Solving Equilibrium Price, Quantity, and Welfare in a Monopolistic and Competitive Market (Econ Follow-Along)
Understanding and calculating how firms choose output and price, as well as how these choices affect net welfare, is a cornerstone of micro, and something you’ll come across regularly as a student or an enthusiast.
In this article, we’ll work our way through a classic monopoly problem with a linear demand curve and a quadratic cost function. We’ll derive the profit-maximizing price and quantity, as well as consumer surplus, producer surplus, and deadweight loss. We’ll then shift to a nearly identical problem (save for a modified firm-level cost function) in a competitive market to determine the long-run competitive equilibrium and welfare.
Graphing is very useful for problems like these, so graphs are present alongside the math and conceptual explanations.
We’ll start with a production function, a cost function, and the first question of our monopoly problem: what is the equilibrium price and quantity?
- Q = 60 — P (hence P = 60 — Q)
- C = 36 + 2Q²
To do this, we’ll take the first order condition (FOC) by finding the equation for profit, and then take the partial derivative with respect to price.
When calculating an FOC, you take partial derivatives with respect to inputs, and in this model firms face in inverse demand curve whereas they set the price P to adjust quantity. So, price is the input and thus what we’ll take the partial derivative with respect to.
Intuitively, the reason we’re taking the FOC of profit at all is because we’re finding the point at which profit is “flat” on the graph with a slope of 0. At this local maximum, firms have no incentive to adjust their price, because doing so can only decrease profits (note that we’ll confirm this using a second-order concavity check in a second). Here’s that concept explained visually:
Now, let’s set up our profit equation. We know that profit equals revenue minus costs:
- π = R — C
And that revenue equals quantity times price
- π = PQ — C
Now, let’s plug in our cost function, and we’ll swap out both Q’s for 60 — P since we know Q = 60 — P:
- π = P(60-P) — 36 + 2(60 — P)²)
Now we’ll take the derivative with respect to price (this would be a partial derivative, but since we’ve swapped out Q, π is just a function of P, so this is a total first-order derivative). We’ll need to use the chain rule, but note that the value inside the parenthesis equates to 0 once derived:
- dπ/dP = 60–2P + 2*2(60 — P)
Hey! Quick note from Jon. If you’re a student in econ 101 or 201, you’ll find these articles useful (exact problem walkthroughs like the ones you’ll see on exams). Back to the problem!
Now let’s expand this:
- dπ/dP = 60–2P + 4(60 — P)
- dπ/dP = 60–2P + 240 — 4P
- dπ/dP= 300 — 6P
Now, as part of the FOC, we’ll set this value equal to zero and solve:
- 300–6P = 0
- 6P = 300
- P = 50
And we’ll plug that value into our production function to solve for Q:
- Q = 60 — P
- Q = 60–50
- Q = 10
So, our monopoly equilibrium price is $50, and our quantity is 10. Let’s update our graph:
As mentioned, we can take a second-order check to ensure that our point is a maximum and thus that the inverted U-shape in the graph holds (otherwise we wouldn’t be finding an equilibrium quantity, as firms would have an incentive to shifts prices to increase profits):
- dπ/dP = 300–6P
- d²π/dP² = -6
Since this value is negative, we’ve confirmed that profits curve downward around our point and that it holds as the equilibrium price and quantity.
Let’s also quickly calculate the marginal cost before moving on, as this is generally handy and will be useful shortly. We’ll take the first derivative of the cost function to do this:
- C = 36 + 2Q²
- MC = 4Q
Now, given our equilibrium quantity and price, we’ll calculate the consumer and producer surplus. It’s easiest to understand this graphically, so let’s first graph our equilibrium demand function.
We know this to be the case since Q = 60 — P means (0, 60) and (60, 0) are intercepts (and then you can just draw a line to connect those points).
Now, we’ll add both the marginal revenue and the marginal cost curves to this graph. We’ll do that because monopoly firms will not set price at the point where demand equals cost; instead they will set price equal to the intersection of their marginal cost and marginal revenue curves.
We know that marginal revenue is downward sloping since it decreases as the quantity produced increases, and that marginal cost is upward-sloping, since it increases as quantity produced increases.
Additionally, we previously calculated that marginal cost = 4Q. Let’s quickly calculate marginal revenue as well using the PQ component of our profit equation:
- R = QP
- R = Q(60 — Q)
- R = 60Q -Q²
- MR = dR = 60–2Q
So, let’s graph both the MC and MR curves using our equations:
Intuitively, note that P = 60 — Q (the D curve), while MR = 60–2Q, so so the MR curve is twice as steep as the Demand curve.
