Given: U(x1, x2) = x1x2+x1 Find Marshallian demand

Question

a) Calculate the Marshallian demand for x1 and x2

b) Graph the demand for x1 when p2= 20 and I = 100.

c) Show how the graph changes when p2=5.

Answer 1

To calculate the Marshallian demand we need to set up the utility maximization problem and get the answer in terms of the parameters and the prices. The Langrange equation is L=U(x,y)-Lambda(xp1+yp2-I) where p1 is the price of x1, p2 is the price of y and I is income. Next we solve for the first order conditions then setting @U/@X=@U/@Y get x and y in terms of the parameters then substitute those values back into the budget constraint and solve for x and y in terms of the parameters and income. For this problem the optimization problem is:

L=x1x2+x1-lambda(x1p1+x2p2-I)

First Order Conditions:

@L/@x1=X2+1=0

@L/@x2=X1    =0

@L/lam = x1p1+x2p2-I=0

 

Now set the first two first order conditions equal to each other and get both X1 and X2 in terms of each other then substitute back into the budget constraint and solve for the Marshallian demand of each good so we have:

X2+1=X1

X2=X1-1

X1=X2+1

 

So for X1 the Marshallian demand is given by the equation (X1p1+(X1-1)p2-I=0

So Solve for X1 in terms of p1,p2 and I and that is the Marshallian demand.

X1p1+X1p2-p2=I

X1(p1+p2)-p2=I

X1(p1+p2)=I+p2

Thus the Marshallian demand for X1 is X1=(I+p2)/(p1+p2)

 

Now do the same thing for X2

(X2+1)p1+X2p2-I=0

X2p1+p1+X2p2=I

X2(p1+p2)+p1=I

The Marshallian demand for X2 is:

X2=(I-p1)/(p1+p2)

 

b)Graph the demand for x1 when p2= 20 and I = 100:

Simply graph the equation:

X1=(100+20)/(p1+20) or

X1=120/(p1+20) where p1 is the vertical axis and X1 is the horizontal axis.

 

c) Show how the graph changes when p2=5.

Now our equation to graph is

X1=(100+5)/(p1+5)

X1=105/(p1+5)