Question

U(X,Y)=X+Y

(a) Suppose the initial situation is given by PX​=2,PY​=5 and I=10. Determine the optimal consumption of X and Y. Depict this optimal consumption basket in a diagram.

Answer 1

To determine the optimal consumption of goods X and Y, we need to maximize the utility function u(X,Y)=X+Y subject to the budget constraint. Given PX=2, PY=5, and I=10 (the budget), we can solve for the optimal consumption by setting up the relevant equations and maximizing the utility function.

Step-by-step explanation:

To determine the optimal consumption of goods X and Y, we need to maximize the utility function u(X,Y)=X+Y subject to the budget constraint. Given PX=2, PY=5, and I=10 (the budget), we can set up the following equation:

$2X + $5Y = $10

To solve for the optimal consumption, we can rearrange the equation to solve for X in terms of Y:

X = (10 - 5Y)/2

Substituting this expression for X in the utility function, we get:

u(Y) = (10 - 5Y)/2 + Y

Now we can maximize u(Y) by taking the derivative with respect to Y and setting it to zero:

du(Y)/dY = -5/2 + 1 = -3/2 = 0

From this, we can solve for Y:

Y = 2

Substituting Y=2 back into the equation for X, we get:

X = (10 - 5(2))/2 = 0

Therefore, the optimal consumption of X and Y is X=0 and Y=2. This consumption bundle represents the point where the budget constraint line intersects the highest indifference curve in the graph.