You own a company that produces chairs, and you are thinking...

You own a company that produces chairs, and you are thinking about hiring one more employee. Each chair produced gives you revenue of $\$ 10$. There are two potential employees, Fred Ast and Sylvia Low. Fred is a fast worker who produces ten chairs per day, creating revenue for you of $\$ 100$. Fred knows that he is fast and so will work for you only if you pay him more than $\$ 80$ per day. Sylvia is a slow worker who produces only five chairs per day, creating revenue for you of $\$ 50 .$ Sylvia knows that she is slow and so will work for you if you pay her more than \$ 40 per day. Although Sylvia knows she is slow and Fred knows he is fast, you do not know who is fast and who is slow. So this is a situation of adverse selection. a. Since you do not know which type of worker you will get, you think about what the expected value of your revenue will be if you hire one of the two. What is that expected value? b. Suppose you offered to pay a daily wage equal to the expected revenue you calculated in part a. Whom would you be able to hire: Fred, or Sylvia, or both, or neither? c. If you know whether a worker is fast or slow, which one would you prefer to hire and why? Can you devise a compensation scheme to guarantee that you employ only the type of worker you prefer?

You own a company that produces chairs, and you are thinking about hiring one more employee. Each chair produced gives you revenue of $\$ 10$. There are two potential employees, Fred Ast and Sylvia Low. Fred is a fast worker who produces ten chairs per day, creating revenue for you of $\$ 100$. Fred knows that he is fast and so will work for you only if you pay him more than $\$ 80$ per day. Sylvia is a slow worker who produces only five chairs per day, creating revenue for you of $\$ 50 .$ Sylvia knows that she is slow and so will work for you if you pay her more than \$ 40 per day. Although Sylvia knows she is slow and Fred knows he is fast, you do not know who is fast and who is slow. So this is a situation of adverse selection.
a. Since you do not know which type of worker you will get, you think about what the expected value of your revenue will be if you hire one of the two. What is that expected value?
b. Suppose you offered to pay a daily wage equal to the expected revenue you calculated in part a. Whom would you be able to hire: Fred, or Sylvia, or both, or neither?
c. If you know whether a worker is fast or slow, which one would you prefer to hire and why? Can you devise a compensation scheme to guarantee that you employ only the type of worker you prefer?

Key Concepts

Adverse Selection

Adverse selection occurs when one party in a transaction has more or better information than the other, leading to an imbalance in decision-making. In hiring, this concept is crucial because the employer cannot distinguish between high-productivity and low-productivity workers, leading to uncertainty and potential inefficiencies in the hiring decision.

Expected Value Calculation

Expected value calculation is a statistical method used to determine the average outcome when there is uncertainty about which event will occur. This concept is applied in decision-making to evaluate potential gains or losses by weighting the outcomes by their probabilities, which is essential for making informed choices under uncertainty.

Reservation Wage

The concept of a reservation wage refers to the minimum compensation a worker requires to accept a job. It plays a key role in labor market decision-making because workers’ willingness to work depends on offered wages meeting or exceeding their reservation wage, which is often determined by their opportunity cost and individual productivity.

Screening and Incentive Compatibility

Screening mechanisms are strategies used by employers to induce individuals to reveal their private information, such as productivity levels. Designing incentive compatible compensation schemes ensures that workers self-select in a manner that aligns with their true productivity, helping the employer to mitigate issues arising from adverse selection by attracting the preferred type of worker while discouraging others.

Transcript

00:01 In this problem, we're using the given production function for the production of bar stools to figure out how much machines, which is represented by k, and how many units of labor represented by l, we want to use for this project.

00:16 So we're given the given production function.

00:18 We're told that the budget is $10 ,000, and that it costs $50 per unit k or l.

00:24 And in the end, we want the quantity q to equal 10 for our production.

00:30 So the first part, we are saying that the owner reasons that since both labor and machines cost the same, he will use the same amount of each of these two inputs to get to his desired output of 10.

00:44 So in order to figure out how much of each will hire, we're going to set up a system of equations.

00:50 So if he wants k to equal l, we're going to just say that we'll just set these two equal to a new variable x.

01:00 So both of these are equal to x in this problem, since he wants to use the same amount of both.

