Customer‘s safety is the future trend of the foodservice and hospitality industry

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Allocative Efficiency in General Equilibrium

Part 1. Robinson Crusoe

Economics 313

Robinson Crusoe

A questionable tale of individual survival

 Lets start with a strange example, where there is only one consumer and one firm. Sometimes this is called the Robinson Crusoe economy, names for the 18th Century novel by Daniel Defoe.

 The character lives for decades on an island; eventually he is joined by Friday, at which point the model can be used to illustrate the gains from trade.

The Castaway

 We will ignore the novel’s religious message and colonial cultural baggage, and focus on the basic economics.

 In a one person economy, that person owns the firm, so receives any profit, but is also the sole employee, so earns wages from supplying labour.

 Despite this situation, we will assume that the firm and the consumer act competitively, so planning at the two levels is independent. With some restrictions, we could assume a large number of identical firms and consumers.

One consumer, two goods, and one firm.

 The consumer has preferences over the good 𝑞𝑞 and “leisure.” These are the two goods in the economy.

 The consumer is endowed with 24 hours per day to allocate between labour and leisure. To consume the production good 𝑞𝑞 they need to purchase it in the competitive output market.

 The income to purchase the good 𝑞𝑞 comes the profit of the firm (owned by the consumer) and by trading some of the time endowment to supply labour at the market wage rate.

The Economy: Labour / Leisure choice

The consumers problem is therefore subject to

and 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 + 𝐿𝐿 = 24

 In what follows it is convenient to use the time constraint to substitute out the good 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 out of the utility function.

subject to

 This makes it clear that utility decreases in 𝐿𝐿: i.e. labour is a “bad.” This means indifference curves slope upward in (𝐿𝐿, 𝑞𝑞) space.

 The parameters 𝑤𝑤 and 𝑝𝑝 are treated as constants from the perspective of the consumer: Competitive price taking behaviour.

max 𝑞𝑞,𝐿𝐿

𝑈𝑈(𝑞𝑞, 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙) 𝑝𝑝𝑞𝑞 = 𝑤𝑤𝐿𝐿 + 𝜋𝜋(𝑝𝑝, 𝑤𝑤)

The Economy

max 𝑞𝑞,𝐿𝐿

𝑈𝑈(𝑞𝑞, 24 − 𝐿𝐿) 𝑝𝑝𝑞𝑞 = 𝑤𝑤𝐿𝐿 + 𝜋𝜋(𝑝𝑝, 𝑤𝑤)

 The firm owns technology 𝒇𝒇 𝑳𝑳 hires labour 𝑳𝑳 on a competitive market at wage 𝒘𝒘 and sells the resulting output good 𝒒𝒒 on a competitive market at price 𝒑𝒑. It chooses labour to solve

 The technology 𝑓𝑓 𝐿𝐿 is a fundamental of the economy, like preferences or endowments.

 The parameters 𝑤𝑤 and 𝑝𝑝 are treated as constants from the perspective of the firm: Competitive price taking behaviour.

max 𝐿𝐿

𝑝𝑝𝑓𝑓 𝐿𝐿 − 𝑤𝑤𝐿𝐿

The Economy

 This simple economy has two markets. The fact that the same agent are active in both markets makes the connection easier to see than in the partial equilibrium models you have studied previously.

 Given the assumptions of the model, choices in one market directly imply choices in the other market. When firms choose how much labour to hire, they are simultaneously deciding how much output to produce, and given the competitive market, we assume they can sell all of this at the going price.

 Similarly, when the consumer/worker chooses how much to work at the going wage they (partly) determine their income. Because we assume all income is spent, this decision also determines their demand in the goods market. Consumption of leisure is just the residual.

Two Markets: Goods & Labour

 The assumption that markets are competitive means that for both agents the “other side” of the market will always take everything supplied and provide everything demanded at the going price.

 In principle, this would let us replicate a “partial equilibrium” approach. First we choose one market, say the labour market, and fix the consumption price at 𝑝𝑝 and then find the wage that clears the labour market. Then turn to the goods market, fix and arbitrary 𝑤𝑤 and solve for the market clearing price.

Two Markets: Goods and Labour

 What we are interested in though is how the prices in these two markets – that is the wage in the labour market and the price in the goods market – are simultaneously determined in a General Equilibrium. (And of course the efficiency properties of the equilibrium.)

 So we take a different approach, one similar to the structure of the past two lectures.

