Elevated dishwasher (7/7)

How to include an elevated dishwasher into an Ikea kitchen.

  1. Introduction
  2. Design and constraints
  3. Heights and support beams
  4. Materials
  5. Construction
  6. Installation on-site
  7. Downloads and photos

This is the final part, 7.

Final remarks

If the Tower is part of a kitchen where it is standing next to (an)other tall cabinet(s), things get simplied a bit:

  • If the tower is standing between the wall and a tall cabinet, the third (hidden) support beam can be left out, removing a relatively complex part of the construction.

  • If the tower is standing between the countertop and a tall cabinet, the two support beams against the side wall (to the right, in our construction) can be left out.

  • If the tower is surrounded by a tall cabinet on either side, all support beams can be left out.

In both cases, the Tower’s top cabinet can be fixed to the neighbouring tall cabinet(s) with several screws, instead of to the removed support beam(s). It might be a good idea to keep the side cover panels on each side of the tower, so that there is more material to hold the screws that hold up its top cabinet.

Concerning the third (hidden) support beam: you can probably get away with using a 5×5 cm beam. That way, the sections of this beam that need cutting away can be greatly reduced – only the section for the dishwasher remains, as the beam is already thin enough to fit behind the drawer rails and the steam oven.


The Sketchup file is available for download here: construction.skp

More photos

Here are some additional photos:

Elevated dishwasher (6/7)

How to include an elevated dishwasher into an Ikea kitchen.

  1. Introduction
  2. Design and constraints
  3. Heights and support beams
  4. Materials
  5. Construction
  6. Installation on-site
  7. Downloads and photos

This is part 6.

Installation on-site

The second phase, after construction of the cabinets, is the installation on-site.

1. Prepare the location

  • Make sure there are enough power outlets behind the Tower for all the appliances. They should have have a ground (PE) lead.

  • Make sure there is an available tap for the water intake hose and a point where to connect the waste water hose.

2. Build the Tower

  • Fix the 2 ‘uncut’ wooden support beams to the wall to the right, such that they (1) rest on the floor, (2) are vertical, and (3) form a plane that is at right angles to the plane of the back wall. Depending on the wall, this might be easy or nearly impossible.

  • Position the bottom cabinet; adjust the legs so that it stands level. Then, position a 220 cm tall side cover panel (Förbättra) between it and the beams, onto the narrow ledge of the wooden panel that sticks out from under the right side of the bottom cabinet. Screw the cover panel onto the support beams with a few ‘light’ screws; they only need to hold up the cover panel until the top cabinet is screwed in place. Mark where the beams are located behind them with a pen, as shown on the photo. After that (not shown in images below), with the same light screws, fix one of the two 40 cm side cover panels above it, with the exposed/cut side facing up.

  • Fix the third wooden support beam to the back wall, using the bottom cabinet as a guide for its position left-to-right. (The beam must rest against the inside face of the left side panel.) This beam must also be vertical.

  • Position the bottom cabinet at the correct distance from the back wall, and fixate it to each support beam with a screw. In our case, to find the correct distance from the back wall, we needed to place the neighboring cabinet, because this distance was determined by the tiles, higher up. (Note that this cabinet rests with its right side panel on the wooden support panel screwed to the underside of the bottom cabinet.)

  • Position the wooden spacer frame on top of the bottom cabinet, and the top cabinet on top of that. Make sure its panels are vertical / horizontal, and fix it by drilling several screws through the side panels into the support beams.

  • Cut a recess out of the remaining side cover (Förbättra), so that the dishwasher’s water in- and output hoses can enter the Tower from the side. (I made a mistake, which is why the panel in the photo has 2 recesses.) Then, screw the cover onto the bottom and top cabinets with the screws Ikea provides (from the cabinet’s inside), as well as the remaining 40 cm piece. Especially towards the front of the top cabinet, be generous with screws – it’s not inconceivable the cover carries a small part of the load – but consider where appliances will go so that they won’t be in the way. When done, remove the spacer frame.

  • The cabinet to the Tower’s left also needs a cut-out recess for the hoses, and cutting a strip off the bottom panel is also practical. Depending on what you’ll use this cabinet for, make sure these recesses are far enough towards the back not to interfere with e.g. drawers.

