Problem 1: Insurance
Consider the insurance model described in 2.1.1. Chapter 6.
(1) Explain equation (3) and the Arrow-Borch condition.
Left side: Marginal rate of substitution between Leisure and consumption
Right side: Marginal productivity of labor.
 Condition for pareto efficient allocation (first best)
Indifference curve, slope :
W
B
Production function, slope:
A
h
Allocation A is first best pareto efficient, as the slopes of the production function and the agent’s
indifference curve are equal.
If the agent worked more and consumed more goods, she could receive the same utility as before.
However, allocation B is not feasible under the given production function. There is no other
combination of hours worked, h, and consumption, W (=c), that provides the agent with the same
level of utility as allocation A.
Arrow Borch condition
 Left side: Agent’s Marginal Rate of Substitution measured by marginal utilities of
consumption in two different states.
 Right side: Principal’s Marginal Rate of Substation measured by marginal utilities of
consumption in two different states
 Risk sharing is optimal if both MRS are equal.
Intuition:
Imagine if the agent’s ratio of marginal utilities (MRS) equals 3 and the principal’s 1 (risk neutral).
Then the agent would be willing to give up 3 units of consumption in the good state (θ) to receive
one more unit of consumption in the bad state (є). The principal however, only demands 1 unit
for compensation, if he has to give up one unit in state θ.
 Pareto improvement possible
(2) Let the Agent’s utility function be given by 𝑈(𝐶, 𝐿) = 10ln(𝐶) + 5ln⁡(L). Leisure is 24-hours
worked. The production function is given by 𝑓(ℎ, 𝜀) = 10𝜀 + 𝜀ℎ while the Principal’s utility
function is 𝑣(𝜋) = 2√𝜋. Is the following contract optimal?
CONTRACT 1:
Good state: 𝜀 = 𝑔 = 1
W(g)=, 10, h(g)=19
Bad state: 𝜀 = 𝑔 = 0,5
W(b)=10, h(g)=14
Assume now that the Principal’s utility function instead is given by 𝑣(𝜋) = 10𝜋. Is the contract
optimal now?
Optimal contract is characterized by
UL [W(ε), 1 − h(ε)]
⁡ = ⁡ fn [h(ε), ε]⁡,
UC [W(ε), 1 − h(ε)]
UC [W(ε), 1 − h(ε)]
V ′ [π(ε)]
=⁡ ′
⁡,
UC [W(θ), 1 − h(θ)]
V [π(θ)]
∀ε ∈ ε
∀(ε, θ) ∈ ε2
Thus,
UL =
5
10
⁡, UC = ⁡, fh = ε⁡ ∵ L = 24 − h, W = C⁡
24 − h
W
Therefore,
5⁄
24 − h = ε ⋯ (1)
10⁄
W
And, for the optimal risk-sharing, there should be
10⁄
1
W(g) V ′ [π(ε)] ′
= ′
⁡, V (π) =
⁡⋯ (2)
10⁄
V [π(θ)]
√π
W(b)
To be the optimal contract, the both equations (1) and (2) must be satisfied.
Contract 1 :
Good⁡state ∶ ⁡ε = g = 1, W(g) = 10, h(g) = 19
Bad⁡state ∶ ⁡ε = g = 0.5, W(b) = 10, h(g) = 14
Let’s substitute those numbers to the equation (1) and (2),
5
5
= 1⁡(good),
= 0.5(bad) ⋯ (1)
24 − 19
24 − 14
1⁡ ≠ ⁡
1
= √0.5 ⋯ (2)
1⁄
√0.5
The second equation is not satisfied.
Therefore, the Contract 1 is not the optimal contract.
Assume V(π)=10π, then V’(π)=10
For the first equation, there is not any change.
Thus, it is satisfied.
For the second equation,
1=
V ′ [π(ε)] 10
=
=1
V ′ [π(θ)] 10
Therefore, under the assumption that the principal is now risk neutral (V(π)=10π), the contract is
optimal because the agent has the same utility in both states. In the absence of asymmetric
information full insurance is optimal as the principal does not face the trade-off between incentives
and insurance.
(3) The figure below shows profits and wage costs in the Norwegian Maritime sector. Comment the
figure in light of the theoretical model (“Driftsresultat”= profits, “Lønnskostnader”= wage costs).
Corresponding to the assumptions of the model, wages don’t fluctuate as much as profits.
This can reflect the fact that the agent is more risk averse than the principal (or has limited
access to the capital market) and needs insurance. Therefore wages did not increase as much as
profits, especially during the period 2002 – 2008.
Nevertheless in reality there is probably asymmetric information. This means that the principal
faces a trade-off between incentives and insurance.
So wages are not totally fixed. This is a result of the fact that the principal wants to set
incentives.
There are some facts that contradict the model:
-
Wages have never declined significantly. This might be due to rigidities (contracts, unions
and other interest groups, …)
Further, wage costs do not perfectly reflect wages itself. So, rises and declines of wage costs
might be a result of changes in taxes and/or social charges
(4) Use the provided dataset for the US and compare the predictions from the insurance model
(regarding business cycles) to the fluctuations in hours worked and weekly earnings from 1979 to
2010.
Here we have to look at equation 3: The marginal rate of substitution between consumption and
leisure must be equal to the marginal productivity of labour. The change in the marginal utility in
consumption must be equal to the change in the marginal utility of leisure with respect to a constant
marginal productivity of labour.
A decline in working hours means a rise in leisure and the marginal utility of leisure decreases and a
increase in the wage rate leads to higher consumption opportunities and therefore to a lower
marginal utility of consumption. All in all the development of working hours and wages must go in
opposite direction to maintain a constant marginal rate of substitution.
When we look at the data we see that in the years between 1979 and 2010 there are continous upand downward movements. But in almost half of this specific time horizont, these two variables go in
the same direction. In addition the magnitude of the changes are different.
If a positive productivity shock occurs,the marginal productivity of labour increases. Corresponding to
themodel, the marginal rate of substitution has to increase as well. That means that the marginal
utility of leisure must rise relatively to the marginal utlity of consumption. Thus the consumption of
leisure decreases relatively to the consumption of physical goods.
Some facts that contradict the model: assymetric information, leisure is not a normal good,
rigidities..
If you just take into consideration the last six years, the dataset fits approximative with the model.
Hours worked per employee
1860
1840
1820
Hours worked
per employee,
from OECD
Employment
Statistics
1800
1780
1760
1740
1979
1982
1985
1988
1991
1994
1997
2000
1720
Weekly earnings
360
350
340
330
Weekly
earnings (in
1982 USD),
from CPS, BLS
320
310
300
1979
1982
1985
1988
1991
1994
1997
2000
2003
2006
2009
290