 First we will find the equilibrium in the “consumption” sector by solving the consumer’s problem. Next we will find an equilibrium in the goods sector.

 Then we will bring them together to find the General Equilibrium

Two Sectors: Production & Consumption

-- end of part 1 --

Allocative Efficiency in General Equilibrium

Part 2. The Consumption Sector

Economics 313

 Recall that there is only one agent in this model, who acts as a entrepreneur in the production sector, and as a consumer/worker in the consumption sector.

 We first focus on the consumption decision, and assume that the wage rate and profit are determined exogenously in the production sector.

 To be clear, consumption and labour are connected through the time constraint, and the fact that the utility function values the consumption good and leisure.

Consumption

 To be concrete, we will assume that utility takes the Cobb-Douglas form

 This creates “nice” convex indifference curves. Recall that because we have substituted the time constraint into the utility function, when we look in (𝑞𝑞, 𝐿𝐿) space, the indifference curves slope upward. Labour is a “bad.”

 The indifference curve for utility �𝑈𝑈 is given by

The Consumer / Worker’s Problem

U = 𝑞𝑞(24 − 𝐿𝐿)

𝑞𝑞 𝐿𝐿 = �𝑈𝑈

(24 − 𝐿𝐿)

Indifference Map

𝑈𝑈 = 3350

𝑈𝑈 = 500

𝒒𝒒

𝑳𝑳

 The consumer chooses consumption and labour subject to the budget constraint, where income depends on the labour choice, and on the wage 𝑤𝑤 and the firm profit 𝜋𝜋 𝑝𝑝, 𝑤𝑤 These are set in the production sector.

 The necessary condition for utility maximization is that the budget constraint is satisfied, and

𝑀𝑀𝑈𝑈𝐿𝐿 𝑀𝑀𝑈𝑈𝑞𝑞

= 𝑀𝑀𝑀𝑀𝑀𝑀𝑞𝑞𝐿𝐿 = 𝑤𝑤 𝑃𝑃

 Note that the 𝑀𝑀𝑈𝑈𝐿𝐿is negative, which is needed because labour enters in as income, not consumption.

𝑝𝑝𝑞𝑞 = 𝜋𝜋(𝑝𝑝, 𝑤𝑤) + 𝑤𝑤𝐿𝐿

The Consumer / Worker’s Problem

Utility Maximization

𝐿𝐿∗

q = 74 + 6𝐿𝐿

𝒒𝒒

𝑳𝑳

𝑞𝑞∗

At the utility maximum, the ratio of the marginal disutility of labour to the marginal utility of the consumption good is equal to the wage divided by the price.

 The equilibrium depicted assumes that the consumer receives 74𝑝𝑝 of profit income, and faces a wage of 6𝑝𝑝

 Given this income, the consumer chooses to work about 6 hrs a day, and will consume about 100 units of the production good.

Partial Equilibrium in Labour/𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙

 We could have written the model in consumption – leisure space, and both the budget line and indifference curves would have taken a more familiar shape.

 Models like this are used to explore the labour supply decision, and examine policies like progressive taxation, minimum wages, employment insurance, overtime pay, etc.

 The approach we take instead makes is simple to integrate the consumption sector with the production sector

Partial Equilibrium Models of Labour Supply

-- end of part 2 --

Allocative Efficiency in General Equilibrium

Part 3. The Production Sector

Economics 313

 Our model of the choice between consumption and leisure treated the wage rate and profit from owning the firm as fixed and independent of the choices made in the consumption sector.

 As we turn to production, and make a similar competitive assumption. When acting as the firm, the agent assumes that they can hire as much labour as they want at the going wage 𝑤𝑤 and sell the resulting output at the competitive price 𝑝𝑝.

The Agent as Entrepreneur

 The given the competitive prices 𝑝𝑝 and 𝑤𝑤, firm chooses 𝐿𝐿 to maximize profits:

 The necessary condition for an interior solution is:

 The marginal product of labour is equal to the “real wage”

𝑤𝑤 𝑝𝑝 .

The Firm’s Problem

𝜋𝜋 𝑝𝑝, 𝑤𝑤 = max 𝐿𝐿

𝑝𝑝𝑓𝑓 𝐿𝐿 − 𝑤𝑤𝐿𝐿

𝑝𝑝𝑓𝑓′ 𝐿𝐿 = 𝑤𝑤

𝑓𝑓′ 𝐿𝐿 = 𝑤𝑤 𝑝𝑝

 The profit maximizing choice of labour 𝐿𝐿∗ determines the level of production, and since this is the only firm, this is the equilibrium in the market for 𝑞𝑞

 For any given combination of 𝑤𝑤 and 𝑝𝑝, and any particular choice of 𝑞𝑞 and 𝐿𝐿 will result in some amount of “profit” according to the linear function:

 Profit maximisation amounts to making this a large as possible, subject to feasibility.

