<PAGE>
 
                                                                      EXHIBIT 24

                        LYONDELL PETROCHEMICAL COMPANY

                               POWER OF ATTORNEY

     Each person whose signature appears below hereby constitutes and appoints
Bob G. Gower, Joseph M. Putz and Russell S. Young, and each of them, his or her
true and lawful attorneys-in-fact and agents, with full power of substitution
and resubstitution, for him or her and in his or her name, place and stead, in
any and all capacities, in connection with any outstanding securities of
Lyondell Petrochemical Company (the "Company"), or any public offering or other
issuance of any securities of the Company authorized by the Board of Directors
of the Company, or by the Executive Committee thereof pursuant to due
authorization by such Board, (1) to execute and file, or cause to be filed, with
the United States Securities and Exchange Commission (the "Commission"), (A)
Registration Statements and any and all amendments (including post-effective
amendments) thereto and to file, or cause to be filed, all exhibits thereto and
other documents in connection therewith as required by the Commission in
connection with such registration under the Securities Act of 1933, as amended,
and (B) any report or other document required to be

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filed by the Company with the Commission pursuant to the Securities Exchange Act
of 1934, as amended, (2) to execute and file, or cause to be filed, any
application for registration or exemption therefrom, any report or any other
document required to be filed by the Company under the Blue Sky or securities
law of any of the United States and to furnish any other information required in
connection therewith, (3) to execute and file, or cause to be filed, any
application for registration or exemption therefrom under the securities laws of
any jurisdiction outside the United States, including any reports or other
documents required to be filed subsequent to the issuance of such securities,
and (4) to execute and file, or cause to be filed, any application for listing
such securities on the New York Stock Exchange, or any other securities exchange
in any other jurisdiction where any such securities are proposed to be sold,
granting to such attorneys-in-fact and agents, and each of them, full power and
authority to do and perform each and every act required to be done as he or she
might or could do in person, hereby ratifying and confirming all that such
attorneys-in-fact and agents, and each of them, may lawfully do or cause to be
done by virtue of this power of attorney. Each person whose signature appears
below may at any time revoke this power of attorney as to himself or herself
only by an instrument in writing specifying that this power of attorney is
revoked as to him or her as of the date of execution of such instrument or at a
subsequent specified date.  This power of 

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attorney shall be revoked automatically with respect to any person whose
signature appears below effective on the date he or she ceases to be a member of
the Board of Directors or an officer of the Company. Any revocation hereof shall
not void or otherwise affect any acts performed by any attorney-in-fact and
agent named herein pursuant to this power of attorney prior to the effective
date of such revocation.


Dated:  March 8, 1996

<TABLE>
<CAPTION>
 
         SIGNATURE                        TITLE
         ---------                        -----                      
<S>                          <C>
     /s/ Bob G. Gower        Chairman, Chief Executive                      
---------------------------   Officer and Director 
         Bob G. Gower
(Principal Executive Officer)
 
 
    /s/ Russell S. Young     Senior Vice President, Chief Financial             
---------------------------   Officer and Treasurer
        Russell S. Young
(Principal Financial Officer)
 
 
    /s/ Joseph M. Putz        Vice President and Controller
---------------------------
        Joseph M. Putz
(Principal Accounting Officer)
 
</TABLE>

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<TABLE>
<CAPTION>
 
            SIGNATURE                TITLE
            ---------                -----  
 
 
<S>                                 <C>
   /s/ Dr. William T. Butler        Director
---------------------------------
       Dr. William T. Butler
 
 
 
     /s/ Curtis J. Crawford         Director
---------------------------------
         Curtis J. Crawford
 
 
 
      /s/ D. Travis Engen           Director
---------------------------------
          D. Travis Engen
 
 
 
 /s/ Stephen F. Hinchliffe, Jr.     Director
---------------------------------
     Stephen F. Hinchliffe, Jr.
 
 
 
    /s/ Dudley C. Mecum II          Director
---------------------------------
        Dudley C. Mecum II
 
 
 
     /s/ Dan F. Smith               Director
---------------------------------
         Dan F. Smith
 
 
 
      /s/ Paul R. Staley            Director
---------------------------------
          Paul R. Staley
  
</TABLE>

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