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Weeks 1 and 2: Outline

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- 3 conditions for General Equilibrium
- Producer Optimization (Px/Py = MRT)
- Consumer Optimization (Px/Py = MRS)
- Market Clearing
- Goods Markets: Xc = Xp, Yc = Yp
- Factor Markets: Lx + Ly = L, Kx + Ky = K

- Get the production point, consumption point, input allocations and prices.
- Need to determine factor rewards.
- Competitive: Factors paid their VMP.
- Know Prices, MPP. Wage = VMPL = P(MPL)

- Px/Py= MRT = MPLy/MPLx = MPKy/MPKx
- Px/Py= MRS
- Wage = VMPL = Px MPLx = Py MPLy
- Rent = VMPK = Px MPKx = Py MPKy
- Real Wage
- In terms of X: Wage/Px= MPLx = (Py/Px) MPLy
- In terms of Y: Wage/Py= MPLy = (Px/Py) MPLx

- STEP 1: Draw a PPF in XY space
- Maximum Y that can be produced given X

- What does the PPF look like?
(a) Downward Sloping if MPL and MPK>0. Will be satisfied for all production functions we use. Implies if increase good X then need more inputs for X. So must take some inputs from Y. Implies must reduce Y.

(b) Concave if production functions satisfy Diminishing Marginal Returns. Implies as we keep putting more inputs into X production, output of X increases more and more slowly.

dMPL/dL < 0 and dMPK/dK < 0

(c) Linear if production functions satisfy Constant Marginal Returns. Implies as we keep putting more inputs into X production, output of X continues to increase at the same rate.

dMPL/dL = 0 and dMPK/dK = 0

(d) How to draw the PPF?

- Give all inputs to X and none to Y. Plot this point.
- Give all inputs to Y and none to X. Plot this point.
- Now join the 2 points: As a concave curve if the PPF is concave. Or, as a straight line if the PPF is linear.
(e) What is the slope of the PPF? MRT

- STEP 2: Draw community Indifference Curves in XY space.
- Think of a nice Cobb-Douglas utility function and draw this map.

- What is the slope of the ICs? MRS

Y

Slope of IC at point A or the MRS at point A

Indifference Curve

A

X

- From the 3 conditions for a GE
MRT = Px/Py = MRS

- Implies the equilibrium is at the tangency point of the PPF and the IC
- The tangent is the price line.

- Working with PPFs.
- On the PPF. Implies all inputs used. So get factor market clearing.
- IC tangent to PPF (On the PPF). Implies consumed goods equal produced goods. So get goods market clearing.

- 2 Goods: Cloth and Wheat
- Only 1 input L
- Home Country
- 1 worker produces 4 bushels of wheat or 2 yards of cloth.
- Has total 25 workers.

- Foreign Country
- 1 worker produces 1 bushel of wheat or 1 yard of cloth.
- Has total 100 workers.

- How will we get the GE for each economy?

- 1L produces 4W or 2C: w = 4L, c = 2L
- MPLW = 4, MPLC = 2.
- Both MPLW and MPLC positive. So downward sloping PPF.
- MPLW and MPLC same for all workers (given).
dMPLW/dL = 0, dMPLC/dL = 0

So CONSTANT Marginal Returns. So get a Linear PPF.

- Give all L to W. Implies w = 4L = 4(25) = 100.
No inputs left for C. Implies c = 2L = 2(0) = 0.

- Give all L to C. Implies c = 2L = 2(25) = 50.
No inputs left for W. Implies w = 4L = 4(0) = 0.

- Plot (w1,c1)=(100,0) and (w2,c2)=(0,50).

- Now plot the two PPF points. Put W on the x-axis and C on the y-axis. (Just to match FT, no other reason – you can reverse if you want).
- Remember the PPF is downward sloping. Your plotted points should also tells you this.
- Now use the earlier derived fact that the PPF is linear. Draw a straight line to join the 2 points.
- The PPF!

- Slope of the PPF = MRT = MPLc/MPLw
- With linear PPFs, the slope is constant. So the MRT is constant.
- What is the slope? Remember it is a straight line.
- Slope = (w2-w1)/(c2-c1) = (50-0)/(0-100) = - ½
- MRT = ½
- Also note that the Slope = (w2-w1)/(c2-c1) = 2(25)/4(25) = MPLc*L/MPLw*L = MPLc/MPLw.

