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Chapter 7
Competitive Markets and Partial
Equilibrium Analysis
Up until now we have concentrated our efforts on two major topics - consumer theory, which led to the theory of demand, and producer theory, which led to the theory of supply. Next, we will put these two parts together into a market. Specifically, we will begin with competitive markets. The key feature of a competitive market is that producers and consumers are considered price takers. That is, individual actors can buy or sell as much of the output as they want at the market price, but no one can take any unilateral action to affect the price. If this is the case, then the actors take prices as exogenous when making their decisions, which was a key feature in our analysis of consumer and producer behavior. Later, when we study monopoly and oligopoly, we will relax the assumption that firms cannot affect prices. Our main goal here will be to determine how supply and demand interact to determine the way the market allocates society’s resources. In particular, we will be concerned with:
- When does the market allocate resources efficiently?
- When, if the government wants to implement a specific allocation, can the allocation can be implemented using the market (possibly by rearranging people’s initial endowments)?
- Why does the market sometimes fail to allocate resources efficiently, and what can be done in such cases? The third question will be the subject of the next chapter, on externalities and public goods. For now, we focus on the first and second questions, which bring us to the first and second fundamental
theorems of welfare economics, respectively.
7.1 Competitive Equilibrium
The basic idea in the analysis of competitive equilibrium is the “law of supply and demand.” Utility maximization by individual consumers determines individual demand. Summing over individual consumers determines aggregate demand, and the aggregate demand curve slopes downward. Profit maximization by individual firms determines individual supply, and summing over firms determines aggregate supply, which slopes upward. Adam Smith’s invisible hand acts to bring the market to the point where the two curves cross, i.e. supply equals demand. This point is known as a competitive equilibrium, and it tells us how much of the output will be produced and the price that will be charged for it.
Notation
We are going to be dealing with many consumers, many producers, and many commodities. To make things clear, I’ll denote which consumer or producer we are talking about with a superscript. For example, ui^ is the utility function of consumer i, xi^ is the commodity bundle chosen by consumer i, and yj^ is the production plan chosen by firm j. For vectors xi^ and yj^ , I’ll denote the lth^ component with a subscript. Hence xi^ = ¡xi 1 , ..., xiL^ ¢, and yj^ =
xj 1 , ..., xjL
. So xjL refers to consumer j’s consumption of good L. This differs from MWG, which uses double subscripts. But, I think that this is clearer.
7.1.1 Allocations and Pareto Optimality
Our formal analysis of competitive markets begins with defining an allocation, and determining what we mean when we say that an allocation is efficient. Consider an economy consisting of:
- I consumers each with utility function ui
- J firms each maximizing its profit
- L commodities.
u 1
u 2
Figure 7.1: Utility Possibility Frontier
all commodities will in the end find their way into the hands of consumers. A profit-maximizing firm will never buy inputs it doesn’t use or produce output it doesn’t sell, and firms are owned by consumers, so profit eventually becomes consumer wealth. Thus, looking at the utility of consumers fully captures the notion of efficiency. If you draw the utility possibility frontier in two dimensions, as in Figure 7.1, Pareto optimal points are ones that lay on the northeast frontier. Note that Pareto optimality doesn’t say anything about equity. An allocation that gives one person everything and the other nothing may be Pareto optimal. However, it is not at all equitable. Much of the job in policy making is in striking a balance between equity and efficiency — to put it another way, choosing the equitable point from among the efficient points.
