1. The minimum wage does not lead to unemployment due to a substitution of machinery for labor. No doubt this does occur in some instances after the imposition of a minimum wage, but it represents a reaction on the part of the entrepreneurs to maintain their original rate of profit despite the hike in their costs of production. Moreover, this need not be the only reaction of entrepreneurs to a hike in their costs of production. Indeed, in certain sectors of the economy the imposition of a minimum wage might lead to entrepreneurs quitting those sectors, if there is no viable way to maintain the rate of profit via factor substitutions. Plus, note that businesses cannot just pass on these increased costs via hikes in the price of the product. The latter is in turn determined by the prevailing demand and supply conditions in the product market and can only be raised at the expense of reduced profits for the firm.
The minimum wage leads to unemployment because, given the underlying demand and supply schedules (that represent the prevailing maximum buying prices on the part of the buyers and the minimum selling prices of the sellers), an imposition of a wage rate above the market clearing level will lead to fewer workers being hired. This holds true for any and all segments of the labor market; manufacturing and service segments, skilled and unskilled segments. The only way the minimum wage will not lead to unemployment is if the market clearing wage is above the proposed minimum wage rate (Tom Cruise, for instance, is unaffected by the prevailing minimum wage rate).
2. Regarding the argument that the imposition of a minimum wage would lead to increased spending (assuming that the marginal propensity to consume of workers is higher than that of the employers), your brother is right in noting that this is an argument that Keynesians often make to support their claim that the imposition or a hike in the minimum wage rate would stimulate the economy.
This argument, however, erroneously assumes that there is an aggregated market for “labor” out there in the economy. This, of course, is not the case and is just another manifestation of the misuse of aggregates by Keynesians. On the market there exist a number of labor markets. The minimum wage would affect some of these markets, and would lead to a reduction in the number of workers hired in these markets. This, in turn, would lead to a reduction in demand on the part of the fired/unemployed workers for other goods and services. These markets would now witness a decline in demand. If prices are propped up above market clearing levels in these markets, then more unemployment results.
This cumulative process is best explained by William Hutt. See his “Significance of Price Flexibility” (available online on mises.org) and his book “A Rehabilitation of Say’s Law (also available online). For a simple summary of his argument see the following short piece by Richard Ebeling: