11 Cobb Douglas Contract Curve PDF

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lecture slides for the course....

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Consider an exchange economy, where two consumers A and B have initial endowments (1, 3) and (3, 1). The two consumers both have Cobb-Douglas utility functions. In particular, uA (x1 , x2 ) = x1ax21−a and uB (x1 , x2 ) = x1bx1−b 2 , where a, b ∈ (0, 1). Now given this exchange economy, find the contract curve, i.e. the set of all Pareto efficient/optimal allocations. At an Pareto optimal allocation, the indifference curves of the two consumers are tangent to each other. In other words, MRSA (x1A, xA2 ) = MRSB (x1B, xB2 ). Therefore, we must have bx2B ax2A (1) − = − (1 − b)x1B (1 − a)x1A Beside the equality between the two MRS, there is also two market clearing conditions: x1A + xB1 = 4 (2) x2A + xB2 = 4

(3)

Rearrange equation (2) and (3) and we get: x1B = 4 − xA1

(4)

x2B = 4 − xB1

(5)

Now we plug equations (4) and (5) into equation (1) and get −

axA2 b(4 − x2A) = − (1 − a)x1A (1 − b)(4 − x1A)

a(1 − b)(4x2A − xA1 x2A) = (1 − a)b(4x1A − xA1 x2A) 4a(1 − b)x2A + (b − a)x1AxA2 = 4(1 − a)bx1A x2A =

4(1 − a)bx1A (b − a)x1A + 4a(1 − b)

The last line is the equation for the contract curve.

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