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.
![**Instructions for Completing the Bordered Hessian Matrix**
Please, **Fill-In** the *Bordered Hessian Matrix* with the **Format** specified below:
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**Note:** Use the answer provided by a *Bartleby Expert* to accurately complete the matrix.
*Explanation of Matrix:*
The matrix here is a bordered Hessian matrix, which typically is a square matrix used in optimization problems to assess the behavior of a multivariable function. It often includes elements representing second derivatives of a function f with respect to its variables, bordered by first derivatives with respect to constraints. Ensure proper format and order when filling in the matrix, as this impacts the accuracy of your optimization analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6274a625-9a74-4008-b392-f1961eea52c4%2F11d37e4a-9b65-4318-a0f1-1eabbe2bdcc7%2F8zx88br_processed.jpeg&w=3840&q=75)
![**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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