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We consider an experditur minimmation problem with a utility function over thee gords. The ufility frnction is u(x1,x2,x3)=x1x2x3 The prices of goods x=(x1,x2x2) ave p1=(p1,p2,p7) We denote income by M, as usual, with M>0 Assume xR3+( ie, x1>0,x2>0,x3>0 ), unless otlerwise stated. 1. u/x1 and 2u/x2, Is the utility function inreasisg in x1 ? Is the utility function concaue in x1 ? 2. The consuncer mininizes expenditure subject to a utility constraint Let 4 represent the minimum level of utility Te consumer requires write down the expenditure minimiation problem of the consumer win respect to x=(x1,x2,x3). texplain briefly why The utility constraint is satistied with equality. 3. Wite downtre Langrangian turction 4 Write down hist order conditions with respet to x=(x1,x2,x3) and A 5. Solve for the the thicksian denncend functions h1h2 and hk3 as functions of the prices p=(P1,4p2,P3) and the mininum requined utility u4. Hint combine the tirst and second order conditions, Ten combine the first ad pird tist. Order conditions, and frudly plug into the utity constaint). 6. Find h1(p,u)/p2 and h(p,/p3 Basid on This, ane These goods (net) substitutes or (net) complements? 7. Writedown the expenditure tunction as tuvetion of p=(p1,44) ? P2,p3) and n, i.e., what 8. Derive the indiect utility function v(p,M) by substituting a=v(,M) in the expenditue frinction, setting the expenditure function equal to M1 and soling for v (M,M). We consider an experditur minimmation problem with a utility function over thee gords. The ufility frnction is u(x1,x2,x3)=x1x2x3 The prices of goods x=(x1,x2x2) ave p1=(p1,p2,p7) We denote income by M, as usual, with M>0 Assume xR3+( ie, x1>0,x2>0,x3>0 ), unless otlerwise stated. 1. u/x1 and 2u/x2, Is the utility function inreasisg in x1 ? Is the utility function concaue in x1 ? 2. The consuncer mininizes expenditure subject to a utility constraint Let 4 represent the minimum level of utility Te consumer requires write down the expenditure minimiation problem of the consumer win respect to x=(x1,x2,x3). texplain briefly why The utility constraint is satistied with equality. 3. Wite downtre Langrangian turction 4 Write down hist order conditions with respet to x=(x1,x2,x3) and A 5. Solve for the the thicksian denncend functions h1h2 and hk3 as functions of the prices p=(P1,4p2,P3) and the mininum requined utility u4. Hint combine the tirst and second order conditions, Ten combine the first ad pird tist. Order conditions, and frudly plug into the utity constaint). 6. Find h1(p,u)/p2 and h(p,/p3 Basid on This, ane These goods (net) substitutes or (net) complements? 7. Writedown the expenditure tunction as tuvetion of p=(p1,44) ? P2,p3) and n, i.e., what 8. Derive the indiect utility function v(p,M) by substituting a=v(,M) in the expenditue frinction, setting the expenditure function equal to M1 and soling for v (M,M)