Mandatory Assignment ECON 4240 Spring 2017
Consider a general equilibrium model with 2 households (1 and 2), two consumer goods (x
and y), two inputs (capital k and labor l) and two firms X and Y . Each household has an
endowment of capital and labor than can be sold on the market or retained. These endowments
are denoted k 1 , k 2 ,l1 and l2 . Each household also owns half of each firm. Households have
utility:
0.3
0.2
0.4 0.4
0.2
U1 = x0.5
1 y1 (l 1 − l1 ) , and U2 = x2 y2 (l 2 − l2 ) ,
where li is the amount of labor that household i supplies to the market (hence li −li is the amount
of leisure). Firm X (respectively, firm Y ) produces good x (respectively good y) according to:
x = kx0.2 lx0.8 , and y = ky0.8 ly0.2 .
Where kx is the amount of capital used by firm X and so on. The initial endowments are:
0 ≤ k 1 < 50, k 2 = 50 − k 1 , l1 = 24, and l2 = 24.
Suppose there are 4 markets (goods x and y, labor, and capital), and hence 4 prices px , py , pk
and pl . Let’s normalize pl = 1 in what follows.1
(1) Why is it the case that we can normalize a price without loss of generality?
24+k1 pk
3
1 pk
,
y
=
, and supplies an
(2) Show that household 1 has demand x1 = 24+k
1
2px
10
py
amount of labor equal to l1 =
96−k1 pk
.
5
Hints: (a) in equilibrium firms make 0 profits - so
when considering the budget available to the households you can safely “guess” that the
profits from firm-ownership are equal to 0, (b) you should also use a "guess and verify
method" for the constraint li < 24, that is: solve for the equilibrium without considering
the constraint and then (in step (10)) you will check that in equilibrium li∗ < 24. You
should include in your answer a printed version of the code you used.
(3) Find the corresponding x2 , y2 and l2 .
1Note
that this normalization is different from the one suggested in the textbook.
1
2
(4) Consider the optimization problem of firm X. Justify why firm x has demand for labor
(−0.8)
equal to lx = 40.2 p0.2
+ 40.2 p0.2
k x and why the marginal cost of x is equal to 4
k .
(5) Consider the optimization problem of firm Y . Justify why firm y has demand for labor
0.8
0.8 0.8
−0.2
equal to ly = 14
(pk )0.8 y and why the marginal cost of y is 41
+ 14
pk .
(6) Consider the market for good x. Use what you learned in step 4 to define the supply of
good x.2 Construct the demand for good x and use the equilibrium in the market for
good x in order to find (a) the equilibrium relation between price px and pk (equilibrium
equation #1) and (b) equation for the quantity of good x traded in equilibrium (call
this qx ) as a function of pk (equilibrium equation #2).
(7) Redo the last step for the market of good y. By doing so you will get equilibrium
equation #3 and equilibrium equation #4.
(8) Consider the market for labor. Setting quantity of labor supplied (l1 + l2 ) and quantity
of labor demanded (lx + ly ) equal you can get another equilibrium equation (equilibrium
equation #5). This last equation will have variables px , py and pk .
(9) Set k 1 = 30. You now have 5 equations for 5 unknowns (px , py , pk , qx and qy ). Solve the
system using Matlab.3 Files code.m and example4240.m together can be used to solve
an example of a system of 2 equations in 2 unknowns. You are encouraged to use the
files as templates for the code you need. You should include a copy of the code you used
to answer the question.
(10) Check that in equilibrium 0 < l1∗ < 24 and 0 < l2∗ < 24.
(11) Note that we have characterized the equilibrium in all 4 markets by focusing only on
3 markets: good x, good y and labor. What general result ensures that we can fully
characterize the equilibrium in n markets by considering only the equilibrium in n − 1
markets?
2Although
(for simplicity) there is only one firm producing good x, we assume good x is traded in a competitive
market, hence the supply of good x corresponds to the marginal cost curve of firm x.
3The software is available at UiO ProgramKiosk. For a Matlab primer check here.
3
(1) Because all demand and all supply functions are homogeneous of degree 0.
(2) Household 1’s optimization problem
max {x10.5 y10.3 (24 − l1 )0.2 } subject to l1 + k 1 pk = x1 px + y1 py
Substitute constraint:
0.3
0.2
max x0.5
1 y1 (24 − x1 px − y1 py + k 1 pk )
FOC:
x1 :
−.8
.5x1−0.5 (24 − x1 px − y1 py + k 1 pk )0.2 = 0.2x0.5
↔
1 px (24 − x1 px − y1 py + k 1 pk )
x1 = 75 24−y1 ppyx+k1 pk .
y1 :
.3y1−.7 (24 − x1 px − y1 py + k 1 pk )0.2 = 0.2y10.3 py (24 − x1 px − y1 py + k 1 pk )−.8 ↔
24 − x1 px − 53 y1 py + k 1 pk = 0 ↔
3 24−x1 px +k1 pk
.
y1 = 5
py
Using the two FOC, we build demand by household 1:
3
24− 5 (24−x1 px +k1 pk )+k1 pk
5
x1 = 7
↔
px
x1 =
y1 =
24+k1 pk
,
2px
3
5
24−(12+
k1
p )+k1 pk
2 k
py
=
3
10
24+k1 pk
py
.
Plugging x1 and y1 back into the budget constraint we get l1 :
l1 = x1 px + y1 py − k 1 pk =
96−k1 pk
.
