UNIVERSITY OF OXFORD Department of Economics Undergraduate - Microeconomics
Last updated: September 17, 2014
Tutorial/Cla utorial/Class ss 1: General General Equilibrium Equilibrium
Sketch solutions
FOR USE BY TUTORS ONLY DO NOT CIRCULATE, ESPECIALLY TO STUDENTS Note: For each tutorial/class, a collection of questions related to the topic is provided. In general there are too many questions for students to tackle in a single week. Tutors will decide which of these questions their students should do in preparation for the tutorial or class.
Problems and Multipart Questions 1. A Pure Exchange Economy I Consider a pure exchange economy with two consumption goods, 1 and 2, and two consumers, a and b. Consumer a has an endowment of 10 units of good 1 (and none of good 2), and preferences that are represented by the utility function: ua (xa1 , xa2 ) = ln xa1 + ln xa2 ; consumer b has an endowment of 10 units of good 2 (and none of good 1), and preferences that are represented by the utility function: ub (xb1 , xb2 ) = ln xb1 + ln xb2 . Normalise the price of good 1 to 1, and write p for the price of good 2. (a) Draw this economy in an Edgeworth box, putting good 1 on the x-axis and good 2 on the y-axis. Identify the initial endowment. Sketch in a budget line and some indifference curves for each consumer. ← x 1 10
I C b 8
10
p
6
xa 2
↑
xb2 ↓
Budget Line 4
2
I C a
Initial Endowment
0 0
2
4
xa 1
→
6
8
10
(b) Write down the budget constraints for the two consumers. The value of consumer a’s endowment is 10 so her budget constraint is:
xa1 + p xa2
≤ 10 .
The value of consumer b’s endowment is 10 p so her budget constraint is:
xb1 + p xb2
1
≤ 10 p .
(c) Write down the consumers’ constrained optimisation problems. Consumer a chooses xa1
≥ 0
and xa2
≥ 0 to
ua = ln xa1 + ln xa2 Consumer b chooses xb1
≥ 0
maximise
subject to
xa1 + p xa2
≤ 10 .
and xb2
≥ 0 to maximise
ub = ln xb1 + ln xb2
subject to
xb1 + p xb2
≤ 10 p .
(d) Solve the consumers’ problems using Lagrangeans. Evaluate the consumers’ demand curves for goods 1 and 2. What does the Lagrange multiplier represent? Local non-satiation. Lagrangean for consumer a:
L = ln xa1 + ln xa2 + λ[10 − xa1 − p xa2 ] . FOCs:
∂ 1 = ∂x a1 xa1
− λ = 0 ,
1 ∂ = a a ∂x 2 x1
− λp = 0 .
L
L
From these FOCs we have:
λ = Substitute in for xa1 = 10
1 1 = a . a x1 x2 p
− p xa2 (budget λ =
Re-arrange to obtain:
line):
1 1 = . 10 p xa2 xa2 p
−
5 xa2 = . p
Substitute into budget line to obtain:
xa1 = 10 Same method for consumer b.
− p 5 p = 5 .
Gives:
xb2 = 5 , xb1 = 5 p . The Lagrange multiplier is (as usual) the marginal utility of income - here, it tells us how much utility would increase if the consumer had an additional unit of the numeraire good 1.
2
(e) Write down two market clearing conditions. The demand for x1 must equal the supply of x1 :
xa1 + xb1 = 10 . The demand for x2 must equal the supply of x2 :
xa2 + xb2 = 10 .
(f) Hence find the Walrasian equilibrium relative price and allocation of this economy. Sketch it in your diagram. Is the allocation efficient? Substitute answer to part (d) into market clearing conditions from part (e):
5 + 5 p = 10 5 + 5 = 10 . p Hence the Walrasian equilibrium relative price is 1, and the allocations are xa1 , xa2 = (5, 5) and xb1 , xb2 = (5, 5).
← x b1 10
Budget Line 8
6
xa 2
Eq. Allocation
↑
xb2 ↓
4
I C a
2
I C b
Initial Endowment
0 0
2
4
xa 1
→
6
8
10
The allocation is efficient because it exhausts the endowment and lies on the contract curve.
3
2. A Pure Exchange Economy II Consider a pure exchange economy with two goods, x and y, and two consumers, a and b, that trade the goods. The preferences of consumer a are represented by the utility function: ua(xa , ya ) = x a + ln ya , and the preferences of consumer b are represented by the utility function: ub (xb , yb ) = x b + ln yb . (a) With the price of good x normalised to 1, calculate the Walrasian equilibrium when consumer a has an endowment (4, 0) of (x, y) and consumer b has an endowment (0, 4). Consumer a maximises xa + ln ya s.t.
≤ 4. L(xa, ya) = xa + ln ya − λ[xa + pya − 4]
Lagrangean: FOCx : FOCy : CS:
1 λ = 0, λ = 1 1/ya λp = 0, ya = 1/λp = 1/p λ > 0 xa + pya = 4, xa + 1 = 4 , xa = 3
−
−
⇒
Consumer b maximises xb + ln yb s.t.
≤ 4 p. L(xb , yb) = xb + ln yb − µ[xb + pyb − 4 p]
Lagrangean: FOCx : FOCy : CS:
xa + pya
xb + pyb
1 µ = 0, µ = 1 1/yb µp = 0, yb = 1/µp = 1/p µ > 0 xb + pyb = 4 p, xb + 1 = 4 p, xb = 4 p
−
−
⇒
Market clearing (good x): Equilibrium:
xa + xb = 4 + 0,
−1 3 + 4 p − 1 = 4,
p = 1/2
p = 1/2, (xa , ya ) = (3, 2), (xb , yb ) = (1, 2)
(b) Now assume that consumer a has an endowment (0, 4) and consumer b has an endowment (4, 0). What is the Walrasian equilibrium in this case? Relabelling a for b and b for a: equilibrium:
p = 1/2, (xa , ya ) = (1, 2), (xb , yb ) = (3, 2)
4
(c) Illustrate your findings in an Edgeworth box, and clearly indicate all the Pareto efficient allocations.
⇒ 1 + (1/y)y = 0 ⇒ y = −y. a’s indifference curve has a slope = −1 (flattish), but
Slope of the indifference curve:
x + ln y = c
At (0, 1), for example, b’s has a slope = 3 (steep); therefore, no gains from trade.
−
Efficient allocations are:
xa = 4. (xb = 4
− xa ,
yb = 4
ya = 2 (any x);
0
≤ ya ≤ 2,
x a = 0;
2
≤ ya ≤ 4,
− xb.)
(d) Comment briefly. With quasi-linear utility, any efficient outcome will entail the same allocation of the ‘non-linear’ good (provided initial wealth is high enough).
5
3. An Economy with Production I
[Exam 2012, Part A]
Consider an economy with one consumer, one firm (owned by the consumer), and two types of good, x and y. The consumer owns the endowment of the economy, which is 48 units of good x, and her utility function is u(x, y) = ln x + ln y. The firm can transform good x into good y; if it uses X units of good x it produces Y = X 1/2 . (a) Find the consumer’s MRS and the firm’s MRT. The consumer’s MRS is
−y/x.
The firm’s MRT is
− 12 X −1/2
or
− 12 Y −1.
(b) Show that an allocation in which she consumes x = 32 and y = 4 is efficient. When 32 units of good x are consumed, the endowment of 48 is exhausted by letting the firm use the remaining 16 to produce 4 units of good y . In this case, MRS =
−4/32 = −1/8, and MRT = − 12 4−1 = −1/8.