 This function is composed of combinations of 𝑞𝑞 and 𝐿𝐿 that result in the same amount of profit. So this line is called an “isoprofit” line, and can be plotted in 𝑞𝑞, 𝐿𝐿 space.

The Market for 𝑞𝑞.

𝜋𝜋 = 𝑝𝑝𝑞𝑞 − 𝑤𝑤𝐿𝐿

q = �𝜋𝜋 𝑝𝑝

+ 𝑤𝑤 𝑝𝑝 𝐿𝐿

𝒒𝒒

𝑳𝑳

An Isoprofit Line

�𝜋𝜋

The isoprofit line, for market prices 𝑝𝑝 and 𝑤𝑤, is given by �𝜋𝜋 = 𝑝𝑝𝑞𝑞 − 𝑤𝑤𝑙𝑙

solved for 𝑞𝑞

In this example, �𝜋𝜋 = 120𝑝𝑝

Isoprofit Lines for Various Choices of 𝑤𝑤 and 𝑝𝑝.

𝑳𝑳

(𝑤𝑤 = 1, 𝑝𝑝 = 1)

(𝑤𝑤 = 27, 𝑝𝑝 = 3)

(𝑤𝑤 = 25, 𝑝𝑝 = 5)

(𝑤𝑤 = 6, 𝑝𝑝 = 1)

𝜋𝜋 𝑝𝑝

𝜋𝜋 = 350

𝜋𝜋 = 1500

𝜋𝜋 = 660

𝜋𝜋 = 60

Isoprofit lines are just definitions so they can cross, unlike isoquants or indifference curves. Holding 𝒑𝒑 and 𝒘𝒘 constant, higher isoprofit lines represent higher levels of profit.

q = 𝜋𝜋 𝑝𝑝

+ 𝑤𝑤 𝑝𝑝 𝐿𝐿

𝒒𝒒

 This economy produces the good 𝑞𝑞 and leisure. The trade off between these is given by the marginal product of labour.

 The output choices for 𝑞𝑞 depend on the production function 𝑞𝑞 = 𝑓𝑓(𝐿𝐿) that describes the technology for transforming labour into the consumption good.

 We’ll use a simple function

 This familiar function is strictly concave. The points below the function are all feasible for the firm to produce, but not “technically efficient.”

The Production Possibility Set

𝑞𝑞 = 𝑓𝑓 𝐿𝐿 = 100 ln(𝐿𝐿)

The Production Function

𝑳𝑳 Notice that the feasible set is “strictly convex” even though the function is strictly concave. This is “nice” technology.

𝑞𝑞 = 𝑓𝑓 𝐿𝐿 = 100 ln(𝐿𝐿)

feasible set

𝒒𝒒

 Notice that the production function fully determines the (efficient) trade off between output and leisure.

 So it is the PPF for this simple one firm economy, where productive efficiency only requires that the resources are used efficiently by this one firm.

 The connection between 𝑓𝑓(𝐿𝐿) and the PPF is clearer if it plotted in (𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙, 𝑞𝑞) space, remembering the constraint:

𝑃𝑃𝑃𝑃𝑃𝑃 = 𝑓𝑓(24 − 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙)

The Production Possibility Set

The Production Possibility Set:

𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍

Notice that the feasible set is “strictly convex.” It doesn’t go all the way to 24 because with Log technology, less that one hour of labour produces negative output.

𝑞𝑞 = 100 ln(24 − 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙)

feasible set

𝒒𝒒

 The profit maximum for a give 𝑝𝑝 and 𝑤𝑤 is found by choosing the highest isoprofit curve that has at least one point -- i.e. combination of 𝐿𝐿 and 𝑞𝑞 -- in the feasible set.

 Because the production function is strictly concave there is only one profit maximising choice of 𝐿𝐿. The highest isoprofit line is tangent to the production function.

Partial Equilibrium in Production

Maximizing Profit:

𝑳𝑳 Notice that these isoprofit lines are parallel because they all depend on the competitive prices 𝑝𝑝 and 𝑤𝑤: here ⁄𝑤𝑤 𝑝𝑝 = 9. .

