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.
![**Original Question:**
Q2. Suppose a consumer seeks to maximize the utility function
\[ U(x, y) = (x + 2)(y + 1), \]
where \( x \) and \( y \) represent the quantities of the two goods consumed. The prices of the two goods and the consumer's income are \( p_x, p_y, \) and \( I \).
(b) State the first-order conditions for utility maximization. Find the consumer’s demand functions, \( x^* \) and \( y^* \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6274a625-9a74-4008-b392-f1961eea52c4%2F11d37e4a-9b65-4318-a0f1-1eabbe2bdcc7%2Fxhszc9a_processed.jpeg&w=3840&q=75)
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