Question

pts) Let U(X,Y,Z) = Xayb z a,b,c > 0 Find the Marshallian demand functions. Calculate og opp om and Interpret the results of these partial derivatives. Interpret | Check the second order conditions.

Answer 1

U(X,Y,Z) = X​​​​​​a​ Y​​​​​​b Z​​​​​​c

Let the prices be Px, Py, Pz respectively, now the budget Constraint will be : XPx+YPy+ ZPz= M , where M is the income.

To find the Marshallian demand functions ,we set up the lagrangean problem as :

L = X^a Y^b Z^c + $\lambda$ ( M- XPx-YPy-ZPz)

dL/dX = (a)X^(a-1) Y^b Z^c - $\lambda$ (Px) = 0 ..1)

dL/dY = (b) X^a Y^(b-1) Z^c - $\lambda$ (Py)= 0 ..2)

dL/dZ = (c) X^a Y^b Z^(c-1) - $\lambda$ (Pz) = 0 ..3)

dL/d$\lambda$ = M- XPx - YPy - ZPz = 0 ..4)

1) and 2) implies : aY/bX = Px/Py ==> YPy = bXPx/a

1) and 3) implies : aZ/cX = Px/Pz ==> ZPz = cXPx/a

Now Substituting it in the budget constraint/ (4) , we get :

M = XPx + bXPx/a + cXPx/a ==> (a+b+c)(XPx) = Ma

==> X* = Ma/(Px)(a+b+c) This gives Y* = Mb /(Py)(a+b+c) and Z* = Mc/(Pz)(a+b+c)

The above are the expressions for Marshallian Demand for X, Y, and Z.

B) dX*/dPx = -Ma/(Px​​​​​2​)(a+b+c)

dY*/dPy = -Mb/(Py​​​2)(a+b+c)

dZ*/dZ = -Mc/(Pz​​​​​2​)(a+b+c)

dX*/dM = a/(Px)(a+b+c)

C) d2L/dX2 = (a)(a-1)X^(a-2) Y^b Z^c  

d2L/dY2 = (b)(b-1) X^a Y^(b-2)Z^c

d2L/dZ2 = (c)(c-1) X^a Y^b Z^(c-2)

The above expression will be less than 0 when a<1 , b<1 and c<1 and the demand will indeed be utility maximizing.