𝑓𝑓′ 𝐿𝐿 = 𝑤𝑤 𝑝𝑝

𝜋𝜋 = 70

𝜋𝜋 = 25

𝜋𝜋 = 183 Normalizing prices so that 𝑝𝑝 = 1, then the maximum profit is 𝜋𝜋∗ = $142.50

𝜋𝜋∗ = $142.50

𝑞𝑞∗

𝐿𝐿∗

𝒒𝒒

Maximizing Profit:

𝑳𝑳

𝑓𝑓′ 𝐿𝐿 = 𝑤𝑤 𝑝𝑝

= 9

𝜋𝜋∗ = 142.50p

𝑞𝑞∗

𝐿𝐿∗

With a strictly convex technology set, there is a unique profit maximizing choice of 𝐿𝐿 for any combination of 𝑝𝑝 and 𝑤𝑤

feasible set

𝒒𝒒

 The equilibrium depicted assumes that the firm will employ about 11 hours of labour a day at the wage rate of 9𝑝𝑝, which results in a daily profit of 142.50 𝑝𝑝

 This creates about 240 units of the production good.

Partial Equilibrium in Production

-- end of part 3 --

Allocative Efficiency in General Equilibrium

Part 4. General Equilibrium

Economics 313

 Clearly, something isn’t adding up. The income level depicted in the consumers problem is inconsistent with the equilibrium of the firm.

 At the given wage and prices faced by the consumer, the firm does not have the profit required to general the income the budget line assumes to exist.

 In other words, assuming this high level of income in the form of profit, the consumer is only wanting to work about 5 hours a day.

General Equilibrium

 Because all of our diagrams and calculations are in 𝐿𝐿, 𝑞𝑞 space it is straightforward to bring the two sectors together.

 Also, you know that the slope of the budget line and the slope of the isoprofit line are both 𝑤𝑤/𝑝𝑝.

 So have we found a General Equilibrium? Clearly not, as we have not ensured that the prices and wages that maximized utility were the same as those that maximized profit.

General Equilibrium

Utility Maximization and Profit Maximization

π 𝑝𝑝

= $142.50

U = 3600

The consumption and production sectors are in partial equilibrium, but not General Equilibrium.

𝑞𝑞∗∗

𝐿𝐿∗∗𝐿𝐿∗

𝑞𝑞∗

𝑤𝑤 𝑝𝑝

= $9

The consumer budget assumes they work 10 hours a day and earn $140 profit (if 𝑝𝑝 is normalized to $1).

But they only want to work a little more than 4 hours a day at that income.

𝜋𝜋∗

𝑝𝑝

𝑤𝑤 𝑝𝑝

𝒒𝒒

𝑳𝑳

 General equilibrium requires that

1. The consumer must be maximizing utility, given the price 𝑝𝑝 and wage 𝑤𝑤

2. The firm must be maximizing profit, given the price 𝑝𝑝 and wage 𝑤𝑤

3. Both the labour market and the goods market must clear.

General Equilibrium

Utility Maximization and Profit Maximization

q = 𝜋𝜋 𝑝𝑝

+ 𝑤𝑤 𝑝𝑝 𝐿𝐿

𝐿𝐿∗∗∗

𝑞𝑞 = 100 ln(𝐿𝐿)

U = 𝑞𝑞(24 − 𝐿𝐿)

𝑞𝑞∗∗∗

𝒒𝒒

𝑳𝑳

Utility Maximization and Profit Maximization

q = 113 + 12𝐿𝐿

𝐿𝐿∗∗∗=7.25

𝑞𝑞 = 100ln(𝐿𝐿) 𝑞𝑞∗∗∗ =198

π 𝑝𝑝

= $113

U = 3350

𝑤𝑤 𝑝𝑝

= $12

𝒒𝒒

𝑳𝑳

3350 = 𝑞𝑞(24 − 𝐿𝐿)

 The general equilibrium requires that 𝑤𝑤 𝑝𝑝

= 12.

 At this “real wage” firms demand about 7.25 hrs of labour a day and earn $113𝑝𝑝 in profit. The worker/owners facing a wage of $12𝑝𝑝 and earning $113𝑝𝑝 in profit income, optimally chooses to supply 7.25 hrs of labour. The labour market clears.