7.1.2 Competitive Equilibria
We now turn to investigating competitive equilibria with the goal of determining whether or not the allocations determined by the market will be Pareto optimal. Again, we are concerned with competitive markets. Thus buyers and sellers are price takers in the L commodities. Further, we make the assumption that the firms in the market are owned by the consumers. Thus all profits from operation of the firms are redistributed back to the consumers. Consumers can then use this wealth to increase their consumption. In this way we “close” the model - it’s entirely self contained. Although our formal analysis will be of a partial equilibrium system, where we study only one or two markets, we will define an competitive equilibrium over all L commodities. In a competitive
economy, a market exists for each of the L goods, and all consumer and producers act as price takers. As usual, we’ll let the vector of the L commodity prices be given by p, and suppose consumer i has endowment wil of good i. We’ll denote a consumer’s entire endowment vector by wi, and the total endowment of the good is given by P i wil = wl. We formalize the fact that consumers own the firms by letting θij (0 ≤ θij ≤ 1) be the share of firm j that is owned by consumer i. Thus if firm j chooses production plan yj^ , the profit earned by firm j is πj = p · yj^ , and consumer i’s share of this profit is given by θij^ ¡p · yj^ ¢^. Consequently, consumer i’s total wealth is given by p·wi +P^ θij πj. Note that this means that all wealth is either in the form of endowment or firm share; there is no longer any exogenous wealth w. Of course, this depends on firms’ decisions, but part of the idea of the equilibrium is that production, consumption, and prices will all be simultaneously determined. We now turn to the formal description of a competitive equilibrium. There are three requirements for a competitive equilibrium, corresponding to the requirements that producers optimize, consumers optimize, and that “markets clear” at the equilibrium prices. An equilibrium will then consist of a production plan yj∗^ for each^ firm, a consumption vector^ xi∗ for each consumer, and a price vector p∗. Actually, the producer and consumer parts are just what we have been studying for the first half of the course. The market clearing condition says that at the equilibrium price, it must be that the aggregate supply of each commodity equals the aggregate demand for that commodity, when producers and consumers optimize. Formally, these requirements are:
- Profit Maximization: For each firm, yj^ (p) solves
max pyj^ subject to yj^ ∈ Yj.
- Utility Maximization: For each consumer, xi^ (p) solves
max ui^ ¡xi¢^ subject to p · xi^ ≤ p · wi^ +
X
θij^ ¡p · yj^ (p)¢^ ,
where θij is consumer i’s ownership share in firm j. Note: this is just the normal UMP with the addition of the idea that the consumer has ultimate claim on the profit of the firm. 2 (^2) There is something a little strange here. Note that we won’t know the firm’s profit until after the price vector is determined. But, if we don’t know the firm’s profit, we can’t derive consumers’ demand functions, and so we can’t
This lemma is a direct consequence of the idea that total wealth must be preserved in the economy. The nice thing about it is that when you are only studying two markets, as we do in the partial equilibrium approach, you know that if one market clears, the other must clear as well. Hence the study of two markets really reduces to the study of one market.
7.2 Partial Equilibrium Analysis
7.2.1 Set-Up of the Quasilinear Model
We now turn away from the general model to a simple case, known as Partial Equilibrium. It is ‘partial’ because we focus on a small part of the total economy, often on a two commodity world. We laid the groundwork for this type of approach in our discussion of consumer theory. If we are interested in studying a particular market, say the market for apples, we can make the assumption that the prices of all other commodities move in tandem. This justifies, through use of the composite commodity theorem, treating consumers as if they have preferences over apples and “everything else.” Hence we have justified a two-commodity model for this situation. Next, since each consumer’s expenditure on apples is likely to be only a small part of her total wealth, it is reasonable to think of there being no wealth effects on consumers’ demand for apples. And, recall, that quasilinear preferences correspond to the case where there are no wealth effects in the non-numeraire good. So, basically what we’ll do in our partial equilibrium approach (and what is implicitly underlying the approach you took in intermediate micro) is assume that there are two goods: a composite commodity (the numeraire) whose price is set equal to 1, and the good of interest. We’ll call the numeraire m (for “money”) and the good we are interested in x. Now, we can set up the following simple model. Let xi and mi be consumer i’s consumption of the commodity of interest and the numeraire commodity, respectively.^3 Assume that each consumer has quasilinear utility of the form:
ui^ (mi, xi) = mi + φi (xi).
Further, we normalize ui^ (0, 0) = φi (0) = 0, and assume that φ^0 i > 0 and φ^00 i < 0 for all xi ≥ 0. That is, we assume that the consumer’s utility is increasing in the consumption of x and that her marginal utility of consumption is decreasing. (^3) This is a change in notation from the set-up at the beginning of the chapter. Now, the subscripts refer to the consumer / firm, rather than the commodity.
Since we already set the price of m equal to 1 , we only need to worry about the price of x. Denote it by p. There are J firms in the economy. Each firm can transform m into x according to cost function cj (qj ), where qj is the quantity of x that firm j produces, and cj (qj ) is the number of units of the numeraire commodity needed to produce qj units of x. Thus, letting zj denote firm j’s use of good m as an input, its technology set is therefore
Yj = {(−zj , qj ) |qj ≥ 0 and zj ≥ cj (qj )}.