01:05 And then we're also going to say that q is equal to 10, right? so 10 is equal to 0 .1.

01:12 And then since k and l are both equal to x, it's going to be 0 .1.

01:15 Let me put these in front, 0 .1 times x, is going to be raised with this little carrot to the point two, and then also multiplied by x again, which is equal to l, which is going to be raised to the 0 .8 power.

01:34 All right, so now we have our quantity 10, right, is equal to 0 .1 times x to 0 .2, times x to the 0 .8.

01:42 And then if we simplify this and we combine the two x's raised to the point two and point eight power, we're going to get that 10 is equal to 0 .1 times x to the first, right? because when you simplify this, you're going to add up the 0 .2 plus 0 .8 to simplify these powers.

02:02 So we're going to get 10 is 0 .1 times x.

02:04 If we divide 10 by 0 .1, we're going to get 100 is equal to x.

02:08 All right.

02:09 So now we know that we're going to want 100.

02:12 Units of both labor and machines.

02:14 Okay, is 1 equals l is 100.

02:19 So then if we know that each unit costs $50, we're going to say $50 multiplied by 100, it's going to be $5 ,000 worth of both k and l.

02:37 So we're going to buy $5 ,000 worth of machines and also $5 ,000 worth of machines, of labor to get this project done.

02:45 So total cost to produce 10 barstools is going to be equal to his exact budget limit, $10 ,000.

02:56 All right.

02:57 So now let's say that he takes a different approach, and he wants to produce at the point where the marginal productivity of both capital and labor are equal, not the average productivity, right? so how do we figure out what marginal productivity is.

03:13 We're simply just going to take the derivative of this production function with respect to first capital, k, and then labor l.

03:21 So i'm going to call this the marginal productivity of capital for mpk.

03:27 So mpk is going to be equal to.

03:30 So taking the derivative, you're going to do 0 .1 times 0 .2, so it's going to be 0 .02, and then multiplied by k to the 0 .2 minus 1, so k to the negative 0 .08 power multiplied by l to the point 8, right? because if we're taking the derivative with respect to k, l to the point is treated like a constant, right? so l to the point 8.

04:05 Oh, sorry, it's k to the negative point 8.

04:08 Right, so our marginal product of capital is 0 .02 times k to the negative 0 .8 times l to the point 8.

04:16 All right.

04:16 And then if we do marginal product of labor, so mpl, we're going to find that that is equal to, so 0 .08, because 0 .1 times the power that l is raised to originally, right? so 0 .08 times, and we keep k constant, times k to the 0 .2, and then finally times l to the, negative 0 .2.

04:44 All right, so that is our marginal product of labor.

04:49 Okay, so now what we're going to want to do, if we want to produce at the point where these two are equal, we're simply going to set mpk equal to mpl.

04:56 Right, so at 0 .02 is going to be equal to.

05:00 How about we actually just copy and paste this? so this up here, this mpk, is going to be equal to this mpl up here, like that.

05:15 All right.

05:16 And then if we simplify this, so if we bring the k to the raise to the point two power, which is on the right side of the equation, divide both sides and bring it over to the left, or sorry, if we bring the k to the negative point eight over to the right, we're going to get something out like this.

05:34 So let's do, so we're going to have, actually let me just write this out over here.

05:40 It'll be easier to visualize.

05:41 So we have l to the point.

05:43 8 on the left side, and then we divide both sides by the l to the negative 0 .2.

05:50 So negative 0 .2.

05:52 And then we're going to bring the k over to the right side, so it equals k to the 0 .2, divided by k to the negative 0 .8 .8.

06:02 Right? and then we still have the 0 .02 on the left and the 0 .08 on the right.

06:08 So if we divide 0 .08 by 0 .02, we're going to get a constant of 4 on this right side.

06:12 All right, so then what we're seeing here is when we simplify these fractions, we can bring up, so l to the negative point two, if it's in the denominator, you simply take away the negative sign and put it in the numerator, and then l to the point 8 and l to the point two are going to simply give us l to the first power, which is just l.

06:32 And the same thing is going to happen on the other side if we bring that up.

06:36 So it's going to be l to the first equals 4k to the first, so l equals 4k .k.

06:42 So that is our first equation that we're dealing with here.

06:46 So l is going to be equal to 4k...

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