 With an income of $113𝑝𝑝, facing a price of 𝑝𝑝 for good 𝑞𝑞 per day, consumers want to buy 200 units of 𝑞𝑞. Facing a wage of $12𝑝𝑝 the firm optimally chooses to produce 200 units of 𝑞𝑞 with 7.25 hrs of labour per day. The goods market clears.

General Equilibrium

 Only the price ratio 𝑤𝑤 𝑝𝑝

is defined. Multiplying both wages

and prices by any positive number leaves the equilibrium unchanged.

 Had we found a value of 𝑤𝑤 𝑝𝑝

that cleared the labour

market, taking into account optimal profit maximising labour demand at that price and the resulting profit, then this price ratio would also clear the goods market. This is Walras Law.

General Equilibrium: two observations

-- end of part 4 --

Allocative Efficiency in General Equilibrium

Part 5. Allocative Efficiency

Economics 313

 This model is so simple that Distributive and Productive Efficiency are trivial.

 With one consumer, the only issue is whether the goods produced 𝑞𝑞 and 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 are consumed. This is true because by assumption the consumer is maximizing utility.

 Similarly, with one firm, the input of labour can’t be reallocated, and by the assumption of profit maximization, all inputs the firm uses are deployed to maximize output.

Three Degrees of Efficiency

 On the other hand, the simplicity of this model makes clear the third form of efficiency, Allocative Efficiency.

 This asks whether the “right” mix of 𝑞𝑞 and 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 are created in the economy from the perspective of consumers.

 This requires that the rate at which the economy can shift production between goods must be the same as the rate at which consumers are willing to substitute between goods. The MRT = MRS.

Three degrees of Efficiency

 The slope of the indifference curve is the MRS, and competitive behaviour ensures this is equal to

𝑤𝑤 𝑝𝑝

 The MRT is just the Marginal Product of Labour: this shows how labour, the negative of leisure is transformed in to the output good. Competitive behaviour in the labour market ensures this is also equal to

𝑤𝑤 𝑝𝑝

 Therefore, in the General Equilibrium MRT = MRS.

 A look at the diagram for the Competitive Equilibrium illustrated this very clearly.

Allocative Efficiency

General Equilibrium

𝒒𝒒

𝑳𝑳

Distributive Efficiency: With only one consumer, total utility cannot increase if output or leisure is reallocated.

Productive Efficiency: With only one firm, total output cannot increase if inputs are reallocated.

Allocative Efficiency: It is not possible to change the mix of output between 𝑞𝑞 and leisure and increase total utility.

The decentralized competitive general equilibrium achieves all three forms of efficiency

-- end of part 5 --

Allocative Efficiency in General Equilibrium

Part 6. The Planner’s Problem

Economics 313

Recall the Welfare Theorems

 The First Welfare Theorem of Economics  Equilibrium in Competitive Markets is Pareto Optimal

 The Second Theorem of Welfare Economics  Any Pareto Optimal allocation can be sustained as a

competitive equilibrium (if preferences are convex).

 The reallocations required for the Second theorem are not possible for our one agent economy.

 In our simple economy, it is unclear why the consumer/producer would even think of a “labour” market, or ponder the “price of output”, even though, as we have seen, these ideas are well defined.

 Instead, the would act as a “social planner” who chooses how much to work to directly maximize “social welfare,” which here is just the agent’s own utility.

 In an economy with more than one consumer, a social planner might also desire a particular distribution of total consumption.

The Planner’s Problem

 The “Social Planners Problem” is then to  Maximize Total Utility subject to Technology and Resource

Constraints.

 In the present model the planner’s problem is max

𝐿𝐿,𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑈𝑈(𝑞𝑞, 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙)

subject to 𝑞𝑞 = 𝑓𝑓(𝐿𝐿) and 𝐿𝐿 = 24 − 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙

 Again, the same diagram makes it clear that the Competitive Equilibrium solves the Planner’s Problem.

The Planner’s Problem

feasible production

𝑈𝑈 ↑

The decentralized Competitive Equilibrium is the solution to the Planner’s Problem: The First Fundamental Theorem of Welfare Economics

We can again substitute out the constraint, and the problem becomes

max 𝐿𝐿

𝑈𝑈(𝑞𝑞, 24 − 𝐿𝐿) subject to 𝑞𝑞 = 𝑓𝑓(𝐿𝐿), or even simpler:

max 𝐿𝐿

𝑈𝑈(𝑓𝑓(𝐿𝐿), 24 − 𝐿𝐿) The first order conditions are then:

𝜕𝜕𝑈𝑈(𝑞𝑞, 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙) 𝜕𝜕𝑞𝑞

𝑓𝑓′ 𝐿𝐿 + 𝜕𝜕𝑈𝑈 𝑞𝑞, 𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝜕𝜕𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙

−1 = 0

 Our diagram makes it clear that the Competitive Equilibrium solves the Planner’s Problem.

