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okay hello and welcome everyone in this video I'm going to talk about some different contracts and the way that these influence incentives for workers so alright so for our set ups suppose we have the goal of hiring workers to produce gizmos we'll assume each worker has some effort cost function the idea is that though the worker might enjoy the job there's some costs associated with exerting high levels of effort so be given by this quadratic one-half e squared minus 4 e plus 16 we'll assume the workers utility is just going to be whatever is their compensation which could include a fixed and a variable component minus this effort cost will assume the worker will have some outside option so in this case 30,000 we'll assume total gizmo production is just Q is equal to e so there's one-to-one relationship between one unit of effort and one gizmo produced and we'll assume we can sell each gizmo for $16,000 and we'll assume our only cost as labor so maybe our gizmos are some type of service I don't know where we have some really cool production technology either way so suppose we suppose we pay only a fixed wage what's going to be the effort for our worker that's the first part is like alright suppose we pay some flat wage a lot of jobs do this a lot of salary jobs a lot of hourly wages are just gonna be completely invariant to the actual output we're gonna assume away the possibility of any type of bonuses or something that can be conditioned on meeting a particular threshold no in this example we're just thinking about some fixed wage and then what's going to be the effort that results then we can think about what the wage would have to be to get them to agree get the worker to agree to that contract then in C we'll think about what are our profits basically a B and C are all kind of figuring out what's going to happen if we pay a fixed wage D is gonna be investigating what happens if we have a full incentive contracts and now we're gonna vary the fix component where we're gonna break down the wage into a fixed component in a variable component this is a little bit reminiscent of a two-part tariff in terms of like in this case not extracting surplus from consumers but extracting surplus from the workers right so hopefully see that relationship then we can think does the worker or the entrepreneur have some preference between these contracts you should be able to answer that right away which is if we're looking at what our workers outside option is which was 30,000 probably the optimal way to build these contracts is to make them exactly indifferent between their outside option of 30,000 and our fixed wage and hour full incentive contract and indeed that's going to be the result and then we'll think about like which one's gonna be profit maximizing turns out under this type of setup in relatively simple models like this making some assumptions the full incentive contract is the profit maximizing contract so anyway so let's go ahead and see what happens if we pay only a fixed wage well so we need to write down the workers utility function that's going to be given by their wage which is just the entire compensation in this case it's a fixed contract so it'll just be the fixed component minus the effort cost function all right so W minus 1/2 e squared minus 4 e plus 16 and then distributing the sign and then taking the derivative differentiating the utility with respect to effort right so the idea here is we want to find this is the workers utility maximization problem where they're choosing their effort level so take the derivative with respect to our choice variable in this case effort we find the optimal effort level is 4 for the worker when in matter of fact it doesn't matter what the wage is gonna be they're always going to exert an effort level of 4 so we better take that into consideration right so what's going to be the wage we need to give so that they're willing to do this well remember the outside option was 30,000 I'm just gonna drop the K or drop 2,000 and just make it 30 for ease of computation so we'll set 30 equal to their utility in this case the utility is gonna be well the same function above now we're gonna evaluate this at 4 right okay so we have 30 is equal to a minus this expression which boils down to just 8 right because this is 16 minus 16 and positive 16 so this part is just 16 and it's this drops out so this part is 16 divided by 2 is 8 right and then so oops so here I got ahead of myself I think I was going through and I was trying to correct my work things look like my handwriting neater this is actually wrong right here I'm trying to highlight it won't let me do it kind of a little bit so this is this should be 30 still right so where did this 8 come from it has a mistake I meant to write 38 down here so as we're solving here just one more time we have 30 is equal to this stuff evaluate effort at 4 we're gonna find that we'll get 30 is equal to a minus 8 move the 8 to the other side so a has to be 38 right so then thinking about this make this make better sense the effort cost associated with a level of effort equal the level of effort of four it's gonna be is gonna be 8 and we need the worker to get at least 30 otherwise they'd rather take their outside option so we have to pay them 38 that they're compensated for their effort and then they decide to accept this contract right okay so we'd have to pay them 38 and then they're exactly indifferent all things considered between this contract and not working for us because though this 38 looks a lot bigger than the 30 they're getting from their outside option they have an effort cost of 8 when they set a level of effort equal to 4 okay so what about our profits