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Mathematical Method for Economics

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1. The per period sales of a new product, x(t), evolves over time according to

x(t):= A / (1 + b * e - c t ) where A, b and c are positive constants.

a) By taking the limit of x(t) as t tends to infinity, show that per period sales tends to A as t increases.
b) Show that the rate of growth of sales is proportional to the difference between A and x(t).
c) Show that the maximum rate of change occurs when x(t) := A / 2
d) Find the value of t, as a function of the constants b and c, at which this maximum change occurs.

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Find mathematical economic solutions for perfect competition, two firm monopoly, price discriminating monopoly, natural monopoly with constraint.

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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