When we are given an expression for the Short Run Total Cost Curve (for eg: 8 +
3Q - 1.5Q^2 + 0.25Q^3), how do you derive expressions for the following:
1. Average Fixed Costs
2. Average Viarable Costs Curve
3. Marginal Costs Curve
4. Short Run Supply Curve
I also have a second question:
In perfect competition, if you are given the firm's total cost curve (for eg:
450 + 15Q + 2Q^2), and you are also given the Price, how do you find out the
firm's profit maximising level of output and it's profit/loss level?
An answer to either of these questions or preferably both would be great.
Thank you.
Your equation: TC = 8 + 3Q - 1.5Q^2 + 0.25Q^3
1) Total fixed costs are 8. So avg fixed costs are 8/Q.
2) Ergo Avg variable costs are (TC-8)/Q
3) Marginal cost is the first derivitive of total costs. (use calculus)
4) Short run supply is marginal cost curve.
Always, always, maximize when MC=MR
For a perfect competitor, MR=Price.
MC is the first derivitive of TC.
Profit/loss = Total revenue - Total cost. and TR=P*Q. Just plug in your maximizing Q.
mc=d/dq(8+3Q - 1.5Q^2 + 0.25Q^3)
=3+2*1.5q+3*0.25q2
To derive the expressions for the different cost curves and the short run supply curve, you will need to use some basic principles of calculus and economics. Let's go through each question step by step.
1. Average Fixed Costs (AFC):
Average Fixed Costs represent the fixed cost per unit of output. To derive the expression for AFC, you divide the total fixed costs by the quantity of output (Q).
AFC = Total Fixed Costs / Quantity (or AFC = 8 / Q)
2. Average Variable Costs (AVC):
Average Variable Costs represent the variable cost per unit of output. To derive the expression for AVC, you subtract the fixed costs from the total costs and then divide the resulting value by the quantity of output (Q).
AVC = (Total Costs - Total Fixed Costs) / Quantity
= (TC - 8) / Q
3. Marginal Cost (MC):
Marginal Cost represents the change in total cost resulting from producing one additional unit of output. To derive the expression for MC, you need to take the derivative of the total cost function with respect to quantity (Q).
MC = dTC/dQ
= d(8 + 3Q - 1.5Q^2 + 0.25Q^3)/dQ
= 3 - 3Q + 0.75Q^2
4. Short Run Supply Curve:
In the short run, the supply curve of a firm is determined by its marginal cost curve. To derive the short run supply curve, plot the quantities of output corresponding to different prices. The firm will supply output at a price that covers at least its marginal cost.
So, the short run supply curve is the part of the marginal cost curve that lies above the average variable cost curve.
Now, let's move on to your second question about profit maximization in perfect competition.
To find the firm's profit-maximizing level of output and its profit/loss level, follow these steps:
1. Set marginal cost (MC) equal to marginal revenue (MR). In perfect competition, MR is equal to the price.
MC = MR = Price
2. Find the quantity (Q) that satisfies the above equation by setting MC equal to the given price.
3. Once you have the quantity of output, plug it into the total cost function to find the total cost (TC) at that level of output.
4. Calculate the total revenue (TR) by multiplying the price by the quantity (TR = Price * Q).
5. Finally, calculate the firm's profit or loss by subtracting the total cost from the total revenue.
Profit/Loss = Total Revenue - Total Cost
If the result is positive, it indicates a profit; if it is negative, it indicates a loss.