(c) What relative price of good x is required for this allocation to be a competitive equilibrium? Find the firm’s profits in this equilibrium, and verify that the consumer’s budget constraint (which includes her income from the firm’s profits) is satisfied. The price ratio must equal MRS so px /py = 1/8, say px = 1/8, py = 1. The firm makes 4 from selling 4 units of good y but must pay 16/8 = 2 to obtain the 16 units of good x. Its profit is therefore 2.
−
The consumer spends a total of 32 px + 4 py = 8 on the two goods. Her income is the value of her endowment, 48 px = 6, plus the firm’s profit, 2, giving a total of 8. So her budget constraint is indeed satisfied.
(d) Illustrate the equilibrium in a diagram showing the production possibility frontier of the economy. The ppf is y = (48 6. 325 6. 245 6. 8 164 6. 083 6. 000 5. 916 y5. 831 5. 4 745 5. 657 5. 568 5. 477 5. 385 5. 292 5. 0 196 0 5. 099 5. 000
− x)1/2.
7. 000 6. 875 6. 750 6. 625 6. 500 6. 375 6. 250 6. 125 6. 000 5. 875 5. 750 5. 625 5. 500 5. 375 5. 250 5. 125
9. 846 9. 143 8. 533 8. 000 7. 529 7. 111 6. 737 6. 400 6. 095 16818 5. 5. 565
32
1 2 3 4 5 6 7 8 9 10 11 12 PPF 13 14 15 16
x
6
IC
48 endowment
64
4. An Economy with Production II Consider an economy with a single turnip farmer endowed with one unit of time (available for work and leisure) and a field. The farmer has preferences represented by the utility function: u(t, l) = ln t + ln(1 l) ,
−
where t is the number of turnips consumed and l is labour supplied. There is only one industry in this economy turnip production which requires two inputs, labour and fields. Turnips are competitively produced with a production function: f (L, F ) = L F ,
−
−
1
1
2
2
where L is the total labour used and F is the total number of fields used. Assume that all three markets in this economy (for turnips, labour, and fields) are competitive. Normalise the turnip price to 1. Let w be the wage, and r the field’s rental price. (a) Argue that the farmer’s budget constraint may be written t
≤ w l + r.
The farmer cannot spend more on turnips ( t) than he earns from selling his labour (w l) and renting his field ( r).
(b) Hence find the farmer’s optimal demand for turnips and supply of labour in terms of r and w. When is labour supply positive? The farmer chooses t
≥0
and l
≥ 0 to maximise u = ln t + ln(1 − l) subject to t ≤ w l + r .
Local non-satiation. Lagrangean:
L = ln t + ln(1 − l) + λ[w l + r − t] . FOCs:
∂ 1 = ∂t t
L
∂ = ∂l
L − 1 + λ w = 0 . 1−l
− λ = 0 ,
From these FOCs we have:
λ =
1 1 = . t w(1 l)
−
Substitute in for t = w l + r (budget line):
1 1 = . w l + r w(1 l)
−
Re-arrange to obtain the farmer’s labour supply curve:
l =
w r . 2w
7
−
For the farmer’s turnip demand curve, substitute l into budget line to obtain:
t = w
w r r + w + r = . 2w 2
−
Labour supply is positive when w > r.
(c) If the industry employs inputs L and F , what are costs? Costs are C = w L + r F .
(d) Using a Lagrangean with multiplier λ, solve the cost-minimisation problem and deduce the factor demand curves for labour and fields. Interpret λ. The problem is to choose L
≥0
and F
≥ 0 to minimise
C = w L + r F subject to where T is the industry’s output of turnips.
∂ = w ∂L
L
− λ 12 L−
1 2
1 2
∂ = r ∂F
L
1
F = 0 , 2
1
2
2
Lagrangean:
L = w L + r F + λ[T − L FOCs:
1
L F = T ,
1
F ] . 2
− λ 12 L
1 2
1
F − = 0 . 2
Re-arrange the first FOC for w and multiply by L to obtain:
1 w L = λ L F 2 1
1
2
2
λ 12 T L = . w
⇒
Re-arrange the second FOC for r and multiply by F to obtain:
1 r F = λ L F 2 1
1
2
2
1
1
⇒ F = λ 2r T .
