Unconstrained Optimization: level of production that maximizes profits

Unconstrained Optimization

1. Assume that you can sell widgets at $25 per widgets, for as many widgets as you can make. Assume that the marginal cost of producing each widget increases with the number of widgets that are produced. The formula is MC(Y) = Y2. For example, the marginal cost of producing the second widget is 4.

a) Using Excel or similar spreadsheet program, calculate the revenues for producing 1, 2, 3, ... , 10 widgets. Calculate the marginal costs of producing the 1, 2, 3, ..., 10 widgets. Calculate the total cost to produce 1, 2, 3, ... , 10 widgets. Determine your profits for producing 1, 2, 3, ..., 10 widgets. What level of production maximizes your profits?

b) Remember from micro economics that in a perfectly competitive market, a producer will produce the number of products such that MC = Price. Is this the case here? Verify it.

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...$1.00 $1.00 $24.00
2 $50.00 $4.00 $5.00 $45.00
3 $75.00 $9.00 $14.00 $61.00
4 $100.00 $16.00 $30.00 $70.00 Notice how the profit level peaks and stays the same
5 $125.00 $25.00 $55.00 $70.00 for ...