3. Install the appliances

  • Prepare the dishwasher by shortening its feet as much as possible; it should now be just under 82 cm tall and therefore have 1 cm vertical wiggle room.

  • At about 40 cm above the bottom cabinet, install a hook or sling to hold the waste water hose. This is necessary to provide the dishwasher with enough pressure for the waste water pump. To protect against moisture from water vapour, I put some aluminium tape on the underside of the top cabinet (not visible in photo).

  • Slide the dishwasher in place, while continuously feeding the hoses through the hole. Because of the reduced thickness of the support beam here, the dishwasher should fit all the way into the Tower, with its front door (still without front panel) being flush with the cabinets to its top and bottom. After connecting (and possibly extending) the water hoses and the power cable, you should probably do a test run, before it gets harder to remove the device.

Also, place the drawer into the bottom cabinet. The dishwasher door should open flat and not come to within less than a cm of the drawer front panel.

(The front panel of the dishwasher needs more space than that, but we’ll jack up the dishwasher before we attach it.)

  • Prepare the steam oven by crimping an electrical plug onto its power cable, if none is present.

  • Put the steam oven onto the bottom panel of the top cabinet, i.e., directly above the dishwasher, and connect the power cable. Also, place a shelf directly above the oven, so that no vertical space is wasted – the metal brackets and shelf come with the tall refrigerator cabinet. The shelf probably needs a small piece cut out in order to fit around the support beam.

  • Put the microwave on top and also connect the power cable. Also, put a shelf directly above it. This time another type of shelf is used, i.e., one for which no metal brackets are needed. This shelf should be the remaining shelf from the refrigerator cabinet.

  • When all is working, it’s time to install the front panel onto the dishwasher door. First use the adjustment screws to raise all (3) feet of the dishwasher by about 1 cm, so that it sits snugly in its recess. Measure the necessary front panel height; this should be about 72 cm as calculated above, but can probably also be a bit more, because the front panel could extend a little bit (~0.5 cm) beyond the dishwasher’s upper edge. As the steam oven’s door opens downward, make sure it is opened when measuring, so it doesn’t crash into the dishwasher’s front panel later. After shortening the 60×80 cm panel (by circular saw and/or in the hardware store, if possible), follow the dishwasher’s installation instructions to install it. Make sure the cut face is facing downward, so it’s out of sight.

  • And finally, the storage space above the microwave needs a door. For this, too, a 60×80 door panel is shortened, in this case to 63 cm (after measuring the necessary height). This is not the easiest way out – new sunk holes need to be drilled to fix the hinges, for example – but it is the prettiest. Alternatively, a 60×60 cm door could be used and fixed to the cabinet. For the few cm that remain uncovered by the door, a fixed blind would be cut and installed.

And here’s the finished Tower:

Continue to: downloads and photos

Elevated dishwasher (5/7)

How to include an elevated dishwasher into an Ikea kitchen.

  1. Introduction
  2. Design and constraints
  3. Heights and support beams
  4. Materials
  5. Construction
  6. Installation on-site
  7. Downloads and photos

This is part 5.


Building the Tower is done in 2 phases. First, all panels are cut to size and the bottom and top cabinets are built, among others. The image below shows how the materials from the 2 ikea cabinets are used to do this. How the 2 ikea cabinets (on left) are used to form the bottom and top cabinet of the Tower (on right). Parts that are discarded are marked with ‘x’. Note the colors, and how the upper half of the tall cabinet is actually turned upside down to form the top cabinet of the Tower.

1. Cut panels to size

From the tall refrigerator cabinet, take the 200 cm tall side panels, and cut them so as to keep 27 cm (27.2, to be exact) from the bottom (for the bottom cabinet), and 146 cm (or 146.2) from the top (for the top cabinet). Make sure there is a left and a right version of each; marked in light green in the image above.

From the small narrow cabinet, take the 2 side panels and shorten them to 60 cm (they are 80 cm when bought). These will be the top panels for the cabinets; marked in dark green in the image above.

From the Förvättra side cover panels, take one and cut it so as to keep 40 cm from each end. Leave the other 2 untouched. (Not shown in the image above.)

All these cuts are best made with a circular saw or at your local hardware store.

