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Productive Efficiency in General Equilibrium

Part 1. Margins of Efficiency

Economics 313

 Recall three types of efficiency:

I. Distributive efficiency - given a fixed quantity of goods, who should get to consume them?

II. Productive efficiency - given a fixed quantity of goods, who should we have produce them?

III. Allocative efficiency - what quantities of goods should be produced?

 Productive efficiency: applying general definition of efficiency to the specific case of deciding how to produce a given quantity of goods.

Productive Efficiency

→

Efficiency In Production?  We already studied efficiency in an endowment economy,

and saw how competitive markets exhausted all gains from trade

 To begin, we study this question in a way very similar to Distributive Efficiency, except now we are talking about distribution inputs

 Once we have defined Productive Efficiency, we can see related this to the equilibrium in competitive input markets

Efficiency In Production  First imagine what would happen if firms could trade inputs.

Assume two firms

 Each firm employs two freely variable inputs  Capital (K) and labour (L)  Total quantity of each fixed

 They produce good X and good Y

 Assume firm x produces good X and firm y produces Y

 Assume the production technology employed by the two firms gives rise to conventional, convex-shaped isoquants

Efficiency In Production  A necessary condition for efficiency is that firms are

getting as much output as they can from the inputs they use. You might call this technical efficiency, or engineering efficiency

 This amount is represented by the Production Function. For firm Y we can write this as

𝑌𝑌 = 𝑌𝑌 𝐾𝐾, 𝐿𝐿

Efficiency In Production  We can use the production function to define isoquants that map

the relationship between L and K such that output doesn’t change.

𝑌𝑌 𝐾𝐾(𝐿𝐿), 𝐿𝐿 = �𝑌𝑌

 Then the function 𝐾𝐾 𝐿𝐿 is the isoquant for output level �𝑌𝑌. As firm Y adds L, the function 𝐾𝐾 𝐿𝐿 tells us how much K changes to keep the same output.

𝑑𝑑𝑌𝑌 𝑑𝑑𝐿𝐿

= 𝜕𝜕𝑌𝑌 𝜕𝜕𝐾𝐾

𝑑𝑑𝐾𝐾 𝑑𝑑𝐿𝐿

+ 𝜕𝜕𝑌𝑌 𝜕𝜕𝐿𝐿

= 0

 This means that the slope of the isoquant is

𝑑𝑑𝐾𝐾 𝑑𝑑𝐿𝐿

= � −𝜕𝜕𝑌𝑌𝜕𝜕𝐿𝐿

𝜕𝜕𝑌𝑌 𝜕𝜕𝐾𝐾

= −𝑀𝑀𝑀𝑀𝐿𝐿 𝑀𝑀𝑀𝑀𝐾𝐾

-- end of part 1 --

Productive Efficiency in General Equilibrium

Part 2. The Edgeworth Production Box

Economics 313

Efficiency in Production  We analyze the feasible and efficient quantities of two

outputs using fixed quantities of two inputs using the Edgeworth Production Box

 This is similar to usual consumer Edgeworth Box for exchange, but firms use capital and labour and produce output goods not utility

 Instead of consumer A, we put good X on the lower left hand corner and instead of consumer B, we put good Y on the upper right hand corner.

Edgeworth Box of Production

Good X

capital

labor

Good Y

Isoquants of X

Isoquants of Y C

ap it al

fo r

go o d

Labour for good x

Labour for good Y C

apital fo r go

o d Y

Edgeworth Box of Production

Good X

capital

labor

Good Y

Amount of labor devoted to production of good X

Amount of capital devoted to production of good X

labour for good Y

Edgeworth Box of Production

Good X

capital

labor

Good Y

Point A is not production efficient.

Moving to this point from point A we can produce more of both goods.

C ap

it al

fo r

go o d

Labour for good x

Labour for good Y C

apital fo r go

o d YA

Edgeworth Box of Production

Good X

capital

labor

Good Y

This point is production efficient: in order to produce more of X we have to reduce production of Y and vice versa.