1
Substitute in for L and F into L 2 F 2 = T (production function): 1
λ 12 T w
λ 12 T r
2
1 2
= T .
Re-arrange for λ: 1
1
2
2
λ = 2r w . Use to eliminate λ from expressions for L and F . The factor demand curve for labour is:
L =
1
1
2
2
2r w
w
1 2 T
√ r = √ T . w
1 2 T
√ w = √ T . r
The factor demand curve for fields is:
F =
1
1
2
2
2r w
r
8
The Lagrange multiplier is the shadow price of relaxing the constraint, here the marginal cost of producing another turnip.
(e) Remember that the supply of fields is 1. Calculate turnip supply in terms of wr . Using three market clearing conditions, find the Walrasian equilibrium. With the supply of fields fixed at 1, T =
r w
1 2
.
Market clearing: The farmer’s consumption of turnips must equal the endowment (0) plus the industry’s net output of turnips:
r + w = T . 2 The farmer’s consumption of leisure must equal the endowment (1) plus the industry’s net output of leisure:1
1
−r = 1+ − w2w
√ r − √ w T
.
Finally, the farmer’s consumption of fields must equal the endowment (1) plus the industry’s net output of fields:
0=1+ r w
Substituting in for T =
r w
Hence w =
√
3 2
1 2
√ w − √ r T
.
, we have two equations and two unknowns:
1 2
=
r + w 2
and
w r r = . 2w w
−
1 and r = √ . 2 3
Substitute these factor prices into the factor demands to obtain the equilibrium allocation.
−
1 2 , 3, 0 . The consumption bundle is (turnips, leisure, fields ) = √
The production plan is (turnips, leisure, fields ) = The equilibrium price vector is (1,
1
√
3 √ 1 2 , 2 3 ).
3
√ − 1 , 3
1 3,
1 .
The industry’s net output of leisure is ( labour demand). For more explanation see lectures.
−
9
1.0
Budget Line
Turnips 0.8
Prod Function 0.6
Eq. Allocation IC farmer 0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Labour
5. An Economy with Production III Consider a small closed economy with two consumption goods, good 1 (meat) and good 2 (berries). There are two types of agent, h and g, and they have the same preferences over consumption, represented by the utility function: u(x1 , x2 ) = ln x1 + ln x2 , but there are twice as many type-h agents as type-g agents. The only factors of production are their labour: when a type-h agent chooses to spend a fraction α of his day producing meat and the rest producing berries then his output is (y1h , y2h ) = (2α, 2(1 α)); a type-g agent is more productive – when she chooses to spend a fraction β of her day producing meat and the rest producing berries then her output is (y1g , y2g ) = (3β, 12(1 β )).
−
−
Normalise the price of one unit of berries (good 2) to 1, and let p be the price of one unit of meat (good 1). (a) What would the agents choose for α and β if p < 1? What would the agents choose for α and β if p > 4? Argue that those values of p cannot be part of an equilibrium. The income of a type-h agent is 2α p+2(1 α) = 2+2α( p 1) and that of a type-g agent is 3β p + 12(1
− β ) = 12 + 3β ( p − 4).
−
−
If p < 1 then both types of agent would choose to produce only berries, and if
p > 4 then both types of agent would choose to produce only meat. But with the stated preferences over consumption bundles, both meat and berries must be produced in equilibrium.
