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Samenvatting Topics in Public Economics
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PUBLIC ECONOMICS AND POLICY2. EQUILIBRIUM AND EFFICIENCY
2.1 Introduction
According to Adam Smith’s eighteenth century description of the working of the invisible hand there is a link between competition and
efficiency. Indeed, the way in which individuals take motivated decisions being coordinated to produce a socially efficient outcome has found
resonance in policy circles. The model of competition mixes independent decision making of consumers and firms into a complete model of
the economy. While, the equilibrium can be obtained in the economy by price adjusting to equate demand and supply for all goods. Moreover,
the equilibrium so achieved has properties of efficiency.
2.2 Economic Models
Economists apply models to make predictions about the effects of economic policies. They use models because is not easy to conduct
experiments on economic systems that are too large and complex to be fully analysed. Moreover, formal modelling ensures that arguments
are logically consistent with all the underlying assumptions exposed. Most models that deal with policy analysis specify the objectives of the
individual agents and the constraints they face. Then, they aggregate individual decisions to arrive at market demand and supply. Next, the
equilibrium is determined and the effects of government choice variables on this are calculated. When only a single market is studied the
analysis is called PARTIAL EQUILIBRIUM ANALYSIS, while at other times GENERAL EQUILIBRIUM ANALYSIS is used with many markets
analysed simultaneously. In addition, if the model is too highly specified, it may not be capable of capturing important forms of response.
While, if it is too general, it may not be able to provide any clear prediction.
2.3 Competitive Economies
The essential characteristic of competition is that the agents do not consider their actions to have any effect of prices. Consequently, they trat
the prices they observe in the market place as fixed or parametric. Prices measure values and are the signals that guide the decisions of firms
and consumers. The adjustment of prices equates supply and demand to ensure that equilibrium is achieved. Moreover, the role of prices in
coordinating the decisions of independent economic agents is also crucial for the attainment of economic efficiency. The second feature of
economies in this chapter is that all agents have access to the same information: information is symmetric. It does not imply that there is not
uncertainty, but only that when there is it all agents are equally uninformed: no agent is permitted to have an informational advantage.
2.4 The Exchange Economy
The exchange economy is the simplest form of economic activity and implies the trade of commodities between two parties in order to obtain
mutual advantage. Each of the two consumers has an initial stock or endowment of the economy’s two goods. The endowment can be
interpreted as “stocks of goods” or as “human capital” and are the quantities that are available for trade. Given the absence of production
these quantities remain constant. The consumers exchange quantities of the two commodities in order to achieve consumption plans that are
preferred to their initial endowments. The rate at which one commodity can be exchanged for the other is given by the market prices and both
consumers believe that their behaviour can not affect these prices. A consumer is described by their endowments and their preferences.
( )
The endowment of consumer h is denoted by:
h = 1h , 2h
Where wih > or = 0 is h’s initial stock of good i.
When prices are p1 and p2 , a consumption plan for consumer h is xh = (x1h, x2h) and is affordable if it satisfies the budget constraints that is
denoted by:
p1x1h + p2 x2h = p11h + p22h
The preferences of each consumers are described by their utility function. This function should be seen as a representation of the consumer’s
indifference curves and does not imply any comparability of utility levels between consumers.
The utility function for consumer h is denoted by:
Uh = Uh(x1h,x2h)
It is assumed that the consumers enjoy the goods (so the marginal utility of consumption is positive for both goods) and that the indifference
curves have the standard convex shape.
This economy can be pictured in a simple diagram constructed by the total consumption of the two consumers must equal the available stock
of the goods, where the stock is determined by the endowments. Any pair of consumption plans that satisfied this requirement is called a
feasible plan for the economy. The feasible plan can be denoted as:
x1i + xi2 = i1 + i2 , i = 1,2
The simple diagram that can be constructed is called Edgeworth box.
,Given the budget line determined by the prices, the utility-maximizing choices for the two consumers are characterized by the standard
tangency condition between the highest attainable indifference curve the budget line. At an equilibrium of the economy supply is equal to
demand and this is assumed to be achieved via the adjustment of prices. The priced at which supply is equal to demand are called equilibrium
prices.
2
•
x2 x1
•
•
1
The consumer choices in the figure do not constitute an equilibrium for the economy and it can be seen by summing the demands and
comparing these to the level of the endowments. Doing this shows that the demand for good 1 exceeds the endowment but the demand for
good 2 falls short. To achieve an equilibrium position, the relative prices of the goods must change: an increase in the relative price of good 1
raises the absolute value of the gradient -p1/p2 of the budget line, making the budget line steeper. It becomes flatter it the relative price of
good 1 falls. The effect of a relative price change on the budget constraint is shown in the following figure:
In this figure the price of good 1 has increased relative to the price of good 2: this causes the budget constraint to pivot upward around the
endowment point and as a consequence the consumers will now select consumption plans on this new budget constraint. Taking the prices as
given, the consumers choose their consumption plans to reach the highest attainable utility level subject to their budget constraints. Using the
demand functions, we see:
Xi1 (p1, p2) + xi2(p1,p2) = wi1 + wi2
An equilibrium is achieved when the prices lead (conducono) to a budget line on which the indifference curves of the consumers have a
point of common tangency:
2
x2
1•
x
•
1
The two equilibrium conditions ensures that there is only one independent equation. The single price ratio then has to be solved by a single
equation (and not 2), making it possible for there to be always a solution.
,Only the value of p1 relative to p2 matters in determining demands and supplies rather than the absolute values. The economic explanation for
this fact is that the consumers are only concerned with the real purchasing power embodied in their endowment and not only with the level
of prices. Since their nominal income is equal to the value of the endowment, any change in the level of prices raises nominal income just as
much it raises the cost of purchases. This leaves real incomes unchanged. The fact that only relative prices matter is also reflected in the
demand functions:
If xih(p1,p2) is the level of demand at price p1 and p2, then it must be the case that demand is homogenous of degree zero:
xih ( p1 , p2 ) = xih (p1 , p2 ), for 0
The homogeneity shows that only relative prices can be determined at equilibrium and not the level of prices. So, given a set of equilibrium
prices, any scaling up or down of these prices will also be equilibrium prices because the change will not alter the level of demand. In order to
analyse the model, the indeterminacy of the level of prices needs to be removed and it can be possible by adopting a price normalization,
which is a simple method of fixing a scale for prices. The simplest way consist in to select a commodity as numéraire. It means that the price is
fixed at one and measure all other prices relative to this. The numéraire chosen in this wat can be considered the unit of account for the
economy (that is the role usually played by money). Then, we have to demonstrate the dependence between the two equilibrium equations.
Considering other budget lines and indifference curves in the Edgeworth box will show that whenever there is an excess of demand for one
good, there is a corresponding deficit of demand for the other. The level of excess demand for good I is the difference between demand and
supply and is defined by Zi = xi1 + xi2 – wi1 + wi2. By this definition, the value of excess demand can be calculated as:
p1Z1 + p2 Z 2 = p1 x11 + x12 − 11 − 12 + p2 x12 + x22 − 21 − 22 =0
Where the second equality is a consequence of the budget constraint. The relationship in the formula is known as Walras’s Law and states
that the value of excess demand is 0. Walras’s Law is simply an aggregate budget constraint for the economy since all consumers are equating
their expenditure to their income. This Law implies also that the equilibrium equations are interdependent since p1z1 + p2z2 = 0, then z2 = o and
vice versa. That is, if demand is equal to supply for good 1, then demand must also equal supply for good 2. Equilibrium in one market
necessarily implies equilibrium in the other. Moreover, income (in terms of an endowment of many goods) and expenditure (defined in the
same way) must remain equal for each consumer. The demand functions that results from the maximization of utility are homogeneous of
degree zero in prices. Walras’s Law continues to hold, so the value of excess demand remains zero. The number of price ratios and the number
of independent equilibrium conditions are always one less than the number of goods.
2.5 Production and Exchange
Adding production to the exchange model we obtain a complete model of economic activity. Some goods can be present as initial
endowments, e.g. labour, other can be consumption goods produced from the initial endowments, others can be “intermediates” and they can
be produced by one production process and used as input to another. The fully developed model of competition is called the Arrow-Debreu
Economy. An economy with production consists of consumers, or households, and producers, or firms. The firms use inputs to produce
outputs with the intention of maximizing their profits. Each firm has available a production technology that describes the ways in which it can
use inputs to produce outputs. While, the consumers have preferences and initial endowments - as they did in the exchange economy – but
they now also hold shares in the firms. The firms’ profits are distributed as dividends in proportion to the shareholdings. The consumers
receive income from the sale of their initial endowments (e.g. labour time) and from the dividend payments. Each firm is characterized by its
production set, which summarizes the production technology it has available. A production technology can be described as a complete list of
ways that the firm can turn inputs into outputs.
TYPICAL PRODUCTION SET:
This figure employs the standard convention of measuring inputs as negative numbers and output as positive. The reason for adopting this
convention is that the use of a unit of a good as an input represents a SUBTRACTION from the stock of that goof available for consumption.
Considering the figure above and choosing the production plan y1j = -2, y2j = 3. When faced with prices p1 = 2 and p2= 2, the firm’s profit is:
πj = p1y1j + p2y2j = 2 x (-2) + 2 x 3 = 2
Where the negative part is the interpretation of production costs, while the positive part is the interpretation of sales revenue. This is
equivalent to writing profit as the difference between revenue and cost.
PROFIT MAXIMIZATION:
, Under the competitive assumption the firm take prices as given. They are used to construct iso profit curves, which show all production plans
that give a specific level of profit. Since doing nothing (y1j = y2j = 0) earns zero profit, the π = 0 iso profit curve always passes through the origin.
The profit-maximizing firm will choose a production plan that places it upon the highest attainable iso profit curve. What restricts the choice is
the technology that is available as described by the production set. The production plan that maximises the profit is shown by y*, which is
located at a point of tangency between the highest attainable iso profit curve and the production set (there is no other technologically
feasible plan that can attain higher profit). It should be noted how the iso profit curves are determined by the prices. Indeed, the geometry is
that the iso profit curves are at right angles to the price vector. The angle of price vector is determined by the price ratio (p2/p1), so a change in
relative prices will alter the gradient of the iso profit curves. For instance, if p1 increases relative to p2 the price vector will become flatter. This
makes the iso profit curve steeper and the optimal choice must move round the boundary of the production set toward the origin.
Aggregate supply from the production sector of economy is obtained from the supply decisions of the individual firms by summing across the
firms. While, the excess demand is the difference between demand and the sum of initial endowment and firms’ supply.
2.6 Efficiency of Competition
Economics is often defined as the study of scarcity. In the case in which we consider only an individual decision maker: efficiency is achieved
when more can not be obtained so the utility is maximized. This concept become more complicated to explain when there is more than one
decision maker because it is necessary to solve the potentially competing needs of different decision makers.
SOME DEFINITIONS:
• First-best outcome: a first-based outcome is achieved when only the production and the limited endowments restrict the choice of
the decision maker. The first-best is essentially what would be chosen by an omniscient planner with complete command over
resources. It refers to the best you could do if you knew agents’ preferences when you did not have to impose the incentive
compatibility constraint. A first-best equilibrium occurs in a perfectly competitive market when no imperfections or distortions are
present.
• Second-best outcome: a second-best outcome arises whenever constraints other than technology and resources are placed on what
the planner can do. Such constraints could be limits on income redistribution, an inability to remove monopoly power or lack of
information. A second-best equilibrium arises whenever a market includes one or more imperfections or distortions.
2. 6.1 Single Consumer
With a single consumer there is no doubt as what is good and bad from a social perspective: the single individual’s preferences can be taken as
the social preferences in a “Robin Crusoe Economy”. In this case, the “best” outcome must be first-best because there are no constraints on
policy choices and there is no an issue of income distribution to consider. In this case the budget constrain is always at a right angle to the
price vector and is displaced above the origin by the value profit. Utility maximization occurs when the highest indifference curve is reached
given the budget constraint. At the equilibrium, the net consumption plan from the consumer must match the supply for the firm. A simple
characterization of the first-best allocation can be given by using the fact that the gradient of the indifference curve is equal to the ratio of the
marginal utilities of the two goods and is called the marginal rate of substitution. It measures the rate at which good 1 can be traded for good
2 while maintaining constant utility.
The marginal rate of substitution is given by:
MRS1,2 = U1 / U2
Similarly, the gradient of the production possibility set is defined the marginal rate of transformation (MRT1,2). The last one measures the rate
at which good 1 has to be given up to allow an increase in production of good 2. At the tangency point the two gradient are equal, so:
MRS1,2 = MRT1,2
The reason why this equality characterizes the first-best equilibrium is explained by the fact that MRS captures the marginal value of good 1 to
the consumer relative to the marginal value of good 2, while the MRT measures the marginal cost of good 2 relative to the marginal cost of
good 2. The first-best is achieved when the marginal value is equal to the marginal cost.
The market achieved efficiency through the coordinating role of prices. The optimal choice occurs when the budget constraint is tangential to
the highest attainable indifference curve. The condition describing this is that ratio of marginal utilities is qual to the ratio of prices:
MRS1,2 = p1/p2
Similarly, profit maximization by the firms occurs when the production possibility set is tangential to the highest iso profit curve:
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