Labour Market Problem Set Flashcards
Exercise 1: There are 1,600 workers in the economy. Each worker can supply up to 12
hours of labor each day. The utility function of each worker is
U(C, L) = C + 80√S
where C is the amount of consumption and S is spare time (in hours per day). Workers earn
$W per hour and the price of a unit of consumption is $P
(a) Write down the utility maximization problem of the worker
max = C + 80√S
C, S
s.t. P C + W S = 12W
Each workers utility function is given by the above, and each wants to maximize their utility.
There is an income constraint: worker has to consume some goods. Denoted as PC, c is the amount of consumption. P is the price of a unit of consumption.
Each worker can do some activities in their spare time. WS is an opportunity cost because you could work in you spare time.
Exercise 1: There are 1,600 workers in the economy. Each worker can supply up to 12
hours of labor each day. The utility function of each worker is
U(C, L) = C + 80√S
Assuming that P = 1, and the market labor supply curve. How much labor is supplied
at W = 20?
- P = 1 W = 20
max = C + 80√S
C, S
st. PC + WS = 12W
st. 1C + WS = 12W
- Use Lagrangian
L = C + 80√S + λ(12W - C - WS)
L = C + 80√S +12W λ - C λ - WS
- First Order Condition
dL/ dC = 1 - λ = 0
dL/ dS = 40 / √S - Wλ = 0
dL/ dλ = 12W - C - WS = 0
- Solve
- 1 - λ = 0
λ = 1 - Plug λ = 1 into (2)
40 / √S - Wλ = 0
40 / √S - W(1) = 0
W = 40/√S - Solve for S(W), S here
W = 40/√S
√SW = 40
40/W = √S
(40/W )^2 = S(W) - Plug into Constraint
PC + WS - 12W
P = 1, AND S(W) = (40/W)^2
C+ WS = 12W
C + W(40/W)^2 = 12W
We know that each worker can work up to 12 hours, the number of hours(supply) that each worker can give to their employees is given by:
Each Individual Supply:
12 - (40/W)^2
12 - 1600/W^2
To get the market supply, you multiply by 1600, number of suppliers.
1600(12 - 1600/W^2)
L^S(W) = 19200 - 2560/W^2
Now replace W = 20
L^S(W) = 19200 - 2560000/W^2
L^S(W) = 19200 - 2560000/20^2
L^S(W) = 12800H
Divide by the number of workers (1600)
12800H/ 1600 = 8H/Worker
Market labor supply is 12,800h at a wage of $20 or 8h per worker.
Firms in a perfectly competitive market produce using only labor according to
the production function F(L) = √2L. The cost of a unit of labor is $W. The fixed cost of
production is $50. Inverse market demand is
PD(Q) = 100 − 10Q.
Derive the market price as a function of W. How many firms are in the market in the
long run?
Given
- F(L) = √2L,
- Fixed cost of production is $50 - - Inverse: PD(Q) = 100 − 10Q.
- F(L) is a production function rewrite it to solve for L
F(L) = √2L
F(L) = Q(L) = (2L)^1/2
Q(L) = (2L)^1/2
Q^2 = 2L
Solve for L
L = Q^2/2
- The cost function, firm has fixed cost and variable cost in the long run every input can be modified. Fixed cost is $50.
C(Q) = WL + FC
C(Q) = Q^2/2W + 50
- In the long run firms are better off producing at ATC, In a perfectly competitive market:
P = MC = ATC
Using C(Q) Find the following below
MC = WQ
ATC = C(Q)/ Q = Q^2/2W + 50 /Q
= W/2Q + 50/Q
- Solve MC = ATC for Q, the quantity that each firm is going to bring to the market
MC = ATC
WQ = W/2Q + 50/Q
Solve for Q
Q = 10/√W
This is the quantity that each firm is going to bring to the market.
- To find the price by replacing Q with Q = 10/√W in Q
P* = MC
P* = WQ
Plug Q in
P* = W(10/√W)
P* = 10√W
- To find the # of firms, use the inverse demand to find the market quantity
Number of firms
= Qmarket/Qfirm
Need to find the quantity of the market. Use the inverse demand.
P = 100 - 10Q, Plug P* = 10√W
10√W = 100 - 10Q
Solve for Qmarket
Q market = 10 - √W
- Number of Firms
N = Qmarket/Qfirm
N = 10 - √W / 10√W
N = √W - W/10
Firms in a perfectly competitive market produce using only labor according to
the production function F(L) = √2L. The cost of a unit of labor is $W. The fixed cost of
production is $50. Inverse market demand is
PD(Q) = 100 − 10Q.
