Question

1.    Maximize utility U(x, y, z) = 24yz +16xy +16xz

Subject to G (x, y, z) = xyz= 384

Answer 1

Given

$U = 24yz +16xy +16xz\\ G = xyz= 384$

$L=24yz+16xy+16xz\lambda(384-xyz)$

$\frac{\delta L}{\delta x}=16y+16z-\lambda yz=0\\ \frac{\delta L}{\delta x}=24z+16x-\lambda xz=0\\ \frac{\delta L}{\delta z}=24y+16x-\lambda xy=0\\ \frac{\delta L}{\delta \lambda }=384-xyz=0$

$\lambda=\frac{16y+16z}{yz},\lambda=\frac{24z+16x}{xz},\lambda=\frac{24y+16x}{xy}$

$\frac{16y+16z}{yz}=\frac{24z+16x}{xz}\\=2x=3y$

$\frac{24z+16x}{xz}=\frac{24y+16x}{xy}\\=y=z$

$xyz=384\\\frac{3y}{z}.y.y=384=y^3=128×2=256\\y=6.3496\\x=9.5244\\z=6.3496$

$U_{max}=24(6.3496)^2+16(9.5244×6.3496)+16(9.5244×6.3496)\\=6.3496(24×6.3496+32×9.5244)\\U_{max}=2902.0542$