their FWGE model. The authors’ rejection of great research is next week’s story.

Blog Type: Wonkish San Miguel de Allende, Mexico

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