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Pareto Optimality Mathematical Methods & Model Analysis Notes

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PARETO OPTIMALITY ANSWER An economy is composed of consumers with identical incomes and an identical utility function U=x1 0.5x20.5. There is a single input labour, of which 100 units are available. Production of x 1 requires 2 units of labour per unit of output; production of x2 requires 3. (i) Find an expression for the slope of the set of indifference curves of the consumers MRS = -

∂ 
∂U ∂U
= X2÷
∂X 1 ∂X 2 ∂X 1 

= 0.5 X -0.5 X 0.5 1 2
∂X 1

= 0.5 X -0.5 X 1 2
∂X 2
∴MRS = - X 2 / X 1

(ii) What is the slope of the PPF Slope of the PPF is given by 100 = 2 X 1 + 3 X 2 1
∴X 2 = 33 3 -

∂X 2
=∂X 1

2 3


2 3

(PPF Function/ budget constraint for production)

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