2. Make room for support beam

As you’ll see in the photos below, I’ve cut out a rectangular part of the 6 horizontally mounted panels/shelves, just as is shown on the image above. This is to make room for the left support beam. However, I suggest you do the following:

  • For the bottom and top panels of the cabinets (4 pieces; horizontal light and dark green in the image above), shorten their depth by x, where x is the size of the beam in direction perpendicular to the wall (7 cm in our case). This too is done best with a circular saw or in the hardware store. This is less work than cutting out a rectangular piece, and it allows for better ventilation.
  • For the 2 shelves, wait until the top cabinet is hanging, and cut them as necessary by positioning and measuring. You can use a keyhole saw for this.

3. Construct bottom cabinet

  • Connect the 27 cm tall side panels with the bottom panel, as per the manual.

  • Screw the 45 by 65-70 cm wooden panel to the bottom. Locate it such, that it’s sticking out of the right side by 1 cm (by all means, no more than 1.3 cm). Note that the right side is on the left when the cabinet is flipped over. On the other side, there is a larger part sticking out.

  • Screw on the legs, or leg holders. It is here that the legs need to be shortened by the thickness of the wooden panel, compared to the legs carrying all other cabinets. If you use the standard 8 cm Ikea legs, it might be easier to use wooden blocks to fill in the remainder.

  • Flip the cabinet right side up, screw in the rails for the drawer (in image below, right one is missing, and left one is only barely visible), screw in a bracket (including white plastic spacer block) in the right bottom corner (comes with all ikea cabinets, and comes in handy when trying to find the right distance from the back wall).

  • Take an upper panel and screw the metal brackets into it so that their vertical parts are 1.8 cm in from the sides. Their location front-to-back is not really relvant. Their function is to reinforce the panel carrying the dishwarsher.

  • Put the upper panel onto the cabinet and fix it with screws from the inside, as in the image below. Do not use screws with a very large head, as this might obstruct the opening and closing of the drawer. Additionally, one can drill some screws vertically down, through the upper panel into the side panels (I’ve also installed the flimsy back cover in this photo, but that’s not necessary, as the wooden support beams and metal brackets will offer plenty support against deformation.)

4. Construct top cabinet

Connect the 144 cm tall side panels with the remaining panels, in the same way as done for the bottom cabinet. The top panel is screwed onto the side panels by drilling some screws vertically down. No brackets or back cover are necessary, which means however that the cabinet will not resist deformation until it is installed.

5. Construct spacer for dishwasher recess

Using the four 83 cm tall wooden bars, construct a frame as shown in the picture below.

6. Prepare support beam

Take one of the three beams and cut recesses out of it according to the image below. Depending on what you have at your disposal and how thick the beam is, use a keyhole saw or milling machine, or be inventive. Note again that we used 15 cm legs; if you use the standard 8 cm Ikea legs, adjust the location of the recesses accordingly by subtracting 7 cm from the distances mentioned in the top part of the image.

Continue to: installation on-site

Elevated dishwasher (4/7)

How to include an elevated dishwasher into an Ikea kitchen.

  1. Introduction
  2. Design and constraints
  3. Heights and support beams
  4. Materials
  5. Construction
  6. Installation on-site
  7. Downloads and photos

This is part 4.


The following materials were used for the construction of the Tower:

  • From Ikea: a tall refrigerator cabinet, 60×60×200 cm. We chose this one with article number 902.135.68, which comes with 2 distinct shelves:

  • From Ikea: a small kitchen cabinet, 20×60×80 cm, with article number 302.125.62: We keep only the side panels of this cabinet; after cutting them to size, they form the top panels of the cabinets in the Tower.
    Alternatively, buy the cabinet 090.072.81; it’s more expensive, but the side panels are already the right size. You might actually be able to buy this article without legs, door, and shelves.

  • From Ikea: a 60×60 cm maximera drawer. A high one (article number 902.046.39) just fits into the bottom cabinet.

  • From Ikea: 60 cm wide front covers for the drawer, the dishwasher, and the storage space. The drawer cover is 40 cm tall, for the others we’ll need to shorten 80 cm tall door fronts. For our kitchen, the style is Veddinge white, which means we need 302.054.39 (once) and 202.054.30 (twice)

  • From Ikea: 3 side cover panels, 62×220 cm. The Förbättra white ones (902.978.84) are matching our kitchen’s style.

  • From Ikea: the appliances.

