What was the reason for US Fed rate increase in spring 1928?

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I think you are forgetting that the inflation rate during WWI was staggering. The consumer price index at the end of the war was extremely high, and the deflation that you are referring to can be seen as more of a adjustment in pricing back toward pre-war levels. In fact the CPI was relatively flat through the 20s as a 10 year moving average:

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In 1928, equity prices were booming due to the increase in the money supply, and the Federal Reserve feared that this was a speculative bubble driven by low interest rates. The Fed's action was intended to encourage people to move money from investing in the stock market toward bank savings in an effort to stabilize the economy, not to borrow.

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See Timothy Cogley and Christopher Wood.

As far as the price of gold, remember that with a gold standard, the price of gold largely is inflation. Prior to the Glass Steagall Act in 1932, the Federal reserve was required to back 40% of all Federal Reserve notes with gold, and this is consistent with the ratio of Fed holdings during the period. You can see the effect of Glass Steagall in the graph below where it remains relatively flat through 1932 when there is a sudden spike in bond holdings compared to gold:

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That means that the price of gold is driven by the size of the money supply, so changes in interest rates would theoretically drive real gold prices largely to the extent that they expand or contract the money supply. Exchange prices of gold internationally would be more of a measure of the relative strength of the exchanging nations' economies.

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Because in 1928 macroeconomics hadn't been invented; until Keynes, people had odd ideas about money supply, inflation, and economic growth. Both of the other answers point to the fed's fear of liquidity. If you kill liquidity, then you kill speculation and all is well. Mundell-Tobin theory suggested that raising interest rates would raise the velocity of money but somehow reduce the real interest rate (the rate which motivates investors and changes their behavior). By raising interest rates, you discourage speculation.

Furthermore (quoting from the wikipedia page you reference)

. . . if interest rates were low in a different country then its investors would elect to move their funds abroad where interest rates were higher.

As long as currency is pegged to an arbitrary external value rather than to productive capacity, then capital flows with interest rates, not with productive capacity. When interest rates decline, capital moves to countries which offer better interest rates. Lowering interest rates results in a capital outflow; raising interest rates prevents capital outflow.

Right now I cannot find references to back it, but to answer your second question, the Fed had no obligation to announce the reasons for its actions. The Fed's actions were not not transparent. The Fed were technocrats who did what they thought was best with the theory they had. I don't think transparency was required until the Humphrey Hawkins act of 1978

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