A monopolist with the cost function C=1/2Q2 faces a demand curve Q=12-P a. What will be its equilibrium price and quantity? b. If for some reason the firm behaves as if it were in a perfectly competit

a. Equilibrium Price and Quantity under Monopoly

To maximize profit, the monopolist sets marginal cost (MC) equal to marginal revenue (MR).

Marginal Cost:

```
MC = dC/dQ = 1/2 Q
```

Marginal Revenue:

MR = 12 - 2Q (derived from the demand curve)

Setting MC and MR equal:

```
1/2 Q = 12 - 2Q
2.5Q = 12
Q- = 4.8
```

Substituting Q- back into the demand curve:

```
P- = 12 - Q- = 12 - 4.8 = 7.2
```

Therefore, the equilibrium price is P- = $7.20 and the equilibrium quantity is Q- = 4.8 units.

b. Equilibrium Price and Quantity under Perfect Competition

Under perfect competition, the firm is a price taker and faces the market price P. To maximize profit, it produces where price equals marginal cost (P = MC).

Using the firm's cost function, we have:

```
P = MC = 1/2 Q
```

If the market price is P = $6, then:

```
$6 = 1/2 Q
Q = 12
```

Therefore, the equilibrium quantity under perfect competition is 12 units.

However, the monopolist's equilibrium quantity is lower (4.8 units) because it restricts production to maximize its profit.

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