Chapter 25: Multitask, Team, Career-Concern, and Robust Contracts. The Encyclopedia of Economic Principals, interactive companion

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Effort to impress the market

How much effort does an analyst supply with no bonus, only to shape what the market thinks of her?

Effort pays only because the market cannot separate it from ability. The more uncertain the market is about ability, the more it reads into output, and the stronger the implicit incentive.

$Equation, written in LaTeX: \beta=\frac{1}{1+1}=0.5.$

$Equation, written in LaTeX: \frac{0.45^2}{2}=0.10125.$

$Equation, written in LaTeX: \beta=\frac{1/3}{1+1/3}=0.25,$

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Output is ability plus effort plus noise. beta is the weight the market puts on output when it updates ability, set by the prior ability variance and the noise variance (1). delta discounts next year's wage and effort costs e^2/2.

Predict first. As the market learns more about Elena (lower ability variance), does her effort rise or fall?

Your prediction Rise Fall

Ability variance

Discount factor

Posterior weight beta: 0.5000
Effort e*: 0.4500
Effort cost: 0.10125

The market puts weight 0.5000 on Elena's output when it updates its view of her ability, so one more unit of effort raises discounted future pay by 0.4500. With cost e^2/2 she sets effort equal to that return: e* = 0.4500, at a cost of 0.10125, with no current bonus at all. Hand check: beta = 1 / (1 + 1) = 0.5000; e* = 0.9 x 0.5000 = 0.4500; Cost = 0.4500 x 0.4500 / 2 = 0.10125.

Worked steps

beta = 1 / (1 + 1) = 0.5000
e* = 0.9 x 0.5000 = 0.4500
Cost = 0.4500 x 0.4500 / 2 = 0.10125

Use the idea

Expect reputational effort to be strongest early in a career and to fade as a track record builds.

Where the conclusion applies

Normal ability and noise, a competitive wage equal to expected ability, and a market that knows equilibrium effort. Observable effort would remove this incentive.

Check your understanding: With ability variance 3 and delta 0.9, what is effort?

beta = 3 / 4 = 0.75 and e* = 0.9 x 0.75 = 0.675. The book's later case, variance 1/3, gives beta 0.25 and e* 0.225.

Chapter 25 source: section "Career-concerns incentives".

Demonstration 2 of 4

Paying for closed tickets

Can paying more for the measured task lower the value of the work as a whole?

When efforts compete, a stronger reward on the measured task pulls effort out of the unmeasured one. If the unmeasured task is worth more, value falls even as the measured number rises.

$Equation, written in LaTeX: u=bx+4y-\frac12(x^2+y^2)-0.5xy,$

Equation, written in LaTeX: x+0.5y=4, y+0.5x=4

$Equation, written in LaTeX: 2(\frac{20}{3})+10(\frac23)=20,$

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x is ticket-closing effort, rewarded at b per unit; y is diagnostic effort, rewarded at 4. The cross-task parameter rho (0.5 in the book's payoff) makes the two efforts compete for the agent. The firm values x at 2 and y at 10 per unit.

Predict first. Raising b from 4 to 7 when the tasks interact (rho = 0.5): does gross value rise or fall?

Your prediction Rise Fall

Closure incentive b

Cross-task parameter rho

Figure: Paying for closed tickets. Bars for closure effort 2.67 and diagnostic effort 2.67; gross value 32.00 at b = 4 and rho = 0.5. — Closure incentive b: 4, Cross-task parameter rho: 0.5
Constructed example: the chapter's hypothetical support center (b = 4 and 7, rho 0.5 and 0, values 2 and 10); b = 5 and 6 are added.

Closure effort x: 2.67
Diagnostic effort y: 2.67
Gross value: 32.00
Change from b = 4: 0.00

With closure incentive 4 and cross-task parameter 0.5, the agent closes 2.67 units and diagnoses 2.67, for gross value 32.00. This is the starting incentive b = 4. Hand check: x = (4 - 0.5 x 4) / (1 - 0.25) = 2 / 0.75 = 2.6667; y = 4 - 0.5 x 2.6667 = 2.6667; Value = 2 x 2.6667 + 10 x 2.6667 = 5.33 + 26.67 = 32.00.

Worked steps

x = (4 - 0.5 x 4) / (1 - 0.25) = 2 / 0.75 = 2.6667
y = 4 - 0.5 x 2.6667 = 2.6667
Value = 2 x 2.6667 + 10 x 2.6667 = 5.33 + 26.67 = 32.00

Use the idea

Before sharpening a metric, ask which unmeasured work the same people would stop doing.

Where the conclusion applies

Quadratic costs with one cross term and fixed values per unit; compensation and risk costs are left out. The book writes the cross term as a fixed 0.5xy and then sets rho = 0 for its counterfactual; rho multiplies xy here.

Check your understanding: At b = 7 with rho = 0, what is gross value?

x = 7 and y = 4, so value is 14 + 40 = 54, up from 48. With rho = 0.5, x = 20/3, y = 2/3 and value falls to 20 from 32.

Chapter 25 source: section "Multitask incentive problem".

Demonstration 3 of 4

Noise flattens the optimal slope

How steep should a linear commission be when output is noisy and the manager dislikes risk?

A steeper slope buys effort but loads more noise onto a risk-averse manager. The optimum trades the two off, and the more noise there is, the flatter it gets.