    • Dishwasher Välgjord (discontinued). There is a successor, but I don’t know its name. Any standard-size 60 cm wide dishwasher can be used, as long as it has a (removable, shortenable) front cover. Important aspects are the size of the door and the location of its hinge point – see section “Heights” – and its depth – see section “Support beams”. Both sections are found on the previous page.

    • Steam oven Kulinarisk, 303.009.12. Any other built-in oven can be used; just make sure you check the depth; see section “Support beams”, above.

    • Microwave Framtid 903.033.90, though it could be any built-in 60 cm wide microwave. It’s also possible to leave out this appliance; note that it’ll be pretty high up.

  • From the hardware store: 4 metal right angled brackets, about 5 cm on each side.

  • From the hardware store: a wooden panel of 45 by 65-70 cm, about a cm strong. This is fixed to the underside of the bottom cabinet, to carry the weight of the side covers.

  • From the hardware store: 3 wooden beams, 5×7×275 cm.

  • From the hardware store: 4 wooden beams, 83 cm long and several cm × several cm thick, as well as eight 59 cm long laths. These can be scrap wood and won’t be present in the final Tower.

  • From the hardware store: plugs and screws to fix the beams to the wall, as well as a range of wood screws of various sizes, lengths and strengths.

  • From elsewhere (amazon?): legs for the cabinets. As noted, we have 15 cm legs under all our cabinets. Make sure the leg lengths are adjustable and can be shortened by at least 2 cm.

Continue to: construction

Elevated dishwasher (3/7)

How to include an elevated dishwasher into an Ikea kitchen.

  1. Introduction
  2. Design and constraints
  3. Heights and support beams
  4. Materials
  5. Construction
  6. Installation on-site
  7. Downloads and photos

This is part 3.


In this section, we’ll calculate the heights of the 3 parts mentioned above: bottom cabinet, dishwasher recess, top cabinet.

The bottom cabinet is the base on which the dishwasher is standing. As noted above, its front must be 40 cm high, and we must now figure out, how tall the cabinet’s ‘body’ (h2 in the image below), must be. The goal is to make it so, that the dishwasher sits as low as possible. Yes, we want an elevated dishwasher, but elevating it by about 55 cm (40 cm cabinet + 15 cm legs) would be too much, also considering there are 2 appliances above it that we’d like to be able to still use comfortably.

Bottom cabinet and dishwasher

We first look at the front panels. Because the dishwasher is usually installed under the countertop, it is fitted with a panel with a height h1 of 80 cm, just as the other cabinets under the countertop. If we’d use that same panel in this case, the top of the dishwasher would sit 120 cm above the bottom of the cabinet. However, as the door itself is actually less tall, the panel does not need to be 80 cm. In the case of this particular dishwasher, I found that I could to shorten it to h1 = 72 cm and still leave some room when opening the door flat; see the 3 images below.

Choose h1 such that B > A. That way, the drawer panel (not shown) and the dishwasher panel don’t collide when opening the dishwasher.

Having determined that the dishwasher front panel height h1 = 72 cm, we can look at the heights of the cabinet and dishwasher, which, like the front panels, must add to 112 cm. The dishwasher’s height g can be varied between 81.8 and 89.8 cm. We want to choose a low value, so that more is left for the cabinet height h2. If we choose the dishwasher height g = 83 cm, we can create a vertical wiggle room of about 1 cm which helps when installing or removing the dishwasher. Subtracting g from 112 cm, we get the bottom cabinet height h2 = 29 cm.

As the total Tower has a height of 260 cm (excluding the legs), subtracting g and h2 leaves the top cabinet height of 148 cm.

Tower of Power, view from left, showing the heights of the 3 parts: bottom cabinet, dishwasher recess, and top cabinet. The front panels are not shown, nor are the wooden support beams or the side cover panels. This drawing starts at 15 cm from the floor, as it does not show the legs.

Support beams

3 wooden beams are used to support the top cabinet. Their location is shown below; all run the entire height,from the floor to the top of the Tower, and have a height of 275 cm (260 cm + 15 cm legs). Choosing all 3 with a cross-section of 5×7 cm will do fine. The rest of this section is explaining the reasoning behind that choice, as well as showing the cut-outs necessary on one of the beams.

