Introduction:
In this scenario, the price elasticity of demand is given as -2, which means that a 1% increase in price will lead to a 2% decrease in quantity demanded. The consumer initially buys 200 units of the good at a price of $20 per unit. We need to determine at what price the consumer will be willing to purchase 300 units.

Calculating the initial price elasticity of demand:
To calculate the initial price elasticity of demand, we use the formula:
Price elasticity of demand = (% change in quantity demanded) / (% change in price)

Since the consumer initially buys 200 units and the price is $20 per unit, the initial quantity demanded and price are:
Q1 = 200 units
P1 = $20

Now, let's calculate the percentage change in quantity demanded:
% change in quantity demanded = (Q2 - Q1) / Q1 * 100
Since Q2 is not given, we cannot calculate the actual percentage change in quantity demanded. However, we can use the price elasticity of demand to determine the percentage change.

Given that the price elasticity of demand is -2, a 1% increase in price will lead to a 2% decrease in quantity demanded. Therefore, we can assume that the percentage change in quantity demanded is -2 times the percentage change in price.

Now, let's calculate the percentage change in price:
% change in price = (P2 - P1) / P1 * 100
Since P2 is not given, we cannot calculate the actual percentage change in price. However, we can use the price elasticity of demand to determine the percentage change.

Given that the price elasticity of demand is -2, a 1% increase in price will lead to a 2% decrease in quantity demanded. Therefore, we can assume that the percentage change in price is -1/2 times the percentage change in quantity demanded.

Since we want to find the price at which the consumer will be willing to purchase 300 units, we can set up the following equation:
% change in quantity demanded = 2 * (% change in price)

Calculating the price at which the consumer will be willing to purchase 300 units:
Let's solve the equation to find the percentage change in price:
2 * (% change in price) = -2 * (% change in price)
4 * (% change in price) = 0

This equation implies that the percentage change in price is 0, which means there is no change in price. Therefore, the consumer will be willing to purchase 300 units at the same price of $20 per unit.

Conclusion:
Based on the given price elasticity of demand of -2, the consumer will be willing to purchase 300 units at the same price of $20 per unit. The price elasticity of demand indicates that the consumer is relatively unresponsive to price changes, as a 1% increase in price leads to a 2% decrease in quantity demanded.