$Equation, written in LaTeX: S(b)=b-\frac12b^2-\frac12(0.05)(16)b^2=b-0.9b^2.$

Equation, written in LaTeX: S(b)=b-0.5b^2-0.1b^2=b-0.6b^2,

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Wage is a + bX with output X = e + noise of variance var. The manager chooses e = b, effort costs e^2/2 and r is the CARA risk-aversion coefficient. S(b) is surplus net of effort cost and the risk premium.

Predict first. If residual variance falls from 16 to 4, does b* move toward 1?

Your prediction Yes No

Residual variance

CARA coefficient r

Figure: Noise flattens the optimal slope. The surplus curve S(b) on slopes from 0 to 1, peaking at b* = 0.556 with surplus 0.278, and a reference line at b = 1. — Residual variance: 16, CARA coefficient r: 0.05
Constructed example: the chapter's hypothetical route-management contract (variance 16 and 4, r = 0.05); variance 36 and r = 0.01 and 0.10 are added.

Optimal slope b*: 0.556
Effort cost: 0.154
Risk premium: 0.123
Surplus: 0.278

With residual variance 16 and risk aversion 0.05, the risk premium is 0.5 x 0.80 x b^2, so the best slope is 0.556, below the risk-neutral slope of 1. Effort costs 0.154, the risk premium is 0.123 and surplus is 0.278. Hand check: S(b) = b - 0.5b^2 - 0.5 x 0.05 x 16 b^2 = b - 0.9000b^2; b* = 1 / (2 x 0.9000) = 0.5556; Effort cost = 0.5 x 0.55556^2 = 0.1543; Risk premium = 0.5 x 0.05 x 16 x 0.55556^2 = 0.1235; Surplus = b*/2 = 0.5556 / 2 = 0.2778.

Worked steps

S(b) = b - 0.5b^2 - 0.5 x 0.05 x 16 b^2 = b - 0.9000b^2
b* = 1 / (2 x 0.9000) = 0.5556
Effort cost = 0.5 x 0.55556^2 = 0.1543
Risk premium = 0.5 x 0.05 x 16 x 0.55556^2 = 0.1235
Surplus = b*/2 = 0.5556 / 2 = 0.2778

Use the idea

Filter out measurable noise (weather, market moves) before paying on output; it lets the slope rise.

Where the conclusion applies

CARA utility, normal noise and quadratic effort cost. The book rounds the parts to 0.556, 0.154 and 0.123 and notes that they sum to 0.279 against the exact 0.278.

Check your understanding: What are b* and surplus at variance 4?

S = b - 0.6b^2, b* = 1 / 1.2 = 0.833 and surplus = 0.833 / 2 = 0.417.

Chapter 25 source: section "Linear-contract robustness".

Demonstration 4 of 4

Hiding capacity from a buyer who ratchets

When does a supplier hold back output today to keep an easy target tomorrow?

A buyer who cannot commit punishes good news with a tougher target. The supplier weighs the gain from revealing now against the discounted future loss.

Equation, written in LaTeX: 15+0.9(4)=18.6,

Equation, written in LaTeX: 5+0.9(20)=23.

Equation, written in LaTeX: 15+0.9(20)=33,

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Delivering 100 earns 15 now and delivering 80 earns 5. Without commitment the buyer raises the next target after seeing 100, leaving a continuation payoff of 4 instead of 20. delta discounts the second period.

Predict first. Is there a discount factor below which the supplier reveals even without commitment?

Your prediction Yes, below 0.625 No, it always conceals

Discount factor delta

Buyer can commit

Figure: Hiding capacity from a buyer who ratchets. Two stacked bars: reveal totals 18.60 and conceal totals 23.00, each split into payoff now and discounted future payoff, at delta 0.9. — Discount factor delta: 0.9, Buyer can commit: No
Constructed example: the chapter's hypothetical supplier (payoffs 15 and 5, continuation 4 and 20, delta 0.9, with and without commitment); delta 0.5 and 0.6 are added.

Reveal: 18.60
Conceal: 23.00
Choice: conceal

Revealing capacity is worth 18.60 and concealing 23.00, so the supplier chooses to conceal. The buyer would ratchet the target, so concealing gives up 10 now to keep 14.40 of discounted future payoff; that is worth it. Hand check: Reveal = 15 + 0.9 x 4 = 18.60; Conceal = 5 + 0.9 x 20 = 23.00; Future gain from concealing = 0.9 x 16 = 14.40 > 10 given up now; concealing wins only when delta > 10 / 16 = 0.625.

Worked steps

Reveal = 15 + 0.9 x 4 = 18.60
Conceal = 5 + 0.9 x 20 = 23.00
Future gain from concealing = 0.9 x 16 = 14.40 > 10 given up now; concealing wins only when delta > 10 / 16 = 0.625

Use the idea

If you set targets from past performance, commit to the rule in advance or expect held-back output.

Where the conclusion applies

Two periods, known payoffs and a buyer who infers capacity exactly from delivery.

Check your understanding: At delta = 0.5 without commitment, which does the supplier choose?

Reveal gives 15 + 0.5 x 4 = 17 and conceal 5 + 0.5 x 20 = 15, so it reveals. The threshold is delta = 10 / 16 = 0.625.

Chapter 25 source: section "Ratchet effect".

Effort to impress the market

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Paying for closed tickets

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Noise flattens the optimal slope

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies

Hiding capacity from a buyer who ratchets

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies