The U.S. economy expanded at a slightly faster but still extremely weak rate at the end of last year, the government reported Thursday, but there is hope that growth accelerated in early 2013 despite higher taxes and cuts in government spending. The Commerce Department said the economy grew at an annual rate of 0.4 percent in the October-December quarter, slightly better than the previous estimate of 0.1 percent growth. The new revision reflected stronger business investment and export sales. Still, the 0.4 percent growth rate for gross domestic product (GDP) - the total output of goods and services - was the weakest quarterly performance in nearly two years and followed a much faster 3.1 percent increase in the third quarter. For all of 2012, the economy grew 2.2 percent following a 1.8 percent increase in 2011 and a 2.4 percent gain in 2010. Since the recession ended in mid-2009, the economy has been expanding weakly as a series of problems, including high gasoline prices and Europe's debt crisis, have limited growth. Economists believe the economy is growing at an annual rate of about 2.5 percent in the current January-March quarter, as steady hiring has kept consumer spending consistent, companies have increased stockpiling goods, the housing market continues to improve, and businesses plan more investment. Growth seems to be strengthening this year even after taxes increased on January 1 and automatic government spending cuts totaling $85 billion this year began to take effect earlier this month. The non-partisan Congressional Budget Office (CBO) has estimated that the combination of tax increases and spending cuts would reduce economic growth this year by about 1.5 percentage points. The CBO is predicting only 1.5 percent growth for 2013.