That is, you have to spend enough of good m to produce qj units of x. We will assume that cj (qj ) is strictly increasing and convex for all j. In order to solve the model, we also need to specify consumers’ initial endowments. We assume there is no initial endowment of x, but that consumer i has endowment of m equal to wmi > 0 and the total endowment is P i wmi = wm.
7.2.2 Analysis of the Quasilinear Model
That completes the set-up of the model. The next step is to analyze it. Recall that in order to find an equilibrium, we need to derive the firms’ supply functions, the consumers’ demand functions, and find the market-clearing price.
- Profit maximization. Given the equilibrium price p∗, firm j’s equilibrium output q j∗ must maximize max qj pqj − cj (qj ) which has the necessary and sufficient first-order condition
p∗^ ≤ c^0 j^ ¡q j∗^ ¢
with equality if q j∗ > 0.
- Utility Maximization: Consumer i’s equilibrium consumption vector (x∗ i , m∗ i ) must max- imize
max mi + φi (xi) s.t. mi + p∗xi ≤ wmi +
X
θij^ ¡p∗q∗ j − cj^ ¡q j∗^ ¢¢
- For each consumer, derive their Walrasian demand for the consumption good, xi (p). Add across consumers to derive the aggregate demand, x (p) = P i xi (p). Since each demand curve is downward sloping, the aggregate demand curve will be downward sloping. Graphically, this addition is done by adding the demand curves “horizontally” (as in MWG Figure 10.C.1). Since individual demand curves are defined by the relation:
p = φ^0 i (q) , The price at which each individual’s demand curve intersects the vertical axis is φ^0 i (0), and gives that individual’s marginal willingness to pay for the first unit of output. The intercept for the aggregate demand curve is therefore maxiφ^0 i (0). Hence if different consumers have different φi () functions, not all demand curves will have the same intercept, and the demand curve will become flatter as price decreases.
- For each firm, derive the supply curve for the consumption good, yj (p). Add across firms to derive the aggregate supply, y (p) = P j yj (p). For each firm, the supply curve is given by:
p = c^0 j (qj ). Thus each firm’s supply curve is the inverse of its marginal cost curve. Since we have assumed that c^00 j () ≥ 0 , the supply curve will be upward sloping or flat. Again, addition is done by adding the supply curves horizontally, as in MWG Figure 10.C.2. The intercept of the aggregate supply curve will be the smallest c^0 j (0). If firms’ cost functions are strictly convex, aggregate supply will be upward sloping.
- Find the price where supply equals demand: find p∗^ such that x (p∗) = y (p∗). Since the market clears for good l, it must also clear for the numeraire. The equilibrium point will be at the price and quantity where the supply and demand curves cross.
At this point, we can talk a bit about the dynamics of how an equilibrium might come about. This is the story that is frequently told in intermediate micro courses, and I should point out that it is just a story. There is nothing in the model which justifies this approach since we have said nothing at all about how markets will behave if they are out of equilibrium. Nevertheless, I’ll tell the story. Suppose p∗^ is the equilibrium price of x, but that currently the price is equal to p+^ > p∗. At this price, aggregate demand is less than aggregate supply: D (p∗) < S (p∗). Because of this, there
is a “glut” on the market. There are more units of x available for sale than people willing to buy (think about cars sitting on a car lot at the end of the model year). Hence (so the story goes) there will be downward pressure on the price as suppliers lower their price in order to induce people to buy. As the price declines, supply decreases and demand increases until we reach equilibrium at p = p∗. Similarly, if initially the price p−^ is such that p−^ < p∗, then D (p−) > S (p−). There is excess demand (think about the hot toy of the holiday season). The excess demand bids up the price as people fight to get one of the scarce units of x, and as the price rises, supply increases and demand decreases until equilibrium is reached, once again, at p = p∗. Thus we have the “invisible hand” of the market working to bring the market into equilibrium. However, let me emphasize once again that stories such as these are not part of our model.
7.2.3 A Bit on Social Cost and Benefit
The firm’s supply function is yj (p) and satisfies p = c^0 j (yj (p)). Thus at any particular price, firms choose their quantities so that the marginal cost of producing an additional unit of produc- tion is exactly equal to the price. Similarly, the consumer’s demand function is xi (p) such that p = φ^0 i (xi (p)). Thus at any price, consumers choose quantities so that the marginal benefit of consuming an additional unit of x is exactly equal to its price.^4 When both firms and consumers do this, we get that, at equilibrium, the marginal cost of producing an additional unit of x is exactly equal to the marginal utility of consuming an additional unit of x. This is true both individually and in the aggregate. Thus at the equilibrium price, all units where the marginal social cost is less than or equal to the marginal social benefit are produced and consumed, and no other units are. Thus the market acts to produce an efficient allocation. We’ll see more about this in a little while, but I wanted to suggest where we are going before we take a moment to talk about a few other things.
7.2.4 Comparative Statics
As usual, one of the things we will be interested in determining in the partial equilibrium model is how the endogenous parameters of the model vary with changes in the environment. For example, suppose that a consumer’s utility function depends on a vector of exogenous parameters, φi (xi, α), (^4) This is true as long as we assume that for any level of output, consumers with the highest willingness to pay (i.e., marginal benefit) are the ones that are given the units of output to consume, which is a reasonable assumption in many circumstances.
Qt^ is the quantity sold after the tax is implemented. At this quantity, the difference between the price paid by consumers and the price received by firms is exactly equal to the tax. Note that at Qt, the marginal social cost of an additional unit of output is less than the marginal social benefit. Hence society could be made better off if additional units of output were produced and sold, all the way up to the point where Q∗^ units of output are produced. The loss suffered by society due to the fact that these units are not produced and consumed is called the deadweight loss (DWL) of taxation. One question we may be interested in is how the price paid by consumers changes when the size of the tax increases. Let p∗^ (t) be the equilibrium price received by firms. Thus consumers pay p∗^ (t) + t. The following identity holds for any tax rate t :
x (p∗^ (t) + t) ≡ q (p∗^ (t)).
Totally differentiating with respect to t yields:
x^0 (p∗^ + t) ¡p^0 (t) + 1¢^ = q^0 (p∗) p^0 (t) p^0 (t) = −x
(^0) (p∗ (^) + t) x^0 (p∗^ + t) − q^0 (p∗)
The numerator is positive by definition. Since q^0 (p∗) is positive, the absolute value of the denom- inator is larger than the absolute value of the numerator, hence − 1 < p^0 (t) < 0. This implies that as the tax rate increases, the price received by firms decreases, but by less than the full amount of the tax. As a consequence, the price paid by consumers must also increase, but by less than the increase in the tax. Further, the total quantity must decrease as well. Consider the formula we just derived:
p^0 (t) = −x
(^0) (p∗ (^) + t) x^0 (p∗^ + t) − q^0 (p∗).
Evaluate at t = 0, and rewrite this in terms of elasticities:
dp dt =^
− dxdppx dxdppx − dqdppq = − (^) |εd||ε +d| εs
where εd is the elasticity of demand and εs is the elasticity of supply. This says that the proportion of a small tax that is passed onto producers in the form of lower prices is given by (^) |εd|ε|+d|εs. The proportion passed onto consumers in the form of higher prices is given by (^) |εdε|+sεs.
Now, suppose the government is considering two different tax programs: one that taxes a commodity with relatively inelastic demand, and one that taxes a commodity with relatively elastic demand. Which will result in the larger deadweight loss? The answer is, all else being equal, the commodity with the more elastic demand will have a larger deadweight loss. Why? The more elastic demand is, the flatter the demand curve will be. This means that a more elastic demand curve will respond to a tax with a relatively larger decrease in quantity. And, since this quantity distortion is the source of the deadweight loss, the more elastic demand curve will result in the larger deadweight loss. Does this mean that we should only tax inelastic things, since this will make society better off? Not really. The main reason is that even though taxing inelastic things may be better for society as a whole from an efficiency standpoint, it may have undesirable redistributive effects. For example, we could tax cigarette smokers and force them to pay for road construction and schools. Since cigarette demand is relatively inelastic, this would result in a relatively small deadweight loss. However, is it really fair to force smokers to pay for roads and schools, even though they don’t necessarily use the roads and schools any more intensely than other people? Probably not. We recently had a related issue in Massachusetts. The state wanted to increase tolls on the turnpike in order to pay for construction at the airport. Turnpike usage is relatively inelastic, but is it fair to make turnpike users pay for airport construction, even though turnpike users are no more likely to be going to the airport than other drivers? Issues of balancing efficiency and equity concerns such as this arise often in policy decisions.
