this is another example of consumer and producer surplus but this time with a price ceiling imposed so recall that we have the supply and demand functions which in this example are linear but again they don't always have to be we have the supply function of 31 plus 0.5 x an increasing function with a y intercept of 31 and then we have the demand function of 79 minus x that's a decreasing function with a y-intercept at 79. typically what you want to do is begin by finding the equilibrium quantity and the equilibrium price that would naturally occur through market for market forces and you find this equilibrium quantity in equilibrium price by setting the two equations equal to each other so if i set the demand function 79 minus x equal to the supply function 31 plus 0.5 x and then proceed to isolate x and solve the equation value of 32 and if you put that back into either of the original supplier demand functions you will get a price of an equilibrium price value of 47. okay so this is the point 32 comma 47. from this point we would proceed to figure out the consumer surplus and the producer surplus under the conditions of equilibrium but instead we have a price ceiling we have a price ceiling imposed on the situation at 41. now notice that 47 is the equilibrium price so price came in at 47. a price of 41 is going to be below the equilibrium so 41 a price ceiling set at 41 falls below the equilibrium price of 47. what's the impact of this well remember that normally the consumer surplus is the triangle above the revenue rectangle below the demand curve and the producer surplus is the triangle below the revenue rectangle but above the supply curve if we impose a price ceiling which means the price is not allowed to be any higher the price ceiling in here at 41 is of course lower than the equilibrium price of 47. what does that do to the triangles it makes the consumer surplus a bigger area but cuts into the producer triangle making the producer surplus a smaller area this is because the producers have to settle for a price lower than they should have got and consumers are getting a bigger bonus by needing to spend less money than equilibrium dictated that they should spend price ceilings typically get put into effect for different legislative or regulatory reasons in order to protect the consumers for various circumstances all right so now what we have is essentially a new balance point if you will located at the place where the price ceiling has been put into effect and we need to figure out the new x value the quantity that corresponds to the price ceiling intersection now this price ceiling a horizontal line at 41 will be intersecting the supply curve so to figure out this new point of intersection we want to set the horizontal line y equals 41 set that equal to the supply function so we set 41 equal to the supply function which was 31 plus 0.5 x solving for x we get a new intersection value of a new x quantity value of 20 and this new point of intersection is at the point 20 comma 41. this change as i mentioned earlier will provide a new consumer surplus and also a new producer surplus which we now can calculate remember that the area under the demand curve above the x-axis is the integral of the demand function but now we only want to capture the area that's between 0 and the x of the quantity value of the price ceiling so between 0 and 20 we want to integrate between 0 and 20 of the demand function which was 79 minus x and that will give us that area under the demand curve between 0 and x sub c but we only want the portion that's above the re the revenue rectangle so after we get the complete area that's under the demand curve we want to subtract off that portion of the revenue rectangle and remember that was obtained by multiplying length times width which was 41 times 20. so after we get this revenue we subtract off the the revenue rectangle and that should give us the portion that's under the demand curve but above the revenue rectangle you'll go ahead and integrate that demand function and that should give you 79 x minus x squared over 2. evaluating that f of b minus f of a again f of a the lower bound is just going to be zero so just evaluating it at f b putting 20 in everywhere there's an x that's going to give us that's going to give us 1380 and then minus zero is 1380 and then remember we got to subtract that revenue rectangle so 1380 minus 820 gives us 560. so the new consumer surplus the consumer surplus due to the price ceiling is going to be 560. and that's a larger consumer surplus than we than we would have had at equilibrium because of the extended area here now we want to calculate the producer surplus which is the area of the triangle under the horizontal line at the price ceiling but above the supply curve let's go ahead and calculate that so we can find the producer surplus by beginning with the revenue rectangle that's the area under the horizontal line of y equals 41. we calculated that to be 820 subtract the area that's under the supply curve so that would be the integral of supply so the new producer surplus will begin by taking the revenue rectangle area and subtracting the integral from 0 to the quantity at the ceiling which recall was 20. of the supply function 31 plus 0.5 x integrate that 820 minus 31x let's keep this in brackets 31x plus 0.5 our constant will copy through x squared over 2. we still need to integrate this between 0 and 20 prior to the subtraction again putting 0 in is just going to be 0 for our lower bound so evaluating it reduces to just putting the upper bound in so go ahead and put 20 in for x evaluate the integral get the result and subtract that from 8 20. finishing this up gives us a producer a new producer surplus of 100. the other item that's left to calculate is what's known as the deadweight loss and the deadweight loss is indicated by the area of the triangle formed between the demand and the supply curves and the vertical bar from the quantity at the ceiling so we need to figure out the area of this triangle that's what's known as the deadweight loss you can think of that intuitively as surplus that's lost on account of someone due to the fact that you imposed an artificial price at the price ceiling calculating this area will be found by taking the demand curve and the integral the integral of the demand curve and subtracting the integral of the supply curve notice that the area under the demand curve is everything from the demand curve on down to the x-axis under the supply curve on down to the x-axis if we subtract that from the demand curve we will be left with the triangle so again everything under the demand subtract everything under the supply and that leaves you with this small triangle in between which we call the deadweight loss the order of subtraction is such that the top curve is always first subtracting the bottom curve and since demand from a vertical perspective is higher it's demand minus supply notice that the bounds of integration run from x sub c that's the quantity at the ceiling up to x sub e that's the quantity at equilibrium we want to integrate between these two x values so writing that out more formally we have x sub c was 20 and x sub e was 32 of the demand function first and remember that one of the properties of integrals was that we can subtract them separately or combine them into one provided that the bounds are the same and they are in this example so we can simplify this expression inside the integral first prior to integrating let's go ahead and do that 79 minus 31 and then we have minus x and then minus 0.5 now to integrate we will have 48 x minus 1.5 x squared over two and we want to evaluate that between twenty and thirty two f of b minus f of a we will need to evaluate both items because neither one is a zero so we'll first go ahead and evaluate this antiderivative at 32. putting 32 in everywhere there's an x and then we want minus f of a so anti-derivative evaluated at the lower bound of 20. 1.5 x squared over and then after we get these separate evaluations we'll subtract them for our fundamental theorem of calculus so go ahead and work that through so i'm getting 15 36 minus 768 over here 960 minus 300 so that works out to 768 minus 660. a deadweight loss of a hundred and eight