Please Use This Bartleby Expert Solution to Create Bordered Hession Matrix (Addition Image Provided):
Expert Solution
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Step 1
For utility maximization the second-order condition we have to show the utility function is convex (having diminishing MRS). So we use Bordered Hessian matrix to check the second-order condition of utility maximization.
The Bordered Hessian matrix is H=uxxuxypxuyxuyypypxpy0
The second-order condition is satisfied only when |H|=uxxuxypxuyxuyypypxpy0>0
Here
uxx=∂2u∂x2, uyx=uxy=∂2u∂xy uyy=∂2u∂y2
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Step 2
Now
Utility function: u(x,y)=x+2y+1 .......(1)
Differentiate partially from equation 1 with respect to x:
∂u∂x=y+1again differentiate above equation with respect to x∂2u∂x2=0
Similarly
∂u∂y=x+2and ∂2u∂y2=0 and∂2u∂yx=1
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Step 3
Putting value in Bordered Hessian matrix:
|H|=01px10pypxpy0H=-10-pxpy+pxpy-0H=2pxpy>0 because px and py are positive number
So second-order condition is satisfied.
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Solved in 3 steps