10
Now, assume that 1 < p < 4. (Check later that these inequalities are strict.) (b) Taking p as given, calculate what each agent will produce daily. Imagine that the agents sell their output to the market at prices p and 1, producing ‘income’, and then decide on how much of each good they actually want to consume, resulting in ‘expenditure’. Calculate what each agent will demand. Assume that 1 < p < 4 . Since p > 1 , type-h agents will concentrate on hunting and produce 2 units of meat only; since p < 4 , type-g agents will concentrate on gathering and produce 12 units of berries only. Each type-h agent will have an ‘income’ of 2 p, and each type-g agent will have an ‘income’ of 12. Individual demands will be: 1 2 p 2 p 1 12 2 p =
xh1 = xg1 =
= 1,
xh2 =
6/p,
xg2 =
1 2 p 2 1 1 12 2 1
= p ; = 6.
(c) Remembering that there are twice as many type-h agents as type-g agents, find the value of p that equates demand and supply in the meat market, and confirm that 1 < p < 4. Check that with this value of p, demand and supply are equated in the market for berries. Let there be n type-g agents and 2n type-h agents. g Total demand for meat is x1 = 2n xh1 + n x1 = 2n + 6n/p. g Total supply of meat is y1 = 2n y1h + n y1 = 4n + 0. The meat market clears when x1 = y1 , i.e. when 2n + 6n/p = 4n, so p = 3 (and obviously 1 < 3 < 4 ). g
Total supply of berries is y2 = 2n y2h + n y2 = 0 + 12n; total demand for berries g
is x2 = 2n xh2 + n x2 = 2np + 6n, and when p = 3 this equals total supply.
(d) Show that in this equilibrium, type-h agents each consume 1 unit of meat and 3 units of berries, whereas type-g agents each consume 2 units of meat and 6 units of berries. With p = 3, the demands from part (b) become xh = (1, 3), and xg = (2, 6).
The hunter-gatherers now have the possibility of opening up their economy to free trade. In world markets, 1 unit of meat can be exchanged for 2 units of berries, and the country would be a price-taker. (e) Using world prices, calculate what each agent would produce daily. By considering whether each type of agent would become better or worse off, what do you think 11
would be the likely outcome of a referendum on whether or not to open up the economy? If the hunter-gatherers open up their economy to free trade, they would use the ˆ 2 for meat and 1 for berries. world prices of p = As pˆ > 1 , ‘hunters’ would continue to concentrate on hunting but they would be worse off since the relative price of meat would fall. As pˆ < 4 , ‘gatherers’ would continue to concentrate on gathering and they would be better off since the relative price of berries would rise. Since there are twice as many ‘hunters’ as ‘gatherers’, you might conclude that the outcome of a referendum would be to reject opening up the economy. (*) In the closed economy, ug = ln2 + ln 6 = ln 12, and uh = ln1 + ln 3 = ln3 . When p = pˆ, each ‘gatherer’ could use 2 units of her berries to bribe two ‘hunters’ with 1 unit of berries each.
She would then have an ‘income’ of 10 (from her
remaining berries) and demand xg = (2 12 , 5), resulting in ug = ln12 12 > ln12.
In
turn, each ‘hunter’ would have an ‘income’ of 4 (from his meat production) + 1 (from the bribe) and demand xh = (1 14 , 2 12 ), resulting in uh = l n 3 18 > ln 3.
So
both types of agent would be made better off by voting to open up the economy, and subsequently engaging in bribery.
6. The Specific Factors and Ricardian Trade Models Consider an economy which is endowed with L units of a single factor of production, labour, and can produce two goods with production functions y1 = a1 L1, and y2 = a2L2 . Derive the economy’s production possibility frontier. Derive the supply curves for goods 1 and 2 as a function of the goods price ratio, p1 /p2. What is the autarky goods price ratio? This country can trade with another with endowment L∗ and technology y1 = a∗1 L1 , and y = a ∗ L . For what configuration of parameters a , a , a ∗ , and a ∗ does country 1 2
2
2
1
2
1
2
export good 1? Interpret your result. Supply curve for good 1 - see diagram below. Autarky goods price ratio:
( p1 /p2 )Aut = a 2 /a1 .
