J. C. Penney Co. said yesterday its second quarter earnings plummeted 50 percent in the second quarter because of heavy markdowns and higher interest expenses.

The huge retailer said it earned $16 million, or 24 cents per share, for the 13 weeks ended July 28. That compared with $40 million, or 58 cents per share, in the year-earlier quarter.

Penney noted, however, that it had about 22 million fewer shares outstanding in the 1978 second quarter.

For the first half, Penney's profits were down 29 percent, to $51 million off 75 cents per share, from the $73 million or $1.08 per share in the first half of 1978.

Sales in the second quarter rose a meager 2 percent to $2.48 billion from $2.43 billion in the comparable 1978 quarter. Volume for the first half increased 4.5 percent to $4.82 billion from $4.61 billion.

The drop in second-quarter earnings resulted from increased merchandising costs, particularly from heavy markdowns to clear inventories and beef up sales, said chairman Donald V. Steibert and president Walter J. Neppl.

The executives also blamed rising interest expenses -- due to higher borrowing levels and interest rates -- which increased 26 percent to $12 million in the second quarter.

Penney said earnings were affected by its reduction of inventories in the quarter by 8 percent from the year-earlier level. But some industry analysts felt Penney's inventories had to be cut.

"Last year their inventories were too high," said Jeffry Edelman Dillon & Co., Inc. He said analysts were looking for a 40 percent decline in Penney's second-quarter earnings, but the lagging sales and heavy markdowns sent them even lower.

Shearson Hayden Stone Inc., one of the nation's largest brokerage houses, said strength in all its areas of operation helped double its earnings in the fiscal year ended June 30.

Earnings for the fiscal year were $20.1 million, or $3.78 a share, up from $10 million, or $2.11 a share, in fiscal 1978. Revenues rose to $304 million from $221.1 million.

For the three months ended June 30, income was $6.2 million, or $1.15 a share, on revenues of $85.3 million. That compared to income of $6.4 million, or $1.31 a share, on revenues of $73.8 million.