The manager of a monopoly firm obtained the following...

The manager of a monopoly firm obtained the following estimate of the demand function for its output: Q = 3,000 - 100P + 0.25M + 36PR. From an econometric forecasting firm, the manager obtained forecasts for the 2018 values of M and PR as, respectively, $50,000 and $2.50. The manager also estimated the average variable cost function as AVC = 10.10 - 0.07Q + 0.0002Q^2, where AVC was measured in dollars per unit and Q is the number of units sold. The optimal level of production in 2018 is _____ units.

Transcript

00:01 So here we're talking about monopoly.

00:02 We have a production function, or sorry, a demand function.

00:06 3000 minus 100p plus 0 .25m plus 36pr.

00:14 Wow, there's a lot going on here.

00:16 We expect, we have average variable cost, which is equal to 10 .1 minus 0 .07q plus, wow, 0 .0002q squared.

00:32 We want to think about the optimal level.

00:35 So first of all, we know that we are estimating m is equal to 50 ,000.

00:40 We know that pr is equal to 2 .5.

00:44 So if i substitute in, that gives me q is equal to 3000 minus 100p plus 12 ,500 plus 36 outside of 2 .5.

01:00 So this is equal to, q is equal to, this is going to be 72 and this would be 90.

01:08 So this is going to give me 15 ,590 minus 100p.

01:17 100p is equal to 15 ,590 minus q.

01:22 So now what i can do is construct a profit function, right? or a revenue function, right? revenue is equal to price times quantity.

01:33 So this is going to be 15 ,590 minus q all over 100 times q.

01:44 This means that marginal revenue, which is the derivative of revenue with respect to q is going to be 15 ,590 over 100 minus 2q over 100.

02:01 So this is going to be 155 .9 minus q over 50.

02:09 We want to set that equal to marginal cost, but we can get marginal cost is equal to the derivative of cost with respect to quantity, but this is also equal to the derivative.

02:21 So what i need to do is reconstruct the cost function, right? so variable cost is equal to average variable cost times q, which is going to be 10 .1q minus 0 .07q squared plus 0 .0002q cubed, right? total cost is equal to variable cost plus the fixed cost...

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