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* CarMax drops after Q4 results miss

* March producer prices at 2.1% YoY vs 2.2% estimated

* Indexes: Dow down 0.36%, S&P up 0.01%, Nasdaq up 0.47%

April 11 (Reuters) -

The benchmark S&P 500 and the Dow were muted at noon on Thursday after long-dated Treasury yields remained elevated following the latest economic data as well as commentary from Federal Reserve policymakers.

New York Federal Reserve President

John Williams

said that while the U.S. central bank had made considerable progress in lowering inflation, it did not yet need to move to an easier monetary policy setting, given the recent uneven price pressure movements.

Richmond Fed President

Thomas Barkin

said the latest inflation data showed the U.S. Federal Reserve is "not yet where we want to be" and that it was unsure that inflation would continue to ease.

Wall Street sold off sharply in the last session after data showed U.S. consumer prices increased more than expected in March, leading the financial markets to surmise that the central bank might delay cutting interest rates until September this year.

"When you have had such a decline as we did yesterday, then the debate is no longer of when a rate cut is possible, but perhaps maybe a rate hike (instead), that embeds itself within the market perception and so its difficult for the market to rebound quickly," said Peter Cardillo, chief market economist at Spartan Capital Securities.

Yields across long-dated government bonds came back up to elevated levels after easing earlier in the day, with the 10-year note last at 4.562%, hovering near its highest level since November.

Traders now see an over 40% chance of the Fed bringing in the first rate cut in July, according to the CME FedWatch Tool.

Meanwhile, a Labor Department report showed U.S. producer prices increased moderately in March as a rise in the cost of services was softened by a fall in goods prices, which could help calm fears that inflation was re-accelerating.

Gains across chipmakers and megacap growth stocks such as Nvidia and Alphabet helped the Nasdaq rise 0.5%.

Of the 11 S&P 500 sectors, nine ticked lower, with energy leading the losses with a 1.4% fall.

The S&P 500 Financials Index dipped 0.6% a day before the trio of big banks - JPMorgan Chase & Co, Citigroup and Wells Fargo - is slated to post quarterly results.

At 11:48 a.m. ET, the Dow Jones Industrial Average was down 136.67 points, or 0.36%, at 38,324.84, the S&P 500 was up 0.66 points, or 0.01%, at 5,161.30, and the Nasdaq Composite was up 75.31 points, or 0.47%, at 16,245.67.

CarMax dropped 13.2% after the pre-owned vehicles retailer missed analysts' estimates for fourth-quarter results and said it might not meet its long-term vehicle sales target.

Globe Life

slumped 19.5% after Fuzzy Panda Research disclosed a short position in the company, alleging multiple instances of insurance fraud.

Declining issues outnumbered advancers for a 2.01-to-1 ratio on the NYSE and for a 1.31-to-1 ratio on the Nasdaq.

The S&P index recorded two new 52-week highs and six new lows, while the Nasdaq recorded 29 new highs and 89 new lows. (Reporting by Shashwat Chauhan and Shristi Achar A in Bengaluru; Editing by Pooja Desai)