7.3 The Fundamental Welfare Theorems
Recall that we ended our discussion of production by talking about efficiency, and showed that any profit-maximizing production plan is efficient (i.e. the same output cannot be produced using fewer inputs), and that any efficient production plan (under certain circumstances) is the profit maximizing production plan for some price vector. We now turn to ask the same questions of markets. That is, when are the allocations made by markets “efficient,” and is every efficient allocation the market allocation for some initial conditions? Again, the reason we ask these questions has to do with decentralization. When can we decentralize the decisions we make in our society? Do we know that profit-maximizing firms and utility maximizing agents will arrive at a Pareto optimal allocation through the market? If we have a particular Pareto optimal allocation
The top line just says to maximize the sum of the consumers’ utilities. The constraint is that the total consumption of x is the same as the total production. Letting the Lagrange multiplier be μ, the first-order conditions for this problem are:
φ^0 i (x∗ i ) ≤ μ with equality if x∗ i > 0 c^0 j^ ¡q j∗^ ¢^ ≥ μ with equality if q j∗ > 0 X^ I i=
x∗ i =
X^ J
j=
q j∗ Note that these are exactly the conditions as the conditions defining the competitive equilibrium except that p∗^ has been replaced by μ. In other words, we know that the allocation produced by the competitive market satisfies these conditions, and that μ = p∗. Thus the competitive market allocation is Pareto optimal, and the market clearing price p∗^ is the shadow value of the constraint: the additional social benefit generated by consuming one more unit of output or producing one less unit of output. Hence this is just another expression of the fact that at p∗^ the marginal social benefit of additional output equals the marginal social cost. The preceding argument establishes the first fundamental theorem of welfare economics in the partial equilibrium case. If the price p∗^ and the allocation (x∗, q∗) constitute a competitive equilibrium, then this allocation is Pareto optimal. The first theorem is just a formal expression of Adam Smith’s invisible hand — the market acts to allocate commodities in a Pareto optimal manner. Since p∗^ = μ, which is the shadow price of additional units of x, each firm acting in order to maximize its own profits chooses the output that equates the marginal cost of its production to the marginal social benefit, and each consumer, in choosing the quantity to consume in order to maximize utility, is also setting marginal benefit equal to the marginal social cost. Note that while this is a special case, the first welfare theorem will hold quite generally whenever there are complete markets, no matter how many commodities there are. It will fail, however, when there are commodities (things that affect utility) that have no markets (as in the externalities problem we’ll look at soon). As we did with production, we can also look at this problem “backward.” Can any Pareto optimal allocation be generated as the outcome of a competitive market, for some suitable initial endowment vector? The answer to this question is yes. To see why, recall that when all φi ()’s are strictly concave and all cj ()’s are strictly convex, there is a unique allocation of the consumption commodity x that maximizes the sum of the consumers’
utilities. The set of Pareto optimal allocations is derived by allocating the consumption commodity in this manner and varying the amount of the numeraire commodity given to each of the consumers. Thus the set of Pareto optimal allocations is a line with normal vector (1,1,1...,1) (see, for example, Figure 10.D.1 in MWG), since one unit of utility can be transferred from one consumer to another by transferring a unit of the numeraire. Thus any Pareto optimal allocation can be generated by letting the market work and then ap- propriately transferring the numeraire. But, recall that firms’ production decisions and consumers’ consumption decisions do not depend on the initial endowment of the numeraire. Because of this, we could also perform these transfers before the market works. This allows us to implement any point along the Pareto frontier. To see why, let (x∗, q∗) be the Pareto optimal allocation of the consumption commodity, and suppose we want to implement the point where each consumer gets (x∗ i , m∗ i ) after the market works, where P i m∗ i = wm − P j cj
q j∗
. If we want consumer i to have m∗ i units of the numeraire after the transfer, we need him to have m^0 i before the transfer, where
m^0 i +
X
θij^ ¡p∗q∗ j − cj^ ¡q j∗^ ¢¢^ = m∗ i + p∗x∗ i.
Hence if people have wealth m^0 i before the market starts to work, allocation (x∗, q∗, m∗) will result. This yields the second fundamental theorem of welfare economics. Let u∗ i be the utility in a Pareto optimal allocation for some initial endowment vector. There exists a set of transfers Ti (the amount of the numeraire given to consumer i) such that P i Ti = 0 and the allocation generated by the competitive market yields the utility vector u∗. The transfers are given by Ti = m^0 i − wmi.