Production Efficiency  Analogy to the consumption efficiency.

 The slope of the isoquants is the marginal rate of technical substitution (MRTS) of capital for labor (K for L):

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿 𝑌𝑌 =

𝑀𝑀𝑀𝑀𝐿𝐿 𝑀𝑀𝑀𝑀𝐾𝐾

 The MRTS can be used expresses how efficiently firms share a fixed amount of capital and labour

 Production Efficiency occurs when

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾 𝐿𝐿 𝑋𝑋 = 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿

𝑌𝑌

Production Efficiency  Suppose 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿

𝑋𝑋 does not equal 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿 𝑌𝑌

 For example assume that 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿 𝑋𝑋 = 3 and 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾 𝐿𝐿

𝑌𝑌 = 1 4

 For good X, the output of good X increases 3 times as much if we increase labor by one unit than if we increase capital by one unit

 For good Y, labor is only one quarter as productive at the margin as is capital

Production Efficiency  This means the firm producing good X at the current

production level could trade up to 3 units of capital for one additional unit of labour and not reduce output

 The firm producing good Y would can trade one unit of labour for only ¼ a unit of capital and not reduce output

 Therefore the firms are able to trade inputs and both increase their output. Gains from trade!

 Gainful trade can occur whenever MRTS of one firm differs from the other (and each has something to trade)

-- end of part 2 --

Productive Efficiency in General Equilibrium

Part 3. Productive Efficiency: Input Markets

Economics 313

Competitive Input Markets  The analogy to Consumption Exchange is useful for

deriving an efficient allocation of inputs across firms. But how do these trades happen?

 You could think the firms were owned by entrepreneurs, and think of one working part time for the other as trading part of their labour “endowment” for some capital.

 Or you could think of a social planner just reallocating inputs to increase overall output.

Competitive Input Markets  We are interested in Competitive General Equilibrium, so

instead assume that firms can hire labour and rent capital on competitive markets

 Competitive Firms choose inputs to maximize profits, taking as given the prices it faces in input and output markets

 Recall that this can be done in two steps: minimize the cost of producing a given output (which gives the Cost Function) and then choosing the profit maximizing output

Production Efficiency and Behaviour  A firm maximizes its profits must also be minimizing its costs. Let the

hourly rates for labour and capital are 𝑤𝑤 and 𝑟𝑟 respectively.

min 𝐾𝐾𝐿𝐿

𝑤𝑤𝐿𝐿 + 𝑟𝑟𝐾𝐾 subject to 𝑌𝑌 𝐾𝐾, 𝐿𝐿 = �𝑌𝑌

 The first order conditions require that

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿 𝑌𝑌 =

𝑀𝑀𝑀𝑀𝐿𝐿 𝑌𝑌

𝑀𝑀𝑀𝑀𝐾𝐾 𝑌𝑌 =

𝑤𝑤 𝑟𝑟

 Following conditions must be satisfied if the firms are minimizing costs 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿

𝑌𝑌 = 𝑀𝑀𝑀𝑀𝐿𝐿 𝑌𝑌

𝑀𝑀𝑀𝑀𝐾𝐾 𝑌𝑌 =

𝑤𝑤 𝑟𝑟

and 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿 𝑋𝑋 = 𝑀𝑀𝑀𝑀𝐿𝐿

𝑋𝑋

𝑀𝑀𝑀𝑀𝐾𝐾 𝑋𝑋 =

𝑤𝑤 𝑟𝑟

 Competitive general equilibrium is efficient in the allocation of goods for consumption and in the allocation of inputs for production of goods

Production Efficiency and Behaviour  If both firms face the same input prices then this implies

that

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿 𝑌𝑌 = 𝑀𝑀𝑀𝑀𝐿𝐿

𝑌𝑌

𝑀𝑀𝑀𝑀𝐾𝐾 𝑌𝑌 =

𝑤𝑤 𝑟𝑟

= 𝑀𝑀𝑀𝑀𝐿𝐿

𝑋𝑋

𝑀𝑀𝑀𝑀𝐾𝐾 𝑋𝑋 = 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿

𝑋𝑋

 Competitive general equilibrium is efficient in the allocation of goods for consumption and in the allocation of inputs for production of goods