Tower of Power. Top view, showing support beams (brown), top cabinet sides (yellow) and cover panels. Also showing the screws that fix the beams to the wall (green), and those fixing the cabinet to the beams (purple).

Beams on right

The 2 support beams on the right are fixed to the wall to the right. For their size A, there are several considerations:

  • The screws fixing the beam to the wall, as well as those fixing the top cabinet to the beam, are driven in left-to-right. Therefore, A must be large enough to provide some ‘wood thickness’ to hold those screws; e.g. at least 3 cm.
  • Size A determines the Tower’s distance from the right wall. Most people will want to have this as small as possible, but in our case, the presence of a bulky door frame meant the Tower (and therefore, A) needed to be at least 5 cm from the wall, so that value was chosen.
  • In the rare case the walls are not perpendicular to one another, different values for A can be used to create a plane that is at right angles to the back wall. Also, if the right wall is not smooth, the beams are used to level it out; see the images of the construction phase, further below.

The other dimension of these 2 beams, B, has no influence elsewhere and must only be enough for them to be strong. They are 7 cm in our kitchen, mainly so that all beams could be the same size.

Beam on left

The third support beam is located inside the Tower so that it is not visible. It is fixed to the back wall. For this beam, its size C is not critical: it must be enough to providing enough support (to purple screws) and not split (due to green screws). A minimum of ~4 cm seems sensible, but 5 was chosen here to keep in line with the other 2 beams.

The size D provides most challenges. First of all, it must be large enough, because:

  • The larger D, the more bending resistance the beam has.
  • The larger D, the more space there is to place the screws (in purple), so that they are both far enough away from the edge of the (bright-yellow) top cabinet (which itself is already located 1 cm in front of the wall), and far enough away from the front side of the beam so that it doesn’t split. 7 cm seemed a good value.

The problem with this value for D however, is, that it needs more space than is available behind the appliances (dishwasher, steam oven) and drawer (in bottom cabinet). Therefore, some cutting is necessary, which is calculated here:

  • Ikea’s cabinets stand 1 cm from the back wall, and have a depth (i.e. perpendicular to that back wall) of 59 cm, so that there is 60 cm of space in total.
  • The dishwasher is 55 cm deep, plus a (measured) 0.4 cm air gap between the door and the door panel. This means the rear side of the dishwasher stands 4.6 cm from the back wall. This is less than the 7 cm needed for the support beam.
  • The steam oven leaves a few mm more space – its rear side stands 5.4 cm from the back wall – but it’s still not enough for the beam to pass behind it. appliances sizesAppliance sizes. Dishwasher is 55 cm deep (+ 0.4 cm air gap), the steam oven 54.6 cm.

The solution has been to make the support beam behind these appliances thinner. About 6 mm more was removed than mathematically necessary, so that the beam is 4 cm thick behind the dishwasher, and 4.8 cm thick behind the steam oven. Additionally, a small recess was added to accommodate the bottom cabinet’s drawer rails. The exact measures are shown further below.

Tower of Power, view from left. Only the appliances and the ‘hidden’ support beam are shown.

Continue to: materials

Elevated dishwasher (2/7)

How to include an elevated dishwasher into an Ikea kitchen.

  1. Introduction
  2. Design and constraints
  3. Heights and support beams
  4. Materials
  5. Construction
  6. Installation on-site
  7. Downloads and photos

This is part 2.


For reasons of taste, we wanted to have all ‘front-facing appliances’ in one tall cabinet, so that the rest of the kitchen would only have clean fronts. From bottom to top, this ‘Tower of Power’ consist of a drawer, dishwasher (Välgjord), steam oven (Kulinarisk), microwave (Framtid), and a storage cabinet.


The Tower is be used instead of the tall cabinet in the image below.

The design has the following constraints:

  • The single most influential constraint is this. Unlike the steam oven and microwave, which are several cm narrower, the dishwasher has the same outer width as Ikea’s cabinets (60 cm). This means that it’s not possible to simply use a standard tall cabinet and put the dishwasher onto a shelf. Rather, there must be a vertical empty space, beneath and above which there is a 60 cm wide cabinet. The tower therefore consists of 3 parts:
    1. a bottom cabinet consisting of a drawer;
    2. the dishwasher;
    3. a top cabinet consisting of the steam over, microwave, and storage space.
  • The drawer in the bottom cabinet must have a front panel of 40 cm, just like the drawer next to it. As this is too high as a base for the dishwasher, the cabinet containing that drawer will not be 40 cm tall, but considerably less. See the next section “Heights” for clarification.

