MT1 - MT tute 1 answers for FHS microeconomics PDF

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MT tute 1 answers for FHS microeconomics...

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UNIVERSITY OF OXFORD Department of Economics Undergraduate - Microeconomics

Last updated: September 17, 2014

Tutorial/Class 1: General Equilibrium

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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 x2a ; 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 x2b . 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. ← x1b 10

IC b 8 10 p

6

xa2 ↑

x2b ↓

Budget Line

4

2

IC a Initial Endowment

0 0

2

4

xa1

→

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: x1a + p xa2 ≤ 10 . The value of consumer b’s endowment is 10p so her budget constraint is: xb1 + p x b2 ≤ 10p .

1

(c) Write down the consumers’ constrained optimisation problems. Consumer a chooses x1a ≥ 0 and x2a ≥ 0 to maximise ua = ln xa1 + ln xa2

subject to

x1a + p xa2 ≤ 10 .

Consumer b chooses x1b ≥ 0 and xb2 ≥ 0 to maximise ub = ln xb1 + ln xb2

subject to

x1b + p x b2 ≤ 10p .

(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 x2a + λ[10 − x1a − p xa2 ] . FOCs:

1 ∂L = a − λ = 0, ∂xa1 x1 1 ∂L = a − λp = 0 . a ∂x 2 x1

From these FOCs we have:

1 1 = a . x1a x2 p

λ=

Substitute in for x1a = 10 − p x a2 (budget line): λ=

1 1 = a . a x 10 − p x 2 2p

Re-arrange to obtain: x2a =

5 . p

Substitute into budget line to obtain: xa1 = 10 − p Same method for consumer b.

5 = 5. p

Gives: xb2 = 5 , xb1 = 5p .

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 + 5p = 10 5 + 5 = 10 . p Hence the Walrasian equilibrium is⌘ 1, ⇣ ⌘ relative price ⇣ a a b and the allocations are x1 , x 2 = (5, 5) and x1 , x b2 = (5, 5). ← x1b 10

Budget Line 8

6

Eq. Allocation

x2a ↑

xb2 ↓

4

IC a 2

IC b Initial Endowment

0 0

2

4 xa → 6 1

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) = xa + ln ya , and the preferences of consumer b are represented by the utility function: ub (xb , yb ) = xb + 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. Lagrangean: FOCx : FOCy : CS:

L(xa, y a) = xa + ln ya − λ[xa + pya − 4]

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. Lagrangean: FOCx : FOCy : CS:

xa + pya ≤ 4.

xb + pyb ≤ 4p.

L(xb , y b ) = xb + ln yb − µ[xb + pyb − 4p]

1 − µ = 0, µ = 1 1/yb − µp = 0, yb = 1/µp = 1/p µ > 0 ⇒ xb + pyb = 4p, xb + 1 = 4p, xb = 4p − 1

Market clearing (good x): Equilibrium:

xa + xb = 4 + 0, 3 + 4p − 1 = 4, p = 1/2

p = 1/2, (xa, ya) = (3, 2), (xb , y b ) = (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 , y b ) = (3, 2)

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(c) Illustrate your findings in an Edgeworth box, and clearly indicate all the Pareto efficient allocations.

Slope of the indifference curve:

x + ln y = c ⇒ 1 + (1/y)y0 = 0 ⇒ y0 = −y.

At (0, 1), for example, a’s indifference curve has a slope = −1 (flattish), but b’s has a slope = −3 (steep); therefore, no gains from trade. Efficient allocations are: xa = 4. (xb = 4 − xa, yb = 4 − xb .)

ya = 2 (any x);

0 ≤ ya ≤ 2, xa = 0;

2 ≤ ya ≤ 4,

(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).

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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 f irm 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 − 21 X 1/2 or − 21 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 = −21 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 32px + 4py = 8 on the two goods. Her income is the value of her endowment, 48px = 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 − x)1/2 . 8

y 4 IC PPF 0 0

16

32 x

6

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: 1 1 f (L, F ) = L 2 F 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 ≤ wl+r.