-- end of part 3 --

Productive Efficiency in General Equilibrium

Part 4. Production Possibility Frontier

Economics 313

Production Possibility Frontier  The set of Pareto efficient allocations in the Edgeworth box of

production makes all the efficient input combinations visible

 However a different concept, the production possibility frontier (PPF) gives us all the combinations of goods that are production efficient

 The rate at which we can transform good y into good x is given by the slope of the PPF

 This is known as the marginal rate of transformation (MRT)

Edgeworth Box of Production

Good X

capital

labor

Good Y

Contract Curve

The Production Possibility Frontier

Good X Good Y

capital

labor

X=12, Y=28

Good X

Good Y

The Production Possibility Frontier

Good X Good Y

capital

labor

X=24, Y=20

Good X

PPF

Good Y

The Production Possibility Frontier

Good X Good Y

capital

labor

X=30, Y=10

Good X

Good Y

The Production Possibility Frontier

Good X Good Y

capital

labor

X=30, Y=10

X=24, Y=20

X=12, Y=28

Good X

PPF

Good Y

12 24 30

unattainable attainable, efficient

attainable, inefficient

-- end of part 4 --

Productive Efficiency in General Equilibrium

Part 5. Formally Deriving the PPF

Economics 313

Formally Deriving the PPF  The PPF is the maximum Y possible for a given

quantity of X given total available inputs ( �𝐾𝐾, �𝐿𝐿)

PPF = max 𝐾𝐾𝑌𝑌𝐿𝐿𝑌𝑌

𝑌𝑌(𝐾𝐾𝑌𝑌, 𝐿𝐿𝑌𝑌 ) subject to X �𝐾𝐾 − 𝐾𝐾𝑌𝑌, �𝐿𝐿 − 𝐿𝐿𝑌𝑌 = 𝑋𝑋

 You can see the relationship to the CC from the first order conditions of this problem, which require that the :

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾 𝐿𝐿 𝑋𝑋 = 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿

𝑌𝑌

Formally Deriving the PPF  The PPF is the maximum Y possible for a given

quantity of X given total available inputs ( �𝐾𝐾, �𝐿𝐿)

PPF = max 𝐾𝐾𝑌𝑌𝐿𝐿𝑌𝑌

𝑌𝑌(𝐾𝐾𝑌𝑌, 𝐿𝐿𝑌𝑌 ) subject to

X �𝐾𝐾 − 𝐾𝐾𝑌𝑌, �𝐿𝐿 − 𝐿𝐿𝑌𝑌 = 𝑋𝑋

 You can see the relationship to the CC from the first order conditions of this problem, which require that the :

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾 𝐿𝐿 𝑋𝑋 = 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿

𝑌𝑌

Objective Function

Constraint

Formally Deriving the PPF PPF = max

𝐾𝐾𝑌𝑌𝐿𝐿𝑌𝑌 𝑌𝑌(𝐾𝐾𝑌𝑌, 𝐿𝐿𝑌𝑌 ) subject to

X �𝐾𝐾 − 𝐾𝐾𝑌𝑌, �𝐿𝐿 − 𝐿𝐿𝑌𝑌 = 𝑋𝑋

 The solutions to this problem can be written as 𝐾𝐾𝑌𝑌 𝑋𝑋 , 𝐿𝐿𝑌𝑌(𝑋𝑋) and so

PPF = 𝑌𝑌(𝐾𝐾𝑌𝑌(𝑋𝑋), 𝐿𝐿𝑌𝑌 (𝑋𝑋))

 The MRT is the slope of the PPF is 𝑑𝑑𝑌𝑌(𝐾𝐾𝑌𝑌(𝑋𝑋),𝐿𝐿𝑌𝑌(𝑋𝑋))

The Marginal Rate of Transformation  The slope of the PPF is

𝑑𝑑𝑌𝑌(𝐾𝐾𝑌𝑌(𝑋𝑋),𝐿𝐿𝑌𝑌(𝑋𝑋)) 𝑑𝑑𝑋𝑋

𝑑𝑑𝑀𝑀𝑀𝑀𝑃𝑃 𝑑𝑑𝑋𝑋

= 𝜕𝜕𝑌𝑌 𝜕𝜕𝐾𝐾

𝐾𝐾𝑌𝑌 ′ +

𝜕𝜕𝑌𝑌 𝜕𝜕𝐿𝐿

𝐾𝐾𝐿𝐿 ′

 Similarly we can substitute the solutions into the constraint above:

X �𝐾𝐾 − 𝐾𝐾𝑌𝑌(𝑋𝑋), �𝐿𝐿 − 𝐿𝐿𝑌𝑌(𝑋𝑋) = 𝑋𝑋

Differentiating this with resect to 𝑋𝑋 shows that

𝜕𝜕𝑋𝑋 𝜕𝜕𝐾𝐾

(−𝐾𝐾𝑌𝑌 ′ ) +

𝜕𝜕𝑋𝑋 𝜕𝜕𝐿𝐿

(−𝐿𝐿𝑌𝑌 ′ ) = 1

The Marginal Rate of Transformation  This means

𝐾𝐾𝑌𝑌 ′ = −

𝜕𝜕𝑋𝑋 𝜕𝜕𝐿𝐿 𝐿𝐿𝑌𝑌

′ + 1 𝜕𝜕𝑋𝑋 𝜕𝜕𝐾𝐾

Wait, there is a point! Substitution this into the slope of the PPF gives us

𝑑𝑑𝑀𝑀𝑀𝑀𝑃𝑃 𝑑𝑑𝑋𝑋

= − � 𝜕𝜕𝑌𝑌 𝜕𝜕𝐾𝐾

𝜕𝜕𝑋𝑋 𝜕𝜕𝐾𝐾

− 𝜕𝜕𝑌𝑌 𝜕𝜕𝐾𝐾

𝜕𝜕𝑋𝑋 𝜕𝜕𝐿𝐿 𝐿𝐿𝑌𝑌

′

𝜕𝜕𝑋𝑋 𝜕𝜕𝐾𝐾

+ 𝜕𝜕𝑌𝑌 𝜕𝜕𝐿𝐿

𝐿𝐿𝑌𝑌 ′

The Marginal Rate of Transformation

𝑑𝑑𝑀𝑀𝑀𝑀𝑃𝑃 𝑑𝑑𝑋𝑋

= − � 𝜕𝜕𝑌𝑌 𝜕𝜕𝐾𝐾

𝜕𝜕𝑋𝑋 𝜕𝜕𝐾𝐾

+ 𝜕𝜕𝑌𝑌 𝜕𝜕𝐿𝐿

− 𝜕𝜕𝑌𝑌 𝜕𝜕𝐾𝐾

𝜕𝜕𝑋𝑋 𝜕𝜕𝐿𝐿 𝜕𝜕𝑋𝑋 𝜕𝜕𝐾𝐾

𝐿𝐿𝑌𝑌 ′

= − � 𝜕𝜕𝑌𝑌 𝜕𝜕𝐾𝐾

𝜕𝜕𝑋𝑋 𝜕𝜕𝐾𝐾

+ 𝜕𝜕𝑌𝑌 𝜕𝜕𝐿𝐿 𝜕𝜕𝑌𝑌 𝜕𝜕𝐾𝐾

− 𝜕𝜕𝑋𝑋 𝜕𝜕𝐿𝐿 𝜕𝜕𝑋𝑋 𝜕𝜕𝐾𝐾

𝜕𝜕𝑌𝑌 𝜕𝜕𝐾𝐾

𝐿𝐿𝑌𝑌 ′

The term in the square brackets is

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾 𝐿𝐿 𝑌𝑌 − 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿

𝑋𝑋 = 0

Productive Efficiency

The Marginal Rate of Transformation  Finally, this demonstrates that

𝑀𝑀𝑀𝑀𝑀𝑀 = 𝑑𝑑𝑀𝑀𝑀𝑀𝑃𝑃 𝑑𝑑𝑋𝑋

= � 𝜕𝜕𝑌𝑌 𝜕𝜕𝐾𝐾

𝜕𝜕𝑋𝑋 𝜕𝜕𝐾𝐾

= 𝑀𝑀𝑀𝑀𝐾𝐾

𝑌𝑌

𝑀𝑀𝑀𝑀𝐾𝐾 𝑋𝑋

 The MRT is the rate at which output can be transformed. It’s equal to the ratio of the marginal product of K in the two firms. As you shift inputs to move output, this is the trade off you face.

 Similar derivation could be done for L. These are equal because of the efficient distribution of inputs.

MRT and Marginal Cost Recall the profit maximization problem

max 𝐾𝐾,𝐿𝐿

𝑝𝑝𝑌𝑌𝑌𝑌 𝐾𝐾, 𝐿𝐿 − 𝑟𝑟𝐾𝐾 − 𝑤𝑤𝐿𝐿

First Order Condition from Profit maximization requires

𝑝𝑝𝑌𝑌 𝜕𝜕𝑌𝑌 𝜕𝜕𝐿𝐿

= 𝑤𝑤 => 𝜕𝜕𝑌𝑌 𝜕𝜕𝐿𝐿

= 𝑊𝑊 𝑝𝑝𝑌𝑌

and 𝑝𝑝𝑌𝑌

𝜕𝜕𝑌𝑌 𝜕𝜕𝐾𝐾

= 𝑟𝑟 => 𝜕𝜕𝑌𝑌 𝜕𝜕𝐾𝐾

= 𝑟𝑟 𝑝𝑝𝑌𝑌

MRT and Marginal Cost 𝜕𝜕𝑌𝑌 𝜕𝜕𝐾𝐾

= 𝑟𝑟 𝑝𝑝𝑌𝑌

Similarly for firm X: 𝜕𝜕𝑋𝑋 𝜕𝜕𝐾𝐾

= 𝑟𝑟 𝑝𝑝𝑋𝑋

Therefore

𝑑𝑑𝑀𝑀𝑀𝑀𝑃𝑃 𝑑𝑑𝑋𝑋

= − � 𝜕𝜕𝑌𝑌 𝜕𝜕𝐾𝐾

𝜕𝜕𝑋𝑋 𝜕𝜕𝐾𝐾

= − � 𝑟𝑟 𝑝𝑝𝑌𝑌 𝑟𝑟

𝑝𝑝𝑋𝑋 = −

𝑝𝑝𝑋𝑋 𝑝𝑝𝑌𝑌

MRT and Marginal Cost  If we use the Cost Function, profit maximisation

requires max 𝑌𝑌

𝑝𝑝𝑌𝑌𝑌𝑌 − 𝐶𝐶(𝑌𝑌) The FOC is

𝑝𝑝𝑌𝑌 − 𝐶𝐶′ 𝑌𝑌 = 0

 A competitive firm chooses output to equate Marginal Cost and price. Therefore

𝑀𝑀𝑀𝑀𝑀𝑀𝑌𝑌𝑋𝑋 = 𝑑𝑑𝑀𝑀𝑀𝑀𝑃𝑃 𝑑𝑑𝑋𝑋

= 𝑝𝑝𝑋𝑋 𝑝𝑝𝑌𝑌

= 𝑀𝑀𝐶𝐶𝑋𝑋 𝑀𝑀𝐶𝐶𝑌𝑌

MRT and Marginal Cost  So the MRT is just a statement about the marginal

costs of producing goods X and Y

MRT𝑌𝑌𝑋𝑋 = 𝑀𝑀𝐶𝐶𝑋𝑋 𝑀𝑀𝐶𝐶𝑌𝑌

 It costs society 𝑀𝑀𝐶𝐶𝑋𝑋 dollars to produce an additional unit of good X. This money must come out of the production of good Y

 A decrease in the production of good Y by one unit frees up 𝑀𝑀𝐶𝐶𝑌𝑌.

 Thus the amount of good Y that society must sacrifice to produce one more unit of good X is 𝑀𝑀𝐶𝐶𝑋𝑋/𝑀𝑀𝐶𝐶𝑌𝑌.

-- end of part 5 --

Productive Efficiency in General Equilibrium

Part 6. Examples

Economics 313

Functional Form Examples  Calculating a PPF for specific functional forms is

just working through the general steps above. This is conceptually the same, but often leads to tedious algebra.

 The slope is easier to calculate by using the cost function

 In the special case of constant returns to scale, MC is constant, so you get a linear PPF. You can find the intercept by calculating the output when all the inputs are used to produce good Y.

Decreasing Returns to Scale example

 First let’s assume that there is only one input, 𝐿𝐿, and there are �𝐿𝐿 units available. Now there is only one way a given output can be produced at each firm.

 The algebra in the general case is greatly simplified. The isoquants are points, and the Edgeworth box is gone, and optimization is just substitution.

One input PPF example Assume that the production functions are

𝑌𝑌 𝐿𝐿𝑌𝑌 = 𝐿𝐿𝑌𝑌 1/2 and X 𝐿𝐿𝑋𝑋 = 𝐿𝐿𝑋𝑋

1/2

 Then the PPF is found by substituting variables. Because X = �𝐿𝐿 − 𝐿𝐿𝑌𝑌 1/2 we have

𝐿𝐿𝑌𝑌(𝑋𝑋) = �𝐿𝐿 − 𝑋𝑋2 Therefore,

𝑀𝑀𝑀𝑀𝑃𝑃 = 𝑌𝑌(𝐿𝐿𝑌𝑌 𝑋𝑋 ) = �𝐿𝐿 − 𝑋𝑋2 1/2

One input PPF example: MRT  Therefore

𝑀𝑀𝑀𝑀𝑀𝑀𝑌𝑌𝑋𝑋 = 𝑑𝑑𝑀𝑀𝑀𝑀𝑃𝑃 𝑑𝑑𝑋𝑋

= 1 2 �𝐿𝐿 − 𝑋𝑋2 −

1 2 2𝑋𝑋

= 𝑋𝑋 �𝐿𝐿 − 𝑋𝑋2 − 1 2

 But 𝐿𝐿𝑌𝑌 = �𝐿𝐿 − 𝑋𝑋2 so

𝑀𝑀𝑀𝑀𝑀𝑀𝑌𝑌𝑋𝑋 = 𝑋𝑋 �𝐿𝐿 − 𝑋𝑋2 −12 = 𝑋𝑋 𝐿𝐿𝑌𝑌

−12 = 𝑋𝑋 𝑌𝑌

One input PPF example: Diagram Strictly convex technology gives a “nice” strictly convex production possibility set.

For �𝐿𝐿 = 100

feasible

infeasible

One input PPF example: MC  Recall that the cost function for this technology is

min 𝐿𝐿 𝑤𝑤𝐿𝐿 subject to 𝑄𝑄 = 𝐿𝐿1/2 for 𝑄𝑄 = 𝑌𝑌, 𝑋𝑋

 So the cost function is just inverting the production function and multiplying by 𝑤𝑤:

𝑐𝑐 𝑄𝑄 = 𝑤𝑤𝑄𝑄2 and 𝑀𝑀𝐶𝐶 = 2𝑤𝑤𝑄𝑄 for 𝑄𝑄 = 𝑌𝑌, 𝑋𝑋

Therefore,

𝑀𝑀𝑀𝑀𝑀𝑀𝑌𝑌𝑋𝑋 = 𝑀𝑀𝐶𝐶𝑋𝑋 𝑀𝑀𝐶𝐶𝑌𝑌

= 2𝑤𝑤𝑋𝑋 2𝑤𝑤𝑌𝑌

= 𝑋𝑋 𝑌𝑌

Another example

Slide 52

Assume both firms 𝑋𝑋 and 𝑌𝑌 have Cobb-Douglas Constant Returns to Scale technology:

𝑞𝑞 𝐾𝐾, 𝐿𝐿 = 𝐾𝐾1/2𝐿𝐿1/2 for 𝑞𝑞 = 𝑋𝑋, 𝑌𝑌

The cost function 𝑐𝑐(𝑞𝑞)is the solution to

𝑐𝑐 𝑞𝑞; 𝑟𝑟, 𝑤𝑤 = min 𝐾𝐾,𝐿𝐿

𝑟𝑟𝐾𝐾 + 𝑤𝑤𝐿𝐿 Subject to

𝑞𝑞 = 𝐾𝐾1/2𝐿𝐿1/2 FOCs:

𝑟𝑟 − 𝜆𝜆 1 2 𝐾𝐾−

1 2𝐿𝐿

1 2 = 0

𝑤𝑤 − 𝜆𝜆 1 2 𝐾𝐾 1 2𝐿𝐿−

1 2 = 0

𝑞𝑞 = 𝐾𝐾1/2𝐿𝐿1/2

Another example

Slide 53

Therefore 𝑟𝑟 𝑤𝑤

= 𝐿𝐿 𝐾𝐾

→ 𝐾𝐾 = 𝑤𝑤𝐿𝐿 𝑟𝑟

and

𝑞𝑞 = 𝑤𝑤𝐿𝐿 𝑟𝑟

1/2 𝐿𝐿1/2 = 𝑤𝑤

𝑟𝑟

1/2 𝐿𝐿

So 𝐿𝐿 𝑞𝑞; 𝑤𝑤, 𝑟𝑟 = 𝑟𝑟 𝑤𝑤

1/2 𝑞𝑞

Similarly, 𝐾𝐾 𝑞𝑞; 𝑤𝑤, 𝑟𝑟 = 𝑤𝑤 𝑟𝑟

1/2 𝑞𝑞

𝑐𝑐 𝑞𝑞; 𝑟𝑟, 𝑤𝑤 = 𝑟𝑟 𝑤𝑤 𝑟𝑟

1/2 +

𝑟𝑟 𝑤𝑤

1/2 𝑤𝑤 𝑞𝑞 = 2 𝑟𝑟𝑤𝑤 1/2𝑞𝑞

Another example

Slide 54

Then

𝑀𝑀𝐶𝐶 = 𝑑𝑑𝑐𝑐 𝑞𝑞; 𝑟𝑟, 𝑤𝑤

𝑑𝑑𝑞𝑞 = 2 𝑟𝑟𝑤𝑤 1/2

For both firms.

Assume the economy has �𝐾𝐾 units of capital and �𝐿𝐿 units of labour. So it all is devoted to good Y, output is

𝑌𝑌 �𝐾𝐾, �𝐿𝐿 = �𝐾𝐾1/2�𝐿𝐿1/2

Another example

Slide 55

The PPF is

�𝐾𝐾1/2�𝐿𝐿1/2

Slope is − 2 𝑟𝑟𝑤𝑤 1/2

2 𝑟𝑟𝑤𝑤 1/2 = −1

-- end of part 6 --

Productive Efficiency in General Equilibrium

Part 7. Summary

Economics 313

Concept Check  The PPF for an economy is given by Y =18-2X.

 Suppose 4 units of good X are produced in this economy. Calculate the production efficient amount of good Y in this economy.

Summary  We examined the Edgeworth production box and

efficiency

 We discussed how firms will efficiency allocate capital and labour between them

 We also demonstrated how the PPF of an economy can be generated by all the efficient allocations of capital and labour between firms

 We discussed how to interpret the PPF

What I Expect You to Know  To draw and explain the Edgeworth production box and

discuss production efficiency and trade

 Be able describe the relationship between the firm’s contract curve and the production possibilities frontier

 Be able to explain and use the production possibilities frontier and the concept of the marginal rate of transformation

-- end of part 7 --