Country 1 will export good 1 if the relative price of good 1 is lower under autarky, i.e. if a2 /a1 < a∗2 /a∗1 .
12
y1
a1L
p1/ p 2 a2/ a 1
What is the world supply curve of good 1, and what determines the world equilibrium goods price ratio? See diagram.
Equilibrium price (and pattern of specialisation) depends on the level
of demand for good 1.
Three alternative demand curves are shown; the price ratio
is bounded between the autarky price ratios. y1
* *
a1L+a1 L
a1L
p1/ p 2 * * a2 / a 1
a2/ a 1
13
7. Specific Factors Model A small open economy produces two goods, 1 and 2, at output levels yi , i = 1, 2. Output prices are set on world markets and denoted p1 , p2 . The economy is endowed with quantities L of labour and K of capital. Capital is used in sector 2 only, and labour is used in both sectors, Li denoting the use of labour in sector i = 1, 2. The wage rate and the capital rental rate are w and r. The production function for good 1 is y1 = L 1 , and good 2 has production function y 2 = K 1−α Lα2 , 0 < α < 1. (a) What is the value marginal product of labour in each sector? What is the equilibrium wage rate? 1−α K
Sector 1: V M P L1 = p 1 .
Sector 2: V M P L2 = αp 2
.
L2
Wage w = p 1
(b) What is the equilibrium allocation of labour between sectors (expressed as a function of goods’ prices and endowments)? w = p 1 = αp 2
1 α
− K L2
⇒
L2 = K
αp2 1/(1−α) p1
(c) Derive the economy’s supply curves of each good. What is the effect of a change in endowments on production of each good? αp α/(1−α) αp 1/(1−α)
y2 = K
2
p1
,
y1 = L
− K
2
p1
(d) If there is a second economy, identical in all respects except endowments, what is the pattern of trade between these economies? Directly from (iii)
(e) What is the effect of an increase in p 1 on wages, nominal and real? On the return to capital? What is the effect of a decrease in p2 on real factor prices? Equiproportionate, dw/w = dp/p, so (weakly) better off regardless of 1/(1−α) α α/(1−α) α) p2 p1
r = (1 proportionately with p2 , decreases with p1 . consumption bundle.
−
-- increases more than
Change in p2 is trick question;
only relative prices matter for real changes.
(f) Express the value of total output and the total value of factor incomes as functions of goods’ prices and endowments. Comment on your results. 1/(1−α) α α/(1−α)
They are the same:
GNP = GN I = p 1 L + (1
− α)Kp2
p1
(g) Comment on the relationship of this model with (a) the Heckscher-Ohlin model of trade and (b) the specific-factors model. Borderline between the two models.
Both factors are freely mobile as in HO but
one sector happens not to use any.
It is therefore as if capital is specific
to sector 2.
Rybczynski theorem applies -- but without amplification.
14
8. The 2
× 2 Production Model and the 2 × 2 × 2 Trade Model
Country A has two industries, each employing capital and labour to produce output, with constant-returns-to-scale production functions: 1/3
2/3
2/3
y1 = K 1 L1
1/3
y2 = K 2 L2
Factor and product markets are competitive; the prices of the two products are p 1 and p2, and the prices of labour and capital are w and r. (a) By solving the cost minimisation problem for a firm in industry 1 producing an amount of output y1 , show that:
• the optimal capital-labour ratio in industry 1 is: K L = 2rw • and the marginal cost of y1 is: MC 1 = 2/3w 2/3 1/3r 1/3 1/3 K 2w w r • and hence similarly: L = r , M C 2 = 1/3 2/3 2/3 1
1
2
2
Which good is more labour intensive?
The FOCs for the cost minimisation problem with production function y = K a L1−a are: 1 a
−
L r = λa K
w = λ(1
and
−
K a) L
a
where λ is the Lagrange multiplier on the output constraint. w r
Taking the ratio of these we obtain
=
−
1 a K a L,
and substituting a = 1/3 and
a = 2/3 gives the optimal capital-labour ratios in the two industries: K w 2w 2r and L = r .