- ½ yard of Cloth is the OPPORTUNITY COST of Wheat. Increase W by 1. Need ¼ L for 1W. So reduce ¼ L in C. ¼ L can produce ½ C. Reduces C by ½ yard.

- Draw the ICs.
- Draw the tangency point.
- You will note that since the PPF is a straight line, the PPF is also the tangent and hence the price line.
- We already know the slope of the PPF so we know the prices too. The price ratio is Pw/Pc = MRT = ½ = MRS.
- Tangency point is the production and the consumption point. It is on the PPF so get both labor and goods market clearing.

- We have the Prices.
- We have the MPLs.
- So what are the Wages? Remember there is perfect competition.
- Wage = VMPL = Pw MPLw = Pc MPLc
- Real Wage in terms of Wheat = Wage/Pw
- Real Wage in terms of Cloth = Wage/Pc

- Wage = VMPL = Pw MPLw = Pc MPLc
- Wheat: Wage/Pw = MPLw = (Pc/Pw)MPLc
- Cloth: Wage/Pc = MPLc = (Pw/Pc)MPLw
- Wheat: Wage/Pw = 4 = (2)2
- Cloth: Wage/Pc = 2 = (1/2)4

- Repeat the previous steps. Use the same axes. Now MPLw = 1, MPLc = 1.
- You will find that MRS = Pw/Pc = 1 = MRT = MPLc/MPLw = Opportunity Cost of Wheat.
- Real Wages
- Wheat: Wage/Pw = 1 = 1(1)
- Cloth: Wage/Pc = 1 = 1(1)

- Opportunity Cost of Wheat: lower at home
- Home: ½ < 1: Foreign

- Prices: Wheat cheaper at home
- Pw/Pc Home = ½ < 1 = Pw/Pc Foreign

- Real Wages: Both higher at home
- Wage/Pw Home = 4 > 1 = Wage/Pw Foreign
- Wage/Pc Home = 2 > 1 = Wage/Pc Foreign
- Reflects that Labor is more SCARCE at home

- OCW: lower at home
- Home OCW = ½ < 1 = Foreign OCW

- Home has a lower OCW. Home has a CA in Wheat.
- OCC = 1/OCW: lower in foreign
- Home OCC = 2 > 1 = Foreign OCC

- Foreign country has a CA in Cloth.

- Suppose world prices are given to you.
- PPF procedure is the same. Technical.
- IC procedure is the same. Preferences.
- What about the tangency?
- If cannot trade then prices determined at home. Tangency procedure is the same. World prices do not affect you since the home country is a closed economy.

- What about the tangency if the economy opens up to trade?
- Tangency procedure is not the same.
- Draw the post-trade world price line (assume this is given to you for now).
- Remember the PPF and ICs are the same.
- Find the tangency between the New Price Line and the original PPF. Gives the new PRODUCTION point.
- Find the tangency between the new price line and the highest IC of your original IC MAP. Gives the new CONSUMPTION point.

- New Production Point = (Xp,Yp)
- New Consumption Point = (Xc,Yc)
- Look at the x-axis.
- If Xp > Xc, then you are exporting good X.
- If Xp < Xc, then you are importing good X.
- If Xp = Xc, then you are not trading good X.

- Look at the y-axis. Repeat.

- Suppose World Price after Trade = Pw/Pc = 2/3.
- Recall Home Pw/Pc = ½ < Trade Pw/Pc = 2/3 < Foreign Pw/Pc =1.
- New Home Production Point:
- PPF same as earlier.
- New Price Line is steeper.
- New “tangency point” of PPF and the new Price line must be on one of the corners.
- Look for the corner where the new price line is above the PPF.
- This is the New Production point = (100,0)
- Home does not produce any cloth. Produces as much wheat as possible.
- Makes sense because OCW is lower at home.

- New Home Consumption Point:
- IC MAP same as earlier.
- New Price Line is steeper.
- Find the New Tangency Point.
- This is the New Consumption Point.
- With nice preferences, the New Consumption Point will be to the Left of the Autarky Tangency Point.
- Example: New (40,40) Autarky (50,25)

- New Production Point = (100,0)
- New Consumption Point = (40,40)
- Look at the x-axis.
- 100 > 40. So Xp > Xc, then you are exporting good X which is Wheat.