well we said we could sell each gizmo for 16 so we can sell each gizmo for 16 we're gonna get 4 gizmos because there's a one-to-one relationship between gizmos and effort hold this right here Q is equal to e we're assuming no random shock no random variation in the model 16 times 468 or 16 times 4 is 64 minus 38 is 26 so our profits is gonna be 26 all right so what about the full incentive contract now suppose we vary both suppose we break up the compensation to a fixed in a variable component what's gonna happen now well the first thing we have to do is we're going to find the variable component we should get a hint from this from full incentive right full incentive contract means we want to give the worker the full returns of their effort so we're gonna find how does how does profit change with respect to effort and then let's give that amount to the worker so deep IDE is 16 so that becomes B right deep IDE is 16 so that becomes B now we have to solve for the associated effort level when they get through the full returns of their effort so utility is gonna be a plus 16 e minus this expression effort cost and if we differentiate we'll find o 16 is gonna 16 minus e plus 4 means they'll set an effort level of 20 alright so for workers setting an effort level of 20 what's gonna be the fixed component that we're gonna need to set to complete this contract since we're setting both a and B in order to keep the worker exactly indifferent between working for us and not working for us so here's their outside option 30 I haven't replaced it to 38 anywhere of course not so 30 on the left-hand side is just their outside options right hand side is their utility function well is that is the utility which is generated from the wage minus the effort cost function so a plus 320 - 200 this is 400 divided by 2 is 200 here's 4 times 20 is 80 and then here we have minus 16y because this minus sign comes through and we'll find that a we'll end up having to be minus 154 so I should move this line a little bit this is 3 or something the a has to be minus 150 for you can see that obviously because we move this one 84 to the other side there's 30 sitting there so it's negative 154 at first you're like wait a second they're gonna pay us $154,000 whatever well right so you can think of this as like the type of contract where you're a hundred percent commission based so the idea is that they're giving us all this value think of like a car salesperson at the dealership long hours whether they make a sale or not just keeping somebody there for when consumers actually show up right and then so they're giving us all this value and they're not actually transferring us like a hundred fifty four thousand dollars they don't have that money they're generating this value for us by virtue of working for us and then we're compensating them for each sale that they're made that they're making which is facilitated by the use of our capital interacting with their labor and anyway so they get so they're gonna end up getting what the sixteen thousand for every every gizmo they make and as a whole what's gonna happen for the worker well they're gonna walk away with they're gonna walk away with thirty thousand which would keep them exactly and different between the full incentive contract though they're gonna give us full effort and the fixed wage contract though they're they'd only give us an effort of four and not working for us at all where they'd get thirty thousand what's our profits it's just this one fifty four you could actually write out the profit function well of course it's gonna be 16 times 20 which is revenue minus our our labor cost while a labor cost is gonna be 16 times 20 for every unit that's right 20 is gonna be the effort as well as the quantity produced that's just gonna subtract off this 154 they're giving to us so 150 for one minus minus is positive 154 great and then let me move this up a little bit does either the entrepreneur or the worker have a preference between these contracts well clearly the workers got to be indifferent by construction otherwise we haven't done a good job the idea is at least in this model is to set the worker exactly indifferent between accepting any of these contracts and that'll work in the situation where there's no risk involved and we don't have any random variation and these sorts of things so by construction the workers actually gonna get $30,000 I should put okay here $30,000 across either of their three options it doesn't matter whether they do the full incentive contract whether they do the fixed wage contract or whether they do whatever was there outside option which was not working for us they're gonna get thirty thousand either way what about us what do we prefer well we prefer the full incentive contract because our profits there are 154 versus 38 indicate what verses 20 26 actually versus 26 in this case right in the fixed wage fixed wage contract our profits are 26 so I need to change this I'll do that when I upload this in the full incentive contract our profits are 154 we clearly like that better what's the difference well in the full incentive contract we're able to take care of the moral hazard problem so the workers no longer shirking right the worker would have stopped at an effort level of four if we're not compensating their effort fully but when we fully compensate their effort they give us 16 more units of effort they work a lot harder why well because they're being paid on Commission only right so they're gonna or the equivalent and so they're only they're gonna work really hard to make sure that they're because they're receiving the full benefits of their fully internalizing the benefits of their of their effort okay go ahead and conclude here like/subscribe