K 1 L1
=
2
2
Substituting the optimal K/L back into one of the FOCs and solving for λ gives
M C =
the marginal costs: Hence M C 1 =
2/3
w 2/3
r 1/3
1/3
1 a
− − w
r a a .
1 a
and M C 2 =
1/3
2/3
w 1/3
r 2/3
.
Good 1 is more labour intensive -- it has a lower capital-labour ratio for given
w and r .
(b) The country has an endowment of 160 units of capital, and 200 units of labour. Use the optimal capital-labour ratios above to determine how much capital and labour will be used in each industry, as functions of the factor price ratio. Illustrate the factor market equilibrium in an Edgeworth box for the case when w/r = 1. How would the output of the two industries change if the endowment of labour decreased? The resource constraints are K 1 + K 2 = 160 and L1 + L2 = 200; these can be combined with the K/L ratios above: eliminating K 1 and K 2 : 21 wr L1 + 2 wr L2 = 160; eliminating L2 : wr L21 + wr 2(200 L1 ) = 160; L1 = 32 400 160 rw , L2 = 32 160 rw 100 ; and K 1 = 31 400 wr 160 , K 2 = 34 160 100 wr .
⇒
−
−
−
− −
15
When w/r = 1:
L1 = 160, L2 = 40, K 1 = K 2 = 80.
In the Edgeworth box with K and L on the vertical and horizontal axes respectively, and industry 1 at the bottom left, the equilibrium occurs at the intersection of the two lines representing the optimal K/L ratios.
When the
endowment of labour decreases the right hand edge of the box moves to the left; the equilibrium has lower output of good 1 (less of both factors employed), and higher output of good 2 (more of both factors).
This is the standard Rybczynski
diagram (see lectures).
(c) Use the conditions that price equals marginal cost in both industries to show that there is a one-to-one relationship between relative factor prices and relative product prices: 3 w p1 = . r p2
How does the wage change if demand for product 1 increases? Explain this result intuitively. Taking the ratio of the expressions for marginal costs:
p1 M C 1 = = p2 M C 2
w r
1/3
⇒
w p1 = r p2
3
.
The relative wage rises if the demand for product 1 increases.
An increase in
relative demand for product 1 raises the demand for the factor used intensively in its production.
(d) How do your results relate to the Stolper-Samuelson and Rybczynski Theorems? (c) is Stolper-Samuelson; (b) is Rybczynski.
There is a second country, B, identical to A except that it has a smaller endowment of labour. Consumers in the two countries have identical preferences: the relative demand for the two goods, yy , is a function only of the relative prices, pp . 1
1
2
2
(e) Initially both countries are both in autarkic (no trade) equilibrium. Suppose that product prices in country A (determined by consumer demand) are equal: p1 /p2 = 1. What can you say about relative product prices and relative factor prices in country B? Suppose p1 /p2 = 1 in country B as well. demand for good 1.
The countries have the same relative
But from (c), w/r = 1 in both countries, and from (b), since
country B has a lower endowment of labour, its relative supply of good 1 would be lower.
So the relative price of good 1 must rise in country B: in autarkic
equilibrium p1 /p2 > 1 in country B.
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(f) Now suppose that the countries can trade freely with each other. Describe what happens to relative product and factor prices in the two countries. Which goods will be imported and exported by country A? What happens in country A to employment and output in the two industries? In a free-trade equilibrium the relative price must be the same in the two countries; so it rises in country A and falls in country B. Country A moves both labour and capital into industry 1, but the demand for labour increases more than the demand for capital.
The relative factor price w/r rises.
more of good 1, and exports some of it; it imports good 2. opposite happens. (from (c)).
Country A produces
In Country B the
The relative factor price is equalised across the two countries
A ppf diagram could be used to illustrate.
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