The Planner’s Problem

The solution to the planner’s problem ensures the “right” mix of the output good and leisure.

The first order conditions show that

𝑀𝑀𝑀𝑀𝑀𝑀 =

𝜕𝜕𝑈𝑈 𝜕𝜕𝑞𝑞 𝜕𝜕𝑈𝑈

𝜕𝜕𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙

= 𝑓𝑓′ 𝐿𝐿 = 𝑀𝑀𝑀𝑀𝑀𝑀

 Let’s look at this in (𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙, 𝑞𝑞) space:

Allocative Efficiency

The Planner’s Problem

feasible production

𝑈𝑈 ↑

The “social welfare function” is tangent to the PPF

𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍𝒍

𝒒𝒒 MRS=MRT

 In the competitive equilibrium, the agent, acting as a consumer equated the marginal rate of substitution between labour (the negative of leisure) and the consumption good to the wage price ratio. When acting as a producer, they equated the marginal product of labour to that same wage price ration. As we have seen, the condition for efficiency is that the MRS=MRT.

 This is the substance of the first welfare theorem in this simple economy: A Competitive Equilibrium is Pareto Efficient.

 In an economy with more than one consumer, a social planner might also desire a particular Pareto Efficient distribution of total consumption. The Second Welfare Theorem says when this can be done through the “use” of competitive markets. To explore this, we need more consumers.

The First Welfare Theorem

-- end of part 6 --

Allocative Efficiency in General Equilibrium

Part 7. 2 consumers, 2 inputs, 2 firms

Economics 313

A second person arrives on the island

Slide 63

 Our real interest is many consumers and many firms. But it turns out that the key issues missed in the Robinson Crusoe story, namely Distributive and Productive Efficiency, can be explored with the arrival of one additional agent. One Friday this happens.

 We could still think of leisure, and some examples do so, but to keep things general we will revert to earlier notation of Consumers A and B, and Firms X and Y. The underlying input resources, Capital and Labour, belong to the consumers.

Recall the definitions:

Slide 64

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?

The PPF is the set of Productively Efficient input allocations.

Productive Efficiency

PPF

The PPF is the set of Productively Efficient input allocations.

Each unit less of good 𝑋𝑋 we produce at the point 𝑍𝑍 frees up resources worth 𝑀𝑀𝐶𝐶𝑥𝑥, which can move to the production of 𝑌𝑌, where they can be used to produce 𝑀𝑀𝐶𝐶𝑦𝑦 of output.

Productive Efficiency

PPF

𝑀𝑀𝑀𝑀𝑀𝑀𝑌𝑌𝑌𝑌 = − Δ𝑌𝑌 Δ𝑋𝑋

= 𝑀𝑀𝐶𝐶𝑌𝑌 𝑀𝑀𝐶𝐶𝑌𝑌

Δ𝑌𝑌 Δ𝑋𝑋

The PPF is the set of Productively Efficient input allocations.

The point Z is the total supply of the two goods in the economy.

Productive Efficiency

PPF

The Edgeworth Box of exchange showed that for the distribution of a fixed set of goods to be efficient (at an interior), the MRS for the two consumers must be the same.

Distributive Efficiency

B Origin Y

A Origin

The Edgeworth Box of exchange showed that for the distribution of a fixed set of goods to be efficient (at an interior), the MRS for the two consumers must be the same.

Distributive Efficiency

B Origin Y

A Origin

The Edgeworth Box of exchange showed that for the distribution of a fixed set of goods to be efficient (at an interior), the MRS for the two consumers must be the same.

When distribution occurs in a competitive market, the MRS for each consumer is equal to the price ratio:

𝑀𝑀𝑀𝑀𝑀𝑀𝑦𝑦𝑥𝑥𝐴𝐴 = 𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

= 𝑀𝑀𝑀𝑀𝑀𝑀𝑦𝑦𝑥𝑥𝐵𝐵

Distributive Efficiency

B Origin Y

A Origin

The Edgeworth Box of exchange showed that for the distribution of a fixed set of goods to be efficient (at an interior), the MRS for the two consumers must be the same.