Derive the market labor demand function:
From Other Question:
N* = √W - W/10
P* = 10√W
- Set up the maximization problem
Given: F(L) = Q(L) = √2L
max PQ - WL
L
(WL is the cost of labour to produce) - First Order Condition
dπ/dL = P * 1/2(2L)^1/2 * 2 - W= 0
Solve For W
dπ/dL = P/ √2L - W = 0
P/ √2L - W = 0
W = P/ √2L
- Rearrange to find the labor demand of each firm: LdFirm(W)
W = P/ √2L
Solve for L
2L = (P/W)^2
LdFirm(W) = 1/2(P/W)^2
- Found P* that is associate with the Qfirm, Sup p* into Ldfirm
P* = 10√W
LdFirm(W) = 1/2(P/W)^2
LdFirm(W) = 1/2(10√W/W)^2
LdFirm(W) = 50/W
- Find the Labor market demand: multiply by N*
N* = √W - W/10
Ldmarket = 50/W(√W - W/10)
Ldmarket = 50/ √W - 5
A monopsonist faces inverse labor supply WS (L) = L/2
and production function
F(L) = 2√L. Suppose that the price of an output unit fixed at $1.
(a) What amount of labor L
M does the monopsonist demands? What wage W M does it pay?
Given =
- WS (L) = L/2
- F(L) = Q(L) = 2√L.
- The monoponist tries to maximize:
max π. PQ - C
L
max π. (F(L) - WS (L)L)
L
max π. ( 2√L - L/2L)
L
(where WS is the wage function)
Solve for the FOC:
max π. ( 2√L - L/2*L)
L
max π. ( 2√L - L^2/2)
L
dπ/dL = 1/√L - L = 0
1/√L - L = 0
Solve for Lm
1 - L^3/2 = 0
Lm = (1)^1.5
Lm = 1
- Plug Lm = 1 into the inverse labor supply function
Ws(L) = L/2
Ws(L) = 1/2
Therefore Lm = 1, Ws(1) = 1/2
A monopsonist faces inverse labor supply WS (L) = L/2
and production function
F(L) = 2√L. Suppose that the price of an output unit fixed at $1.
Compute the worker surplus, the surplus of the monopsonist as well as the deadweight
loss.
- Worker Surplus
Draw a graph and you plot Lm and Wm
Lm = 1, Ws(1) = 1/2
To plot the graph draw the supply curve and the MRP (Marginal Revenue of Productivity), where MRP is the derivative of F(L)
Marginal Product of Labor:
d/dL = 2(L)^1/2 = 0
d/dL = 1/√L
- Plot Lm = 1
- Plot Wm = 1/2
Now Worker surplus is above the supply curve and under the price of monoposonist. The triangle area between Lm and Wm
Worker Surplus = 1/2(1)(1/2) = 1/4
- To find the surplus of the monopsonist
Plot Wm = 1/2
Lm = 1
Surplus is the the integral:
- 0 to Lm is the bounds
- MRP - Wm in the inside of the integral
Monopsonist’s Surplus:
lm
∫ (MRP - Wm) dL
0
1
∫ ( 1/√L - 1/2) dL
0
1
∫ ( 1/√L - 1/2) dL
0
1 [ 2√L - 1/2L]
0
[ 1.5 - 0] = 1.5
- To get DWL, Find the Total Surplus First. We have to know what is the equilibrium with a competitive market which happens when P = MC
In the labor market P = W and
MC = MRP
Therefore,
W = MRP, NOW plug values in
L/2 = 1/√L
Solve for new Lm*
L/2 *√L = 1
(L)^1/2 * L/2 = 1
(L)^1/2 * L/2 = 1
L^1.5 = 2
Lm* = 2^2/3
Plug Lm* into W* = L/2
W* = L/2
W= 2^2/3/2
Wm = 2^2/3 * 2^-1 = 2^-1/3
Next Graph The Total Surplus of the Monoposonist
Plot Lm* = 2^2/3
Plot Wm* = 2^-1/3
Plot the MRP
Plot the Supply Curve
Plot Wm = 1/2
Plot Lm = 1
Compute the Integral:
Total Surplus
lm
∫ (MRP -1/2(B*H)) dL
0
2^2/3
∫ (1/√L -1/2(2^2/3 * 2^-1/3)) dL
0
Solve the Integral:
[ 2√1.587] = 0.629 = 1.89
Now that we have Total Surplus, Monoposonist Surplus, and Worker Surplus we can find DWL
Monoposonist DWL
DWL = TS - WS - MS
DWL = 1.89 - 0.25 - 1.5 = 0.14