7.3.1 Welfare Analysis and Partial Equilibrium
Recall in our discussion of consumer theory we said that equivalent variation is the proper measure of the impact of a policy change on consumers, and that EV is given by the area to the left of the Hicksian demand curve between the initial and final prices. However, since there are no wealth effects for the consumption good here, we know that the Hicksian and Walrasian demand curves are the same. So, the area to the left of the Walrasian demand curve is a proper measure of consumer welfare. Further, since utility is quasilinear, it makes sense to look at aggregate demand, and there is a normative representative consumer whose preferences are captured by the aggregate demand curve. Hence the area to the left of the Walrasian demand curve is a good measure of changes in
- (a) Some surplus is gained by firms in the form of revenue, P j pqj (b) Some surplus is lost by firms in the form of production cost, P j cj (qj ) - P j pqj − cj (qj ) is the aggregate producer surplus, which is then redistributed to con- sumers in the form of dividends, θij (pqj − cj (qj )).
So, consumers receive part of the benefit through consumption of the non-numeraire good, X i
φi (xi) − pxi
and part of the benefit through consumption of the dividends, which are measured in units of the numeraire: X j
pqj − cj (qj ).
Aggregate surplus is found by adding these two together, and noting that P i pxi = P j pqj In our quasilinear model, the set of utility vectors that can be achieved by a feasible allocation is given by: (^) ⎧ ⎨ ⎩(u^1 , ..., uI^ )^ |^
X
i
ui ≤ wm +
ÃX
i
φi (xi)
⎝X
j
cj (qj )
Suppose that the government (or you, or anybody) has a view of society that says the total welfare in society is given by: W (u 1 , ..., uI )
Thus this function gives a level of welfare associated with any utility vector — it allows us to compare any two distributions of utility in terms of their overall social welfare. The problem of the social planner would be to choose u in order to maximize W (u) subject to the constraint that u lies in the utility possibility set. Clearly, then, the optimized level of W will be higher when the utility possibility set is larger, which occurs when the MAS is larger. Thus the total societal welfare achievable is increasing in the MAS. See MWG Figure 10.E.1. In other words, if you want to maximize social welfare, you should first choose the production and consumption vectors that maximize MAS, and then redistribute the numeraire in order to maximize the welfare function. This gives us another separation result: If utility can be perfectly transferred between consumers (as in the quasilinear model), then social welfare is maximized by first choosing production and consumption plans that maximize MAS, and then choosing transfers such that ui = φi (xi) + mi + ti maximizes W (u).
Now, how does MAS change when the quantity produced and consumed changes? Let S (x, q) be the MAS, formally defined as follows:
S (x, q) =
ÃX
i
φi (xi)
⎝X
j
cj (qj )
Consider a differential increase in consumption and production: (dx 1 , dx 2 , ..., dxI , dq 1 , ...dqJ ) sat- isfying P i dxi = P j dqj. Note that under such a change, we increase total production and total consumption by the same amount. The differential in S is given by
dS =
ÃX
i
φ^0 i (xi) dxi
⎝X
j
c^0 j (qj ) dqj
Since consumers maximize utility, φ^0 i (xi) = p (x) for all i, and since producers maximize profit, c^0 j (qj ) = c^0 (q) for all j. Thus:
dS =
Ã
p (x)
X
i
dxi
⎝c^0 (q) X j
dqj
which by definition of our changes (and market clearing) implies
dS = ¡p (x) − c^0 (q)¢^ dx
And, integrating this from 0 to x¯ yields that MAS equals
S (x) =
Z (^) x 0
¡p (s) − c (^0) (s)¢ (^) ds.
Thus the total surplus is the area between the supply and demand curves between 0 and the quantity sold, x.¯
Example: Welfare Effects of a Tax
We return to the idea of a commodity tax that we first considered in the context of consumer theory. Suppose there is a government that attempts to maximize the welfare of its citizens. The government keeps a balanced budget, and tax revenues are returned to consumers in the form of a lump sum transfer. What are the welfare effects of this tax? Define x∗ 1 (t) , ..., x∗ I (t) and q∗ 1 (t) , ..., q∗ J (t) and p∗^ (t), respectively, to be the consumptions, productions, and price paid by consumers when the per-unit