When distribution occurs in a competitive market, the MRS for each consumer is equal to the price ratio:

𝑀𝑀𝑀𝑀𝑀𝑀𝑦𝑦𝑥𝑥𝐴𝐴 = 𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

= 𝑀𝑀𝑀𝑀𝑀𝑀𝑦𝑦𝑥𝑥𝐵𝐵

This defines the Contract Curve

Distributive Efficiency

B Origin Y

A Origin

The Edgeworth Box of exchange showed that for the distribution of a fixed set of goods to be efficient (at an interior), the MRS for the two consumers must be the same.

The origin for B is where A is allocated all of the goods in the economy. That is, read from A’s origin, this is the production of X and Y.

Which is Z!

Distributive Efficiency

B Origin Y

A Origin

𝑀𝑀𝑀𝑀𝑀𝑀𝑦𝑦𝑥𝑥𝐴𝐴 = 𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

= 𝑀𝑀𝑀𝑀𝑀𝑀𝑦𝑦𝑥𝑥𝐵𝐵

This means we can integrate the two diagrams

Productive efficiency ensures that 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿

𝑌𝑌 = 𝑤𝑤 𝑙𝑙

= 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝐾𝐾𝐿𝐿 𝑌𝑌

Competitive equilibrium in the output market ensure that

𝑀𝑀𝑀𝑀𝑀𝑀𝑌𝑌𝑌𝑌 = 𝑝𝑝𝑌𝑌 𝑝𝑝𝑦𝑦

= 𝑀𝑀𝐶𝐶𝑌𝑌 𝑀𝑀𝐶𝐶𝑌𝑌

Competitive Equilibrium in the output market ensures that 𝑀𝑀𝑀𝑀𝑀𝑀𝑦𝑦𝑥𝑥𝐴𝐴 =

𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

= 𝑀𝑀𝑀𝑀𝑀𝑀𝑦𝑦𝑥𝑥𝐵𝐵

Allocative Efficiency

PPF

What Would Happen if 𝑀𝑀𝑀𝑀𝑀𝑀 ≠ 𝑀𝑀𝑀𝑀𝑀𝑀?

Slide 74

 For example assume that 𝑀𝑀𝑀𝑀𝑀𝑀 = 2 and 𝑀𝑀𝑀𝑀𝑀𝑀 = 1.  Both consumers are the equally well off if they give up

two units of good Y and receive one more unit of good X

 But to produce one more unit of good X, we only have to give up one unit of good Y.

 Both consumers better off if more units of good X are produced.

 This would continue until 𝑀𝑀𝑀𝑀𝑀𝑀 = 𝑀𝑀𝑀𝑀𝑀𝑀

What Would Happen if 𝑀𝑀𝑀𝑀𝑀𝑀 ≠ 𝑀𝑀𝑀𝑀𝑀𝑀?

Slide 75

 Consider how the market works in when 𝑀𝑀𝑀𝑀𝑀𝑀 = 2 and 𝑀𝑀𝑀𝑀𝑀𝑀 = 1.

 Competition in the output market ensures that

𝑀𝑀𝑀𝑀𝑀𝑀 = 1 = 𝑀𝑀𝐶𝐶𝑌𝑌 𝑀𝑀𝐶𝐶𝑌𝑌

= 𝑝𝑝𝑌𝑌 𝑝𝑝𝑦𝑦

 Consumers will increase their consumption of X relative to Y. This reduces the MRS and also increases the price of X.

 This process continues until 𝑀𝑀𝑀𝑀𝑀𝑀 = 𝑀𝑀𝑀𝑀𝑀𝑀

-- end of part 7 --

Allocative Efficiency in General Equilibrium

Part 8. Wealth and Income

Economics 313

The condition that

𝑀𝑀𝑀𝑀𝑀𝑀𝑦𝑦𝑥𝑥𝐴𝐴 = 𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

= 𝑀𝑀𝑀𝑀𝑀𝑀𝑦𝑦𝑥𝑥𝐵𝐵

holds because competitive consumers maximize utility subject to their budget, given by

where 𝐿𝐿 and 𝐾𝐾 are the consumers endowment of Capital and Labour, and 𝜋𝜋 is the profit they earn from owning shares in firm X and Y.

What determines the Budget Line?

