[Edited June 14, 2022 to add material on marginal analysis, complete Dave’s presentation, and point out a mistake made by Mr. Burge. As far as I know, this is his first error ever.]
If you never understood basic economics, David Burge (@IowahawkBlog) has put together a pretty convincing tutorial. I’ve taken the liberty of creating graphs that illustrate Dave’s algebra. I’ve also added some material using marginal analysis. Dave knows his math. Econ, not so much. His complete thread is at the end of this article. To download a copy of my Excel workbook, click here. Forthwith, Iowahawk teaches econ 1.
Dave’s scenario is Dave’s Taco Stand. Here’s his description of the economics of his stand.
T = A + B*P is the demand curve for tacos. B is negative meaning the demand curve slopes downward, just like every one of the millions of demand curves economists have statistically estimated using real world data. But Dave’s Taco Stand faces a downward-sloping demand curve meaning the market for tacos is not perfectly competitive. (A firm competing in a perfectly competitive market faces a horizontal demand curve. The firm believes it can sell all the output it wants without having any impact on the market price. Firms in perfectly competitive markets are price-takers because they take the market prices as given and constant.)
Here are the relevant parameter values with an explanation of each. I’ll just add that Dave has assumed marginal cost is constant, a common assumption in applied economics.
Analysis the Hard Way
So his model becomes
Drawing graphs of the demand, marginal cost, total profit, and total cost functions leads us to the profit-maximizing price and quantity. The demand curve is a straight line sloping downward. Following the convention of the economics profession, the demand curve is actually inverted with the independent variable on the vertical axis. The equation is P = 10 – 0.1T.
Total profit is the quadratic equation shown in the equations above. Total cost is C*T. Note that profit becomes negative if I sell more than 80 tacos. That implies a market price of $2, equal to marginal cost. Since marginal cost is constant, average variable cost and total cost are also $2. When price is less than average total cost, profits are negative.
Profit is maximized at T = 40.
To sell 40 tacos, the price must be $6 per taco. Dave does a little example showing that profits will be lower for any other value of P.
But now we get to engage in the dream of every economist. We get to do some math. To find maximum profit, we need to look for a point on the total profit function that is flat (the slope is zero). That can happen at two points for a quadratic function: the maximum and the minimum.
Mathematically, the slope is the first derivative of a function. Without going into details, the derivative of Dave’s profit function is
We want to find the price that makes this function equal to zero.
The P* refers to the single value of P that solves this equation. The solution is straightforward. We can then use the value of P* to find Q*, the profit-maximizing number of tacos sold.
Dave will sell 40 tacos at a price of $6 each. From the graph, we can see profit will be $150.
But there’s one more issue. The derivative can be zero at either a maximum or minimum. How can we be sure mathematically that we’ve found a maximum?
The answer is found in the second derivative which measures the rate of change of the slope. At the maximum, the slope is gradually getting flatter, changing in a negative direction. At a minimum, the slope is getting steeper, changing in a positive direction. A negative second derivative we’ve found a maximum. A positive second derivative means we’ve found a minimum. Here’s the result for Dave’s Taco Stand:Voila, we’ve found a maximum.
Analysis the Easy Way
Economists know that profits are maximized at the quantity that makes marginal revenue equal to marginal cost (MR=MC). We also know that, for any linear demand curve, the marginal revenue curve will have the same vertical intercept with twice the slope. Here are the calculations. R is total revenue, price per taco (P) times the number of tacos sold (T). Marginal revenue is the change in total revenue when one more taco is sold. The easiest way to find the marginal revenue function is to take the derivative of the total revenue function with respect to output quantity.
Now Dave assumes an increase in demand. Mathematically, this shifts the demand curve up and to the right. But the slope of the demand curve does not change — the new demand curve will be parallel to the original. Mr. Hawk’s commentary is entertaining.
The demand curve shifts up and to the right with a vertical intercept of 12 and horizontal intercept of 120.
Sure enough, price, quantity, and profits all increase.
The graph is similar to the earlier version with the vertical intercept shifted up to 12 and the horizontal intercept of 60.
Marginal Cost Increases
Since Dave has assumed constant marginal cost, there is no capacity for increasing unit cost within the model. Economists will typically use increasing total cost, U-shaped per-unit cost, and upward sloping marginal cost. Instead, Dave shifts the marginal cost curve up to $2.50:
By now, the math should be familiar:
How about that? Increasing cost reduces profit. But note that it also reduces the number of tacos sold. This is a principle of imperfect competition: changes in demand or cost will cause both the profit-maximizing price and quantity to change. Under perfect competition, the taco stand would charge the market price and assume that price will not change if any single taco stand increases or decreases output. Therefore, the response to a change in cost must be a change in quantity only.
Here’s the profit-maximizing graph:
I hope this has been instructive. Thanks to David Burge for dreaming up this little example. Comments are more than welcome.
Here’s Dave’s full article.