  • The entire tower must be 260 cm tall (excluding the legs), so as to keep in line with the rest of the kitchen.

  • We are tall people and chose to use 15 cm legs for all kitchen cabinets, instead of using Ikea’s 8 cm legs. This means the tower is 275 cm tall including the legs.

  • The side of the dishwasher should not be visible. In order to achieve that, cover panels on both sides of the tower were added.

  • The dishwasher should not bear load from the appliances/cabinet located above it. As the cover panels can’t/shouldn’t bear much load either, and as the tower isn’t neighboring other tall cabinets, this means that reinforcements are necessary to carry the weight of the top cabinet. For this, wooden support beams are used near 3 of the 4 vertical edges of the cabinet; 2 on the wall to the tower’s right, and 1 on the wall behind it.

This resulted in a preliminary sketch, which was later digitised in the image below.

Tower of Power. Front and side cover panels are not shown.

Continue to: heights and support beams

Elevated dishwasher

We wanted an Ikea kitchen. We also wanted to have a dishwasher at an elevation, i.e., not under the countertop, but rather integrated in a tall cabinet tower. As Ikea does not offer this, I’ve had to come up with a solution, which I’m sharing here in case someone would like to do the same.


  1. Introduction
  2. Design and constraints
  3. Heights and support beams
  4. Materials
  5. Construction
  6. Installation on-site
  7. Downloads and photos

This is the first part.


Here you see the finished product, with the dishwasher door opening flat at about 55 cm from the floor.

The finished product.

As long as you don’t forget to think for yourself, this guide should help you build this too – even if you might not want to duplicate this exact design and reduce the total height, use other or less appliances, and/or a different length for the legs.

Signpost: if you’re interested in the reasoning and calculations behind this project, follow the link below to start with the part about the design. If you’re only interested in how to build this exact tower, feel free to skip ahead to the list of materials. A download for the design file (in Sketchup) is found on the final page, at downloads and photos.

Continue to: design and constraints

Interpreting the demand curve

This is becoming a bit of an overkill, but it’s an interesting exercise to see if we can up with the correct answer ourselves.

What are we trying to accomplish?

We have

  • an infinitely divisible good; and
  • a consumer with demand function $q = Q(p)$ for this good, with corresponding inverse demand function $p=P(q)$

Both functions described the same curve (the demand curve) in the $q,p$-plane, and we are trying to establish whether a point $(q,p)$ on the demand curve describes

(a) the average price per unit $p$ that the consumer is willing to pay when buying the total quantity $q$, or
(b) the price per unit $p$ that the consumer is willing to pay for an additional amount $\text{d}q$, given a possession of $q$ units.

Let our thesis be that it is the latter, and let’s see if we run into a contradiction.

Consumer surplus

Let’s take the simple straight line $p+q=25$, where a constant unit-price of 5 prescribes a demand of 20:


The area between the demand curve, $p=5$ and $q=0$, is called the Consumer Surplus CS, and we can calculate it by integration:
$$CS = \int_5^\infty Q(p)~\text{d}p=\int_5^{25} (25-p)~\text{d}p$$
This is the same as
$$CS = \int_0^{20} (P(q)-5)~\text{d}q=\int_0^{20} \big((25-q)-5\big)~\text{d}q$$
The latter shows the area might be interpreted as the difference between what the consumer is willing to pay and what he is paying — at least, if our thesis is valid and interpretation (b) holds.

We find that the CS in this case is 200.

Our thesis can help to interpret this. If the price is 5, the consumer keeps on buying until the additional quantity he could buy does not bring that additional utility of 5. That happens to be at a quantity of 20. Because he was able to buy all units at a price of 5, this represents an advantage: he would have paid more for an additional unit when he still had fewer units. E.g. when he still only had 15 units, he would have paid 10 per unit for additional quantity.

