It is relevant not because economics discuss barter any longer but because they assume a barter-based economy, in which goods are traded for goods.
This is the basis of Say's law, which even Say came to recognize is not a "law" in a monetary economy. Say's law is the basis of the modern assumption of general equilibrium, that all markets clear in the long run.
Related to this is the assumption that money is a neutral veil so the monetary aspects of a modern economy don't really matter in the long run, although there can be short run effects.
Moreover, it is is at the foundation of Friedman's monetarism, which is based on Hume's analysis holding that at increase in the money supply will drive up the price level cet. par. Monetarism assumes cet. par, even though empirical evidence runs counter to it.
The neutrality of money view also underpins the assumption of conventional economics that finance can be ignored in econometric analysis. The banking system just intermediates between borrowers and savers, so the interest rate is determinative in economic activity. While it can be used as a lever by the monetary authority, the central bank is reactive rather than proactive, responding to markets rather than determining them.
Keynes's General Theory of Employment, Interest and Money is grounded in effective demand, which is income and consumption based. Neoclassical economics is grounded in production, hence investment based. Keynes agreed that production (supply) is investment-based and that investment is the driver of growth. But he pointed out that investment is consumption-driven and consumption is demand-driven.
Firms produce goods to sell, and when sales lag owing to lagging demand, they unplanned inventories rise and firms reduce quantity rather than price, as neoclassical economics wrongly assumes. Modern economies are "monetary production economies." they don't produce just to produce because they can. They produce in order to make a money profit. Marx had observed this as the difference between C-M-C' as classical economists assumed and M-C-M' as he theorized. See Wray, "Theories of Value and the Monetary Theory of Production" (Levy Working Paper No. 261).
The insight of Keynes was that lagging demand, that is, effective demand insufficient to purchase the quantity of goods that can be output using available resources efficiently, results from demand leakage to saving rather than being initiated by a market failure involving the factors of production, e.g., an exogenous shock. This vitiates Say's law "in the short run" as long as liquidity preference remains high enough to inhibit spending and there is no monetary offset. And "in the long run we are all dead." Waiting for the system to autocorrect is folly when the government sector can offset lagging demand by accommodating liquidity preference (saving desire) to return the economy to capacity and full employment "in the short run".
There are other reasons that the history of money is still important in the study of economics and finance, which are joined at the hip in a monetary economy. But just from the point of view of the fundamental assumptions of conventional economics — even Krugman and DeLong self-identity as neoclassical economics — the history of money is important in understanding the basis of fundamental neoclassical assumptions that run counter to the operation of a modern monetary economy, as well as the failure of neoclassical economists to correctly understand and incorporate finance and monetary operations in their models.
While may not be necessary to know how the erroneous assumptions were arrived at based on the assumption of money arising from barter and economic exchange being essentially barter, it accounts for what might otherwise be surprising. Why would intelligent people think that monetary operations, banking, and finance were irrelevant to economic analysis, miss a major crisis and still be mystified about policy to correct for it. The conclusion is that they are either morons, or are using the wrong model.
Keynesian, Post Keynesian and MMT analysis may not depend on the history of money, but the history of money illuminates the analysis.
It also shows the value of studying the history of money and theory of money, Neoclassical economics is a development of classical economics and also a response to issues previous raised by Smith and Ricardo in particular. The foundations for the neoclassical view of money were laid by the classical economists and Hume. Menger's analysis of money, that is, money as barter-based and gold as the basis of commodity money (numeraire), dominated early neoclassical thinking and influenced neoclassical assumptions.
Keynes apparently developed his views on the monetary theory of production from both Marx (see Wray above) and also Knapp's Chartalism or state money. Post Keynesian later developed understanding of the monetary circuit that corresponds to the circular flow of production-distribution-consumption that underlies neoclassical thinking. This led to an accounting approach and sectoral balance stock-flow consistent analysis by Tobin and Godley. Neoclassical economists have largely either ignored this development, or rejected it, claiming that methodological issues are settled, so history is irrelevant.
A Postscript On Barter
A Postscript On Barter