Budget Constraint

6 Key excerpts on "Budget Constraint"

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  eBook - ePub

  Alternative Principles of Economics

  • Stanley Bober(Author)
  • 2016(Publication Date)
  • Routledge

    (Publisher)

  Now that we have the foundation of this indifference–ordinal ranking approach, we move to arrive at the rule for utility maximization based on this, and then to derive the consumer demand curve. The construction of this curve is merely a reflection of a consumer adjusting to an external event so as to maintain one’s maximizing position, and the event is a change in the price of the good for which we want to derive the curve. Before the event occurs, the consumer has chosen a particular basket of goods that presumably mirrors the highest attainable level of utility, and the basic elements of this choice are obviously the individual’s set of preferences as represented by an indifference map, the prices of the individual goods composing the basket, and the individual’s money income. Before setting out the solution to this choice problem, it should be reiterated and emphasized that the solution here (and in the context of the marginal utility cardinal analysis as well) is based on what we can consider as “independence of preferences.” Each consumer is concerned only with one’s own consumption pattern; the utility or satisfaction derived from a particular purchase is attributable to the individual’s own experience. In the solution to the choice problem the consumer is presumed not to be concerned or indeed influenced by what is going on elsewhere, that is, by the choices being made by other consumers. In Principles texts the student is shown an aggregate or market demand curve adjacent to individual demand curves and is told that points on the aggregate curve are found by summing horizontally the individual demand curves. The individual consumers behind the individual curves are considered to be in their own box, making their own decisions, and not at all being influenced by the behavior of the others. Certainly in an understanding of consumer demand this is one perception that we will want to question. But let us get back to the mainstream indifference model.

  Figure 4.8

  The consumer faces a host of combinations of goods that one may want to purchase and that we see as being reflected in the indifference map. What one would like to do is but one half the problem. The other and more telling half is what the consumer can do at the time of making the choice. The essence of the consumption choice is to narrow down the choices from all possible choices to those that are economically attainable. And what is attainable is dependent on the constraints facing the consumer—those being the prices of the goods and one’s available income. Figure 4.9 illustrates such a Budget Constraint.

  Figure 4.9

  We set out the constraint as

  telling us that the amount of spending on goods A and B cannot exceed the consumer’s available income. The expenditure of less than one’s full income generates the purchase of sets of baskets that lie within the dotted-lines area under the constraint line. The use of one’s total available income (given prices) places one on the Budget Constraint line. This line evidences what is attainable (what the consumer can do), for it shows the set of combinations of A and B that can be purchased given the “objective” data of prices and income. The placement of the budget line is determined by finding the intercepts on the A and B axes. For example, if the consumer were to spend all income on good B , then the amount purchased would be given by

  M/pB

  ; likewise, if we were to set B = 0, we determine the good A budget intercept as

  M/pA .

  There are some features of this line that we should mention. An increase in money income with unchanged prices clearly shifts the line to the right, enlarging the available set of baskets. Clearly a change in the price of one of the goods, given unchanged money income, alters the slope of the constraint line as it changes one of the intercepts. Furthermore, if we were to double income and also double price, the available set of possible purchases would not change; the intercepts remain the same and the equation of the constraint line, of course, remains unchanged as well. So the set of commodity A and B baskets available to the consumer depends on relative prices (

  pB /pA

  ) and on the real value or purchasing power of money (

  M/pB , M/pA

  ). We should note that the slope of our constraint line is (−

  pA /pB

  ).6

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  eBook - ePub

  Intermediate Microeconomics: Neoclassical and Factually-oriented Models

  Neoclassical and Factually-oriented Models

  • Lester O. Bumas(Author)
  • 2015(Publication Date)
  • Routledge

    (Publisher)

  How much total utility an individual is able to attain is constrained by the money he has to spend during some specified period of time, by the individual’s budget. That budget is the person’s after-tax income which may be augmented by grants of money, borrowings, and past savings, and diminished by gifts, loans, and current savings. To equate budget and income assumes that inflows into income equal the outflows mentioned above.

    Assuming that the individual’s budget, B , is expended on n different goods and services during the budget period, the Budget Constraint is:

   

  where P i is the price of the i th good and Q i is the quantity of it purchased during the budget period.

  The Logic of the Condition-Defining Utility Maximization

  How does the individual spend his budget so that total utility is maximized? The condition which must prevail is that for all goods and services the incremental utility per dollar is equal:  

    At this equilibrium, no change in the combination of goods purchased can increase the satisfaction of the consumer.

    Let us transform the utility maximization condition from the equation (4) form of incremental utility per dollar to the more common one of marginal utility per dollar. First, recall that marginal utility is the slope of the total utility curve: MU = ΔTU /ΔQ . Solve for incremental utility and we have, ΔTU = MU · ΔQ . Now substitute for ΔTU in equation (4) MU · ΔQ .

   

  Set all ΔQs , all changes in the rate of consumption equal to one and we have:

   

  In this form it is seen that the utility maximizing consumer adjusts his purchases so that all items yield the same marginal utility per dollar.