Local non-satiation. Lagrangean: L = ln t + ln(1 − l) + λ[w l + r − t] . FOCs:

∂L ∂L 1 1 +λw = 0. = − λ = 0, =− 1 − l t ∂t ∂l From these FOCs we have: 1 1 λ= = . w(1 − l) t Substitute in for t = w l + r (budget line): 1 1 = . wl+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 2w

◆

+r =

r+w . 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 = wL+rF

subject to

1

1

L2 F 2 = T ,

where T is the industry’s output of turnips.

Lagrangean: 1

1

L = w L + r F + λ[T − L 2 F 2 ] . FOCs:

1 ∂L 1 1 1 1 1 ∂L = r − λ L2F2 = 0. = w − λ L 2 F 2 = 0 , 2 2 ∂F ∂L Re-arrange the first FOC for w and multiply by L to obtain:

λ1 T 1 1 1 wL = λ L2F 2 ⇒ L = 2 . w 2 Re-arrange the second FOC for r and multiply by F to obtain: λ1 T 1 1 1 r F = λ L2 F 2 ⇒ F = 2 . r 2 1

1

Substitute in for L and F into L2 F 2 = T (production function): λ 21 T w

!1

λ 12 T r

2

!1 2

=T.

Re-arrange for λ: 1

1

λ = 2r 2 w 2 . Use to eliminate λ from expressions for L and F . The factor demand curve for labour is: L=

⇣

1

1

2r 2 w 2 w

⌘

1 2

T

√ r = √ T. w

1 2

T

√ w = √ T. r

The factor demand curve for fields is: F =

⇣

1

1

2r 2 w 2 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 1 w

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 √ ! r w−r = 1 + −√ T . 1− 2w w Finally, the farmer’s consumption of fields must equal the endowment (1) plus the industry’s net output of fields: ! √ w 0=1+ −√ T . r

Substituting in for T =

 r1 w

2

✓ ◆1 2

r w

Hence w =

p 3 2

and r =

, we have two equations and two unknowns: =

r+w 2

and

w−r r = . 2w w

1 p . 2 3

Substitute these factor prices into the factor demands to obtain the equilibrium allocation. The consumption bundle is (turnips, leisure, fields) = The production plan is (turnips, leisure, fields) = The equilibrium price vector is (1,

1

p 1 3 p 2 , 2 3 ).

⇣

⇣

p1 , 2 , 0 3 3

p1 , − 1 , −1 3 3

⌘

⌘

.

.

The industry’s net output of leisure is (−labour demand). For more explanation see lectures.

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1.0

Budget Line

Turnips 0.8

Prod Function 0.6

Eq. Allocation IC f armer 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, yh2 ) = (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 g g output is (y1 , y 2 ) = (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 2p, and each type-g agent will have an ‘income’ of 12. Individual demands will be: 1 2p 2 p 1 12 2 p =

x1h = x1g

=

= 1,

x h2 =

6/p,

xg2

=

1 2p 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 yh1 + n y 1 = 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 y 2 = 0 + 12n; total demand for berries g is x2 = 2n x h2 + 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 world prices of pˆ = 2 for meat and 1 for berries. 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 = ln 2 + ln 6 = ln 12, and uh = ln 1 + ln 3 = ln 3. 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 = ln 12 21 > ln 12. In turn, each ‘hunter’ would have an ‘income’ of 4 (from his meat production) + 1 (from the bribe) and demand xh = (114 , 2 12 ), resulting in uh = ln 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 = a2 L2 . 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 = a1⇤L1 , and y2 = a2⇤L2 . For what conf iguration of parameters a1 , a2 , a1 ,⇤ and a2⇤ does country 1 export good 1? Interpret your result. Supply curve for good 1 - see diagram below. Autarky goods price ratio:

(p1 /p2 )Aut = a2 /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

a1 L

p1/p2 a2/a1

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/p2 * * a2 /a1

a2/a1

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 goo...