Mathematical economic solutions via lagrange method and algebra
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1) consider a consumer with the following utility function U=f(x,y)=4xy
a) derive demand functions for both commodities.
b) Px=2, Py=2.5, I=40, find the utility maximizing consumption combination.
2) A firm produces two commodities, Q1 and Q2, in pure competition. P1=15 and P2=18. C=2Q1^2 + 2Q1Q2 + 3Q2^2
a) form the profit function.
b) Determine the profit maximizing levels of output for both commodities.
c) Evaluate the Hessian for the second order condition.
3) A discriminating monopolist sells in two markets where the demand respectively is
Q1 = 24 - .2P1
Q2 = 10 - .05P2
TC = 35 + 40(Q1 + Q2)
a) Determine the Q sold and the P charged in each market.
b) find the price elasticity of demand at equilibrium in each market.
4)
a) Find the critical values for a firm producing two goods, x and y when TC = 8x^2 - xy + 12y^2
and the firm is bound by contract to produce a cost minimizing combination of both goods totaling 42.
b) what will it cost the firm to produce one additional unit of its output?
5) suppose a competitive firm sells its goods at the market determined P=60. If the firm's TC = 128 + 69Q - 14Q^2 +Q^3
a) determine the profit maximizing level of output.
b) find the shut down point.
c) find the firms total profit.
d) evaluate the second order condition to verify that profit is maximized.
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Solution Summary
This problem involves several scenarios. In the first, I use the lagrange method to derive demand curves from the utility function and budget constraint. In the second, I determine the profit function, profit maximizing level of output and evaluate the hessian matrix for a perfect competitor producing two goods. In the third I determine the output and price decisions of a monopolist producing two goods and find elasticity of demand given demand functions and cost functions. In the fourth, I find the cost minimizing level of output for a firm producing two goods given an output constraint. I also derive marginal costs. For the fifth, I determine the profit maximizing level of output for a competitive firm given a cost function and prices. I also identify the shut down point, compute profit, and evaluate the second order condition.
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