- Look at the y-axis.
- 0 < 40. So Yp < Yc, then you are importing good Y which is Cloth.

- Makes sense. Buying Cloth from the foreign country since its OCC is lower there. Selling Wheat to the foreign country since your OCW is lower.

- Is the IC higher with Trade or in Autarky?
- If higher, then consumers are better off.
- So economy has got a welfare gain by opening the economy to trade.
- What is the GFT?

- Real Wages in Autarky at Home:
- Wage = VMPLw = Pw MPLw = Pw 4 = VMPLc = Pc MPLc = Pc 2
- Wage/Pw = 4
- Wage/Pc = 2

- New Trade Price is Pw/Pc = 2/3
- New VMPLw / VMPLc = (Pw MPLw)/(Pc MPLc) = (Pw/Pc)(MPLw/MPLc) = (2/3)(4/2) = 4/3 = 1.3
- VMPLw is higher that VMPLc. So all workers want to move to the Wheat industry.

- Home only producing Wheat now. So we can only define VMPLw and not VMPLc.
- Wage = VMPLw = Pw MPLw = Pw 4
- Real Wage
- Wheat: Wage/Pw = VMPLw/Pw
= Pw MPLw/ Pw = MPLw = 4.

- Cloth: Wage/Pc = VMPLw/Pc = Pw MPLw/Pc = (Pw/Pc) MPLw = (2/3) 4 = 8/3.

- Wheat: Wage/Pw = VMPLw/Pw

- Trade Wage/Pw = 4 = Autarky Wage/Pw
- Trade Wage/Pc = 2.67 > 2 = Aut Wage/Pc
- Workers are at least as well-off after trade.
- In fact, here they are better-off.
- Note that the GFT comes from the higher ability to purchase Cloth (the imported good which was cheaper in the Foreign country).

- Repeat the same for the foreign country.
- New Production Point = (100,0)
- New Consumption Point is to the right of the Autarky Tangency Point. Say, new (60,60) and old (50,50).
- BOTH countries Gain from trade.
- Imports Wheat and Exports Cloth.
- Wage = VMPLc
- Real Wage
- Wheat: Wage/Pw = VMPLc/Pw = (Pc/Pw) MPLc = (3/2) 1 = 3/2 = 1.5
- Cloth: Wage/Pc = VMPLc/Pc = MPLc = 1 = 1

- Trade Wage/Pw = 1.5 > 1 = Aut Wage/Pw
- Trade Wage/Pc = 1 = Aut Wage/Pc
- Workers are at least as well-off after trade.
- In fact, here they are better-off.
- Higher ability to purchase Wheat.

- At Aut Home Price: (Pr,X) = (1/2, 0)
- At Trade Price: (Pr,X) = (2/3, 60)
- Think of an increase in Trade Price. The price line is steeper now and consumption moves more to the left. So Home’s Export Supply Curve is Upward sloping.
- Why? Cloth cheaper. Substitute into cloth. (Assuming that extra income from wheat does not dominate the substitution). Also Foreign country wants to sell more Cloth since they are getting a better price.

- Home’s Export Supply Curve is Convex.
- Why? If Price (Pw/Pc) increases by the same amount again, then want to consume less Wheat. Due to the law of Diminishing Marginal Utility, you now don’t want to give up as much of Wheat as earlier because you’re already consuming less of it.
- Similarly, by the law of Diminishing Marginal Returns, if you keep putting in more resources into Wheat production, the output of Wheat will rise more slowly.

- What about linear PPFs (with CONSTANT Marginal Returns)?

- At Aut Home Price: (Pr,X) = (1/2, 0)
- At Trade Price: (Pr,X) = (2/3, 60)
- At Price > 2/3: Say, (Pr,X) = (1, 80)
- At Price between ½ and 2/3: Say, (Pr,X) = (0.55, 55). Check this yourself.

- What else at Autarky price ½ ?
- When Price is ½ , workers get the same wages in both Wheat and Cloth (Just like in Autarky). This is because the slope of a linear PPF is ½ at all points.
- So can produce anywhere on the PPF and continue to consume at the desirable Consumption Point (which is the original consumption).

