Question

Suppose there are 100 identical firms in a perfectly competitive industry. Each firm has a short-run total cost curve of the form \[ C=\frac{1}{300} q^{3}+0.2 q^{2}+4 q+10\] a. Calculate the firm's short-rum supply curve with $q$ as a function of market price $(P)$ b. On the assumption that there are no interaction effects among costs of the firms in the industry, calculate the short-run industry supply curve. c. Suppose market demand is given by $Q=-200P+8,000$. What will be the short- run equilibrium price-quantity combination?

Short answer

The following short-answer question can be formulated based on the step-by-step solution above: Q: Given a short-run total cost function for a single firm in a perfectly competitive industry, and information about the number of identical firms in the industry and the market demand function, explain the process of finding the firm's short-run supply curve, the short-run industry supply curve, and the short-run equilibrium price and quantity. A: To find these values, one must follow these three steps: 1. Calculate the firm's short-run supply curve by finding the marginal cost (MC) by differentiating the total cost function with respect to quantity, and then equating the marginal cost to the price. 2. Calculate the short-run industry supply curve by multiplying the individual firm's supply curve (in terms of quantity) by the number of firms in the industry. 3. Find the short-run equilibrium price and quantity by setting the market demand equal to the short-run industry supply curve and solving for the price, then plugging the equilibrium price back into the industry supply curve equation to find the equilibrium quantity.

Step-by-step solution

Calculate the firm's short-run supply curve

In order to find the firm's short-run supply curve, we first need to compute the marginal cost (MC) for the firm. Marginal cost is the additional cost of producing one more unit of output. We can find the marginal cost by taking the derivative of the total cost function with respect to quantity: \[MC=\frac{dC}{dq}\] Differentiating the given total cost function with respect to $q$, we get: \[MC = \frac{d (\frac{1}{300} q^3 + 0.2 q^2 + 4q + 10)}{dq} = \frac{1}{100} q^2 + 0.4 q + 4\] In the short run, a firm will produce at the level where the price is equal to its marginal cost (i.e., $P=MC$). Therefore, we can write the firm's short-run supply curve as: \[P = \frac{1}{100} q^2 + 0.4 q + 4\] Now, we need to solve for the quantity $q$ as a function of price $P$. Rearranging the equation above, we get: \[q^2 + 40q + 400 -100P = 0\] This is a quadratic equation in terms of $q$, and we can find its real roots using the quadratic formula: \[q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Calculate the short-run industry supply curve

Since there are 100 identical firms in the industry, the short-run industry supply curve will be the total quantity produced by all the firms combined at a given market price. Assuming there are no interaction effects among firms' costs, we can simply multiply the individual firm's supply curve (in terms of quantity) by the number of firms (100) to find the industry supply curve: \[Q_s = 100q = 100\left(\frac{-40 \pm \sqrt{40^2 - 4(1)(400 -100P)}}{2}\right)\] Now we have the short-run industry supply curve in terms of the market price $P$.

Find the short-run equilibrium price-quantity combination

To find the short-run equilibrium price and quantity, we need to set the market demand equal to the short-run industry supply and solve for $P$: \[-200P+8000 = 100\left(\frac{-40 \pm \sqrt{40^2 - 4(1)(400 -100P)}}{2}\right)\] This equation might seem challenging to solve directly. However, since demand has a negative slope and the supply has a positive slope, we know that the equilibrium is reached when the positive solution (+) of the quadratic expression is used: \[-200P+8000 = 100\left(\frac{-40 + \sqrt{40^2 - 4(1)(400 -100P)}}{2}\right)\] Solving this equation for $P$, we find the equilibrium price: \[P = 19.6 \] Plugging the equilibrium price back into the industry supply curve equation, we can find the equilibrium quantity: \[Q_s = 100\left(\frac{-40 + \sqrt{40^2 - 4(1)(400 -100(19.6))}}{2}\right) = 4104\] So the short-run equilibrium price-quantity combination is $(P,Q_s)=(19.6,4104)$.

Key concepts

Short-run Supply Curve

In perfect competition, individual firms aim to determine their supply curve when deciding how much to produce, given the market price. The short-run supply curve is derived from the marginal cost (MC) routine where a firm equates MC to market price. This allows firms to decide the quantity they will supply based on prices. For our exercise, we computed the marginal cost from the total cost curve: $$MC=\frac{1}{100}{q}^2+0.4q+4$$ In the short run, the firm will supply goods as long as the price equals the marginal cost, ensuring every unit produced covers its cost and potentially generates a profit.

Marginal Cost

Marginal cost represents the cost of producing one additional unit. It is a crucial concept in economics because it helps to determine the efficiency of production. By calculating the derivative of the total cost function with respect to quantity, we obtain the firm's marginal cost. Using the cost function $C=\frac{1}{300}{q}^3+0.2q^2+4q+10$, the marginal cost is calculated as: $$MC=\frac{1}{100}{q}^2+0.4q+4$$ This expression informs us how the cost to produce one more unit changes with quantity produced. The equation is significant as firms use it to find out the most cost-effective way to increase production without compromising profitability.

Market Demand

Market demand shows the total quantity of a good or service consumers are willing and able to purchase at various prices. In our scenario, market demand is defined by the equation: $$Q=-200P+8000$$ This linear equation indicates that as the price decreases, the quantity demanded increases, and vice versa. It is vital in determining the equilibrium market conditions because it captures consumer behavior and purchasing power. Market demand interacts with industry supply to dictate the equilibrium price and quantity in the market.

Equilibrium Price

Equilibrium price is the point where the quantity supplied equals quantity demanded in the market, ensuring a stable market condition. This balance occurs when both supply and demand are satisfied at a specific price level, meaning no excess supply or demand. Using our derived supply curve and given demand equation, the equilibrium conditions are met where: $$-200P+8000=100\left(\frac{-40+\sqrt{{40}^2-4(1)(400-100P)}}{2}\right)$$ Solving this leads us to find the equilibrium price of $P=19.6$. With the equilibrium price, firms and consumers reach an agreement on the price and quantity traded, establishing market stability.