PPF

𝑝𝑝𝑥𝑥𝑋𝑋 + 𝑝𝑝𝑦𝑦𝑌𝑌 = 𝜋𝜋 𝑝𝑝, 𝑤𝑤 + 𝑤𝑤𝐿𝐿 + 𝑙𝑙𝐾𝐾

Price taking behaviour means that at any point in time, total income, 𝑀𝑀 𝑝𝑝, 𝑤𝑤, 𝑙𝑙 = 𝜋𝜋 + 𝑤𝑤𝐿𝐿 + 𝑙𝑙𝐾𝐾

is fixed.

This determines the location of the budget line, but not its slope. To find it’s location, you just need one point. For example, assume the consumer A spends their total income on good Y.

They could buy �𝑌𝑌 units, where �𝑌𝑌 =

𝑀𝑀(𝑝𝑝, 𝑤𝑤, 𝑙𝑙) 𝑝𝑝𝑌𝑌

What determines the Budget Line?

PPF

Price taking behaviour means that at any point in time, total income, 𝑀𝑀 𝑝𝑝, 𝑤𝑤, 𝑙𝑙 = 𝜋𝜋 + 𝑤𝑤𝐿𝐿 + 𝑙𝑙𝐾𝐾

is fixed.

This determines the location of the budget line, but not its slope. To find it’s location, you just need one point. For example, assume the consumer A spends their total income on good Y.

They could buy �𝑌𝑌 units, where �𝑌𝑌 =

𝑀𝑀(𝑝𝑝, 𝑤𝑤, 𝑙𝑙) 𝑝𝑝𝑌𝑌

What determines the Budget Line?

PPF

�𝑌𝑌

If consumer A had more income, they would be on a higher budget line, like this depicted to the right.

In general, the MRS changes along the contract curve, and for this reason, the equilibrium in the economy may depend on the distribution of initial wealth endowments.

Economists often make specific assumptions about preference to overcome these “income effects” and allow aggregation to a “representative consumer”

What determines the Budget Line?

PPF

It is clear that the Competitive Equilibrium is Pareto Optimal.

This picture also hints at how a social planner could use markets to achieve a specific efficient outcome.

By moving the endowment point with lump sum transfers, the planner can let markets work to find an efficient outcome.

Unlike the FWT, the second welfare theorem requires more assumptions about preferences (specifically convexity).

The Second Welfare Theorem

PPF

-- end of part 8 --

Allocative Efficiency in General Equilibrium

Part 9. Summary and Expectations

Economics 313

Summary  We have now constructed a 2x2x2 market in General

Competitive Equilibrium.

 The “supply chain” links the (2) Input factor markets for Capital and Labour hired by (2) Firms X and Y to the output market where (2) Consumers A and B purchase goods for consumption

 We have shown how competitive behaviour in these markets ensures that the general equilibrium will be Productively, Allocatively and Distributively Efficient.

Summary of Efficiency

Slide 86

 Distributive efficiency  Ensures goods go to the right consumers  We are on the contract curve; often means 𝑀𝑀𝑀𝑀𝑀𝑀𝐴𝐴 = 𝑀𝑀𝑀𝑀𝑀𝑀𝐵𝐵

 Productive efficiency  Ensures the right producers are producing those goods  We are on the aggregate PPF; often means 𝑀𝑀𝑀𝑀𝑀𝑀1 = 𝑀𝑀𝑀𝑀𝑀𝑀2

 Allocative efficiency  Ensures we are producing the right quantity of goods  We are on the right place on the PPF, and hence have the right

contract curve; often means 𝑀𝑀𝑀𝑀𝑀𝑀 = 𝑀𝑀𝑀𝑀𝑀𝑀

Summary  The connection between Pareto Efficiency and

Competitive General Equilibrium is summarized in the two welfare theorems:

 The First Welfare Theorem of Economics  Equilibrium in Competitive Markets is Pareto Optimal

 The Second Theorem of Welfare Economics  Any Pareto Optimal Allocation can be sustained as a

competitive equilibrium (if preferences are convex).

Summary  These welfare theorems are seen both as a “positive”

endorsement of Adam Smith’s notion of the “invisible” hand, and a “normative” endorsement of markets as a tool for the distribution of resources.

 As informative as these results are they also show how stringent the restrictions on “real markets” must be for them to hold. Perfect competition requires many things, some of which we explore further in the rest of this class.

What I Expect You to Know  To know when distributive, productive and allocative

efficiency are satisfied both mathematically and graphically

 To know what distributive efficiency, productive efficiency and allocative efficiency are not the same thing and to be able to identify all both mathematically and graphically

 Be able to explain in words why the condition for allocative efficiency link competitive market behaviour to Pareto Efficiency.

-- end of part 9 --