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**Optimal Commodity Bundles**

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Optimal Bundle is a point where the budget line is tangential to Indifference Curves. We usually use lagrangian function to derive optimal commodity bundles. But lagrangian method cannot be used in the case of corner solutions. So, Lagrangian method cannot be used for Perfect Substitutes and Perfect Complements.

Method to find optimal commodity bundles for cobb douglas utility functions:

**Example:** Suppose that utility is given by the Cobb-Douglas form U(x; y) = x^ay^b and prices are px , py for x and y respectively. Income is given to be fixed at I.

Step 1: To find out the optimal commodity bundles we maximize U with respect to the budget constraint . Budget Constraint in this case is xpx + ypy = I

Step 2: Set up the lagrange to get the optimal commodity bundles

Max x^ay^b subject to xpx + ypy = I

L = x^ay^b – λ (pxx + pyy – I)

ax^a-1y^b/bx^ay^b-1 =a/b(y/x) =px/py

(Note : ax^a-1y^b/bx^ay^b-1 = MRS of utility function)

a/b (y/x) = px/py

Step 3 :Now use the above equation to get y in terms of x and px, py

y= x(px/py)a/b

Step 4: Put this value of y in budget constraint to get the value of x

Px.x + xpx(a/b) = I

Or x* = a/a+b (I /px)

Step 5: Use this value of x* find out the value of y*

We get y* as b/a+b (I/py)

Note : ( Formula for MRS = dU/dx/dU/dy)

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