Total Willingness To Pay

The total willingness to pay TWTP for a quantity $q$ (i.e., the maximum accepted price for that quantity) can be calculated as the sum of the willingness to pay for each subsequent unit until $q$ (i.e., the maximum accepted price per unit for additional units):
$$TWTP(q) = \int_0^{q} WTP(q’)~\text{d}q’$$

So, in order to calculate how much our buyer would maximally have paid (in total) for those 20 units, we must add the maximum prices for each individual unit. If our thesis is correct and interpretation (b) holds, this is exactly the inverse demand function $P$, so that
$$TWTP = \int_0^{20} P(q’)~\text{d}q’$$
This Total Willingness To Pay for 20 units happens to be 300 in our case.

Now, under normal circumstances the buyer does not need to pay that amount for 20 units, but rather $5\cdot20=100$. The difference between the two is the CS of 200. This is the area above the $p=5$ line, which is what we would expect.

Should we have perfect price discrimination, the seller of the good would know the buyer’s demand curve, and sell him each unit of the good at exactly the maximum price he’d be willing to pay for it; gradually dropping the price with the buyer’s marginal utility: $p=25-q$. That way, the seller is able to capture all of the CS, and the buyer would thus pay 300 for the 20 units.

If our thesis is wrong

This interpretation only works if our thesis is correct and the inverse demand function $P$ describes the willingness to pay for each additional unit. If it describes the willingness to pay per unit for that and all previous units, i.e. interpretation (a), things are different. In that case, the TWTP for $q$ units is simply the multiplication of $q$ and $P(q)$:
$$TWTP(q) = q\cdot P(q)$$
In order to figure out how much would be bought at non-uniform pricing, we need to find the willingness to pay for each unit. That WTP is, as can be seen from the first equation, the derivative of the TWTP, so, in this case:
So, the first quantity is sold at a unit price of 25, just like before. This makes sense, as the buyer does not have any units yet, so the marginal price equals the average price. Then, however, under perfect price discrimination, the price of the good should drop twice as fast as we have previously calculated. That is due to the fact that the additional unit $\text{d}q$, that the seller is trying to sell, does not have a marginal utility given by its price (as is the case in interpretation (b)) but by the increase in total price.1
Moreover, in this situation, the seller sells his last unit for a price of 5, which is when he has sold only 10 units (compared to 20 if interpretation (b) is correct). The buyer has then spent $\int_0^{10} (25-2q)~\text{d}q=150$, which is his TWTP, but for 10 units. This too makes sense: the demand curve prescribes a maximally accepted (average) unit price of 15 – which is exactly what is being paid.

What we cannot see anywhere, is the figure of 200 which is the area above the $p=5$ line. In fact, the Consumer Surplus that was 200 in the case of interpretation (b), is actually 0 in the case of interpretation (a) — simply because of the way the inverse demand function is defined to be the maximum average unit price: if we have a uniform price, the price is the average price, and the buyer will have an incentive to buy more as long as his willingness to pay is higher than the price. Exactly when he buys the quantity $q$ that, on his demand curve, corresponds to the offered price $p$, is the average price he’s willing to pay equal to the offered price. Because the average price he’s willing to pay, times the quantity, is the total price he’s willing to pay for that quantity, and because that also equals the price he is paying at that point in the curve, his CS is 0.


In order to have a sensible interpretation of the area between the demand curve, $p=5$ and $q=0$, called the Consumer Surplus CS, we need to interpret the inverse demand function to mean: “the price per unit $p$ that the consumer is willing to pay for an additional amount $\text{d}q$, given a possession of $q$ units” (b).
The common interpretation (a) as “the price per unit $p$ that the consumer is willing to pay, for each unit, for a the total quantity $q$” is incorrect. It’s easy to see, however, why it is often interpreted that way. Firstly, it is a simpler interpretation that’s easier to visualise, and secondly, in everyday situations — which all have uniform pricing — it still predicts the correct quantity to be traded.


1: This gives us another way to come to the formula. Consider the buyer, which buys a quantity $q$ when the average price per unit is $P(q)$. As he buys a quantity $q+\text{d}q$ when the price per unit is $P(q+\text{d}q)$, the total cost increases by $P(q+\text{d}q) – P(q)$, which means that is the utility of the additional unit $\text{d}q$. So, the marginal utility per unit is $\frac{(q+\text{d}q)\cdot P(q+\text{d}q)-q\cdot P(q)}{\text{d}q}$. Using the inverse demand function $P(q)=25-q$, this is turns out to be $25-2q$.