- What else at Autarky price ½ ?
- Consumption Point (50,50).
- Maximum Wheat Production = 100
- Maximum Wheat X = Prod – Cons = 100 - 50 = 50
- Get a New Point: (Pr,X) = (1/2, 50)
- Already know Aut Point: (Pr,X) = (1/2, 0)
- Can also choose anything in the middle so join these 2 points to get a Straight Line at Autarky Price = ½.

- Usual Excess Demand Function
- Downward sloping for Price not equal to 1.
- Convex for Price not equal to 1. (Law of DMU still holds which gives the convex shape).
- But the Constant MR changes the shape at Foreign Autarky Price = 1

- Straight line segment at Price = 1 = Foreign Autarky Price.
- Special property of Linear PPFs.
- Think of the Autarky Price in the case of a concave PPF: No straight line portion.

At Pw/Pc =1 (Same as Foreign’s Autarky Price):

Continue to consume at A

Can produce anywhere on the PPF

If produce all C, then Import Dd of W

= Cons – Prod

= 50 (A) – 0 (No W) = 50

If produce all W, then Export Supply of W

= Prod – Cons

= 100 (all W) – 50 (A) =50

All C Prod Point

Cloth

Pw/Pc=1

100

Cons Point

A

All W Prod Point

Wheat

50

100

Max IM Dd of W at Pw/Pc = 1

Max Ex Ss of W at Pw/Pc = 1

At Pw/Pc =1 (Autarky Price):

Continue to consume at A

Can produce anywhere on the PPF

If produce all C, then Import Dd of W = Cons – Prod = 50 (A) – 0 (No W) = 50

If produce all W, then Export Supply of W = Prod – Cons = 100 (all W) – 50 (A) =50

At P’<1, Pink Horizontal Line. (Produce all C at green point , Consume at pink tangency point).

At P’’<P’<1, Blue Horizontal Line (Produce all C at green point, Consume at blue tangency point)

New Prod Point for P’ & P’’

New Cons Point for P’

100

New Cons Point for P’’

P’’<P’

A

P’<Pw/Pc

Pw/Pc=1

50

100

IM Dd of W at P’’

IM Dd of W at P’

Plot the points we just derived. The graph should look like this.

Convince yourself that this quadrant looks like this.

Pw/Pc

1

P’

P’’

50

50

Excess Ss of Wheat

Excess Dd of Wheat

- Intersection of Home X Ss and Foreign M Dd of Wheat.
- Home X Ss = Foreign M Dd

- Market Clearing:
- Wheat produced in the world = Wheat consumed in the world.
- Home Prod + Foreign Prod = Home Cons + Foreign Cons
- Home Prod – Home Cons = Foreign Cons – Foreign Prod
- Home X Ss = Foreign M Dd

- ToT = Price of Exports / Price of Imports
- Home exports W so its ToT = Pw/Pc = 2/3
- Can we say something about ToT?
- Recall the Average Cost of producing W at Home = Amount paid to 1 worker = Wage at Home
- Similarly, AC of producing Cloth in Foreign = Wage in Foreign

- Here AC = AVC.
- If Price < AVC then firms Shuts Down.
- Home produces W if Pw >= ACw = Wage/MPLw
- Foreign produces C if Pc >= ACc = Wage/MPLc
- Suppose Pw = 1.
- Wage at Home = Pw MPLw = 4.
- Wage in Foreign = Pw MPLw = 1.
- Home: Pc = ACc = 2. Pw/Pc = ½.
- Foreign: Pc = ACc = 1. Pw/Pc = 1.
- ½ < Pw/Pc < 1.

- ToT = Price of Exports / Price of Imports
- Home exports W so its ToT = Pw/Pc = 2/3
- Can we say something about ToT?
- Recall the Average Cost of producing W at Home = Amount paid to 1 worker = Wage at Home
- Similarly, AC of producing Cloth in Foreign = Wage in Foreign

- Here AC = AVC.
- If Price < AVC then firms Shuts Down.
- Home produces W if Pw >= ACw = Wage/MPLw
- Foreign produces C if Pc >= ACc = Wage/MPLc
- Suppose Pw = 1.
- Wage at Home = Pw MPLw = 4.
- Wage in Foreign = Pw MPLw = 1.
- Home: Pc = ACc = 2. Pw/Pc = ½.
- Foreign: Pc = ACc = 1. Pw/Pc = 1.
- ½ < Pw/Pc < 1.