Next, as stated, the equilibrium price and quantity for a monopolist occurs when MC = MR and shifting vertically upward to the demand curve. We know that this equilibrium occurs at (10,50). Let’s add that point to the graph:
Now, consumer surplus is the area under the demand curve and above the horizontal price line at 50 over the interval (0,10), producer surplus is the area over the MC line and below the price line, and deadweight loss is the triangle between the monopoly equilibrium output and the competitive equilibrium output which would otherwise occur at the intersection of D and MC curves. We’ll also plot variable cost on the graph, which is the area under the MC curve to the left of the equilibrium quantity value:
Though graphing our consumer surplus, producer surplus, and deadweight loss, we can easily calculate the values by looking at CS and PS as rectangles and right triangles:
CS is ((60-50)*10)/2, or 50. For producer surplus, we know the “rectangle” is 50*10, and that the part of the rectangle we want to calculate the area for is PS, not VC. Let’s write that out:
- PS = (50*10) — VC
We know that C = 36 + 2Q², so variable cost (the cost that changes with quantity) is 2Q²:
- PS = (50*10) — 2Q²
Since we have the dimensions of the triangle (really, the interval over which we’re solving PS), we can see that Q = 10. Let’s plug that in and solve:
- PS = (50*10) — 2(10)²
- PS = 500–2(100)
- PS = 300
Finally, to calculate the deadweight loss, we need to figure out the competitive equilibrium. As stated, this occurs when P = MC. We already have both those values from previous calculations, so let’s just set them equal to each other and solve for Q and P:
- P = 60 — Q, MC = 4Q
- P = MC
- 60 — Q = 4Q
- 60 = 5Q
- Q = 12
- P = 60–12 = 48
Let’s update the graph with these competitive equilibrium values, and also with the price value of MC = MR:
Now we can easily calculate the area of the deadweight loss (DL):
- ((50–40)*(12–10))/2 = (10*2)/2 = 20/2 = 10
So, we’ve calculated the consumer surplus as 50, the producer surplus as 300, and the deadweight loss as 10 in a monopolistic market.
Next, we’ll look at this market through competitive dynamics, starting with firm-level competitive supply and cost curves. We’ll use the same production function but change the cost function slightly to C = 36 + 4q².
In our production function, “Q” is market output (equal to firm output in a monopoly), so we’ll denote “q” as firm-level supply.
In a competitive market, we know that P = MC, and MC in our revised cost function is 8q (the derivative of 4Q²). So, P = 8q. This means our inverse supply curve (quantity as a function of price) is q = P/8.
We’ll graph our MC cost curve, and our average cost curve. Average cost is cost divided by quantity:
- AC = (36 + 4q²)/q = 36/q + 4q
To find the minumum of the average cost curve, which is where it intersects the marginal cost curve, we can take the partial derivative of AC with respect to quantity (another FOC, as firms now set quantity instead of price).
- ∂AC/∂q = -36/q² + 4
Let’s set this equal to 0:
- -36/q² + 4 = 0
- -36/q² = -4
- -36 = -4q²
- -36/-4 = q²
- 9 = q²
- q = 3
As follows:
A quick note on firm versus market-level supply: we know that inverse firm-level supply is q = P/8. To denote market-level supply (Q), we can use summation notation:
To say that the market-level supply constitutes the summation of all the firm-level supply functions (Q(q)).
Next, to compute our new (given the revised cost function) long-run competitive equilibrium, we can note that price is equivalent to the minumum of the average cost curve as a function of average cost (basically, we plug 3 into AC).
- P = AC(3) = 36/(3) + 4(3) = 12 + 12 = 24
So, our equilibrium price is 24. Given our production function P = 60 — Q, P = 60–24 = 36. This makes our equilibrium price and quantity 24, 36. Let’s graph that out:
Also note that we know firms produce a quantity of 3 at the minumum of the AC curve and we’ve calculated that the market quantity in the long-run equilibrium is 36. Thus, dividing 36 by 3 identifies 12 as the number of firms in the market.
We’ll now re-calculate producer and consumer surplus. Like previously, we’ll graph it first by adding the supply curve to the graph. Note that the supply curve, being a competitive market, is equivalent to the supply curve. We already derived each firm’s supply as q = P/8. Multiplying this by 12 (for 12 firms in the market), we get Q = (3/2)P, or P = (2/3)Q. This is the market supply, and the equation of the supply (marginal cost ) curve. Plugging the equilibrium quantity (36) into this function yields 24, or the equilibrium price.
The area under the marginal cost curve is variable cost; we know this is 832 because of (24*36)/2, but for sake of understanding: variable cost per-firm is 4q², and given q = 3, (4)(3)² = 36. Market variable cost is thus 36*12 = 432. Total revenue is QP = 24*36 = 864. So, R — VC = 864 – 432 = 432 on the market level, and (3)(24) — 36 = 72–36 = 36 on the firm level (also 432/12).
Note that there is no deadweight loss, as this is a perfectly competitive market.
So, there you go! We’ve run through equilibriums given similar production and cost functions in a monopolistic and competitive market, and calculated welfare in each market setup.
Hope you found this helpful, and leave any questions in the comments! Check out my other articles on economics here (useful for studying, or just learning).