5
(3) Household 2’s optimization problem
0.4
0.2
max x0.4
subject to l2 + k 2 pk = x2 px + y2 py
2 y2 (24 − l2 )
Substitute constraint:
0.4
0.2
max x0.4
2 y2 (24 − x2 px − y2 py + k 2 pk )
FOC:
0.4
−0.8
x2 :0.4x−0.6
y20.4 (24 − x2 px − y2 py + k 2 pk )0.2 = 0.2x0.4
↔
2
2 y2 px (24 − x2 px − y2 py + k 2 pk )
4
24 − 32 x2 px − y2 py + k 2 pk = 0
x2 = 3p2x 24 − y2 py + k 2 pk .
−0.6
0.4
−0.8
↔
y2 : 0.4x0.4
(24 − x2 px − y2 py + k 2 pk )0.2 = 0.2x0.4
2 y2 py (24 − x2 px − y2 py + k 2 pk )
2 y2
24 − x2 px − 32 y2 py + k 2 pk = 0
y2 = 3p2y 24 − x2 px + k 2 pk
Using the two FOC, we build demand by household 2:
x2 = 3p2x 24 − 23 24 − x2 px + k 2 pk + k 2 pk ↔
3px x2 = 48 − 43 24 − x2 px + k 2 pk + 2k 2 pk ↔
9px x2 = 144 − 4 (24 − x2 px ) + 2k 2 pk ↔
x2 =
48+2k2 pk
5px
y2 =
48+2k2 pk
5py
hence:
l2 = x2 px + y2 py − k 2 pk =
96−k2 pk
.
5
(4) Optimization problem of firm X:
Marginal productivity of income:
0.8
kx : 0.2 klxx
,
0.2
lx : 0.8 klxx
,
Set marginal rate of technical substitution equal to the ratio of prices:
M RT Sx = ppkl = pk :
0.8
0.2( klx )
x
= pk ↔ klxx = 4pk ↔ lx = 4pk kx
0.2
0.8( kl x )
x
Hence, x = kx0.2 lx0.8 = kx (4pk )0.8 → kx = x (4pk )−0.8 → lx = 40.2 pk0.2 x.
0.2 0.2
(−0.8)
So total cost of producing x is: kx pk +lx = x4(−0.8) p0.2
+ 40.2 p0.2
k +4 pk x. Marginal cost: 4
k
(5) Optimization problem of firm Y :
Marginal productivity of income:
0.2
ky : 0.8 klyy
,
5
ly : 0.2
0.8
ky
ly
,
Set marginal rate of technical substitution equal to the ratio of prices:
M RT Sy = pk :
0.2
l
0.8 ky
y
0.8
ky
0.2 ly
= pk ↔ 4 klyy = pk ↔ ly = 41 pk ky .
0.2
−0.2
−0.2
0.8
Hence, y = ky0.8 ly0.2 = ky 41 pk
→ ky = 41 pk
y → ly = 14 pk 41 pk
y = 14
(pk ).8 y.
−0.2 0.8
0.8
So total cost of producing y is: ky pk + ly = 41
pk y + 41
(pk ).8 y. Marginal cost:
0.8
1 −0.2
1 0.8
+
pk .
4
4
(6) Market for good x:
The marginal cost of firm X is constant, and therefore supply is a horizontal line, which determines the price.
px = 4(−0.8) + 40.2 p0.2
k (equation #1)
Note that firm X makes no profits (px = marginal cost of firm X).
Demand:
24+k1 pk
2px
1
+ 25 (50 − k 1 ) ppxk = p1x 108
+ (20 + 10
k 1 )pk .
5
−1 −0.2 108
1
+
(20
+
Equilibrium quantity: qx = 4(−0.8) + 40.2
pk
k
)p
(equation #2)
1
k
5
10
x(px , pk ) = x1 + x2 =
+
48
5px
(7) Market for good y:
The marginal cost of firm Y is constant, and therefore supply is a horizontal line, which determines the price.
−0.2
+
py = 41
1 0.8
4
p0.8
k (equation #3)
Note that firm Y makes no profits (py = marginal cost of firm Y ).
Demand:
24+k1 pk
py
1
+ 25 (50 − k 1 ) ppky = p1y 84
+ (20 − 10
k 1 )pk .
5
0.8 −1 −0.8 84
−0.2
1
+ 41
pk
k 1 )pk (equation #4)
Equilibrium quantity: qy = 41
+ (20 − 10
5
y(px , pk ) = y1 + y2 =
(8) Market for labor:
3
10
+
48
5py
6
k
= 192−50p
.
5
0.8
1
1
0.2 0.2
Demand: lx + ly = 40.2 p0.2
x
+
p
y
=
4
p
k
k
4 k
px
0.8 1 1
108
1
equilibrium: 40.2 p0.2
+
23p
+
p
k
k
k
px
5
4
py
Supply l1 + l2 =
96−k1 pk
5
+
96−k2 pk
5
0.8 1 84
+ 23pk + 14 pk
+ 17pk .
py
5
k
+ 17pk = 192−50p
. (equation #5)
5
108
5
84
5
(9) Unknown: px , py , pk , x, y.
Using k 1 = 30, we get:
px = 1.468, py = 1.035, pk = 0.5585, x = 23.4641, y = 25.4055.
(10)
l1∗ =
96
5
l2∗ =
96−20∗0.5585
5
+ (32 − k 1 )pk =
96−30∗0.5585
5
= 17 ∈ (0, 24).
(11) Walras Law
= 15.8 ∈ (0, 24).