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Solving for the Walrasian Equilibrium: Two examples Example 1

Econ 401A October 9, 2016 1 Solving for the Walrasian Equilibrium: Two examples Example 1: Every consumer has the same utility function function 1112()Uxxx . There is an initial endowment of 30 units of commodity 1. Commodity 1 is both consumed and used as an input in the production of commodity 2. The production function is 4qz . The utility function is homothetic since 1( )( )UxUx . Then ( ) ( )UxUy implies that ( ) ( )UxUy . (a) Solve for the production plan that maximizes the utility of the representative consumer. (b) What must be the WE price ratio? Example 2: Every consumer has the same utility function function 412()Uxxx . There is an initial endowment of 81 units of commodity 1. Commodity 2 is a produced using commodity 1 as an input. The production function is 1/26qz . (a) Show that the utility function is homothetic. (b) Solve for the production plan that maximizes the utility of the representative consumer. (c) What must be the WE price ratio?

2. The profit is S)z 21. For any z the firm will produce the maximum possible output to maximize profit. In this example the set of feasible plans is the set S z q q z d{( , )| 4 } so the firm will choose qz 4. Then profit is S z214 The marginal profit to increasing the input is therefore 4 21 d pp dz S . Supporting prices The price vector p

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