Question

In: Economics

For the utility function U(x,y)=x2 + y2 Find dU/dx and dU/dy and express dU/ in terms...

For the utility function U(x,y)=x² + y²

Find dU/dx and dU/dy and express dU/ in terms of dx and dy
Find dU/dx where y=y(x) satisfies px + qy = M
Find the values of x,y where dU/dx =0
Noting that d²U/dx²=d/dx[dU/dx] + d/dy[dU/dx]dy/dx, use your answer in (b) to find d²U/dx²
Does your answer in (c) describe a maximum or a minimum? Draw an indifference curve diagram which depicts your answer.

Expert Solution

The utility is given to be .

(a) We have and . Also, we have or .

(b) For y satisfying , solving for y, we have or . Here y is a function of x, and . Putting it in the required equation, we have or .

(c) For , we have or or . Hence, for all values of x and y for which , we would have .

(d) We have , and hence, or , and since , we have or .

(e) For and , the answer in part-c would describe a minimum.

The graph is as below.

As can be seen, the tangency condition gives the point at P, but since the utility is concave, not convex, the condition is a minimizing one. At point P, it is the minimum of U that is tangent to the constraint.

Rahul Sunny answered 2 years ago

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