<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
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revoked as to him or her as of the date of execution of such instrument or at a
subsequent specified date.  This power of 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 24, 1995



     Signature                                     Title
     ---------                                     -----


   /s/ Bob G. Gower                            
_____________________________     Chairman, Chief Executive 
       Bob G. Gower                Officer and Director
(Principal Executive Officer)

 /s/ Russell S. Young 
______________________________    Senior Vice President,
     Russell S. Young              Chief Financial Officer and
(Principal Financial Officer)      Treasurer


   /s/ Joseph M. Putz 
______________________________       Vice President and
       Joseph M. Putz                 Controller
(Principal Accounting Officer)
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     Signature                                    Title
     ---------                                    -----

 /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 


*  Effective as of April 1, 1995