  The Mathematics of Utility Maximization

  The logic of satisfaction being maximized when all marginal utilities per dollar are equal is handled mathematically here with the inclusion of the Budget Constraint. To simplify the problem it will be assumed that only two different goods are consumed and exhaust the individual’s budget. The utility function is:

   

  where X and Y

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  eBook - ePub

  Microeconomics For Dummies

  • Lynne Pepall, Peter Antonioni, Manzur Rashid(Authors)
  • 2016(Publication Date)
  • For Dummies

    (Publisher)

  the dual.

  What’s most important is the conclusion that you can derive from knowing something about utility (see Chapter 4 ) and something about the Budget Constraints and how the two have to be related. The relation between the two that is important in this case is that the highest level of utility possible for a constrained consumer occurs when the indifference curve is tangent to the Budget Constraint. Therefore the slope of the utility curve and the slope of the Budget Constraint are equal at that point.

  The slope of the indifference curve is the marginal rate of substitution (MRS), as in Chapter 4 , and the slope of the Budget Constraint is the relationship between the two prices (–p 1 /p 2 ). Given that these have to be equal at the optimal choice for the consumer, you know that the optimizing consumer’s best point occurs when

  You need to know this equation for Chapter 6 ’s full discussion on consumer optimization and for Chapter 9 , which looks into the famous supply and demand model.

  Passage contains an image Chapter 6

  Achieving the Optimum in Spite of Constraints

  In This Chapter

  Breaking down the effect of a change in prices into income and substitution effects

  How consumers’ preferences are revealed

  Comparing income and substitution effects

  The single most important part of microeconomics is the constrained optimization model, which is based on the idea that people act to achieve the best they can, given some kind of constraint that limits their choice. This way of looking at people’s decisions runs through most of the microeconomic syllabus, finding its way into all sorts of things from consumer choice to environmental or health economics. Yet its roots lie in the way economists look at individual decision-making.

  To microeconomists, people optimize.

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  eBook - ePub

  Microeconomics For Dummies - UK

  • Peter Antonioni, Manzur Rashid(Authors)
  • 2015(Publication Date)
  • For Dummies

    (Publisher)

  truncated in economics speak.

  To show this, cut a vertical line in above the maximum rationed consumption of good x 1 . To the left of the line, the budget set behaves as normal. To the right, where the maximum consumption is greater than the rationed amount – we call it r for the moment – the set consists of goods that the consumer could afford, but can’t get. We present this example in Figure 5-7 .

  Figure 5-7: Rationing truncates the Budget Constraint.

  Putting the Utility Model to Work

  In essence, a consumer can optimise in the classic choice model in two ways:

  • Utility maximisation approach: She can decide on a Budget Constraint and then find out how much utility she can get for it (as we discuss in the earlier section ‘Taking It to the Limit! Introducing the Budget Constraint ’).
  • Cost minimisation approach: She has a level of utility that she wants to achieve and minimises the cost that she has to pay to get it. As a microeconomist, you treat the amount of utility that she wants to get as fixed, and then work out which Budget Constraint is the lowest possible constraint that allows her to afford a bundle from that utility function.

  Ultimately, the two approaches work out to the same thing, and should arrive at the same answer, but from two different directions. Deciding which approach to use is a matter of computational efficiency – meaning you use the version that’s easiest to do in the time available – and which is more efficient depends on how much information you have about the constraint or the indifference curves. In microeconomics, the insight that they achieve the same answer is known as the dual.

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  Behavioural Economics and Experiments

  • Ananish Chaudhuri(Author)
  • 2021(Publication Date)
  • Routledge

    (Publisher)

  The point here, is that Ana is willing to give up some Y for X and vice versa. But, the inward bowed shape of the indifference curve suggests that this is not 1:1; in fact, the rate at which she is willing to trade off Y for X is not constant. It changes as she moves down the indifference curve from A to B. At A, or around it, she is consuming lots of Y and little X; so, she is willing to give up a rather large amount of Y to get a little more X. But at B, or around it, she is consuming lots of X and little Y; so, she is no longer willing to sacrifice much Y. If you still take away more Y from her, you will need to give her lots more X to compensate her.

  Ana knows what she can afford and how she should choose. What should she buy?

  Now, what should Ana consume? Of course, she wants to consume as much of X and Y as she can, but she is not really free to choose anything she wants. Her choices are constrained by what she can afford. This is the essence of the economic concept of constrained maximization ; that is, Ana must choose that consumption bundle which gives her maximum happiness, subject to the fact that this is something that she can actually afford, which in turn is decided by her Budget Constraint. Figure 1.17 illustrates Ana’s choice problem.

  Figure 1.17  

  Finding Ana’s most preferred consumption bundle out of those that are affordable

  Left to herself, Ana wants to pick a consumption bundle on the highest possible indifference curve; so, anything on I2 is better than anything on I1 , which, in turn, is better than I0 . But Ana is not free to choose anything she wants; she can only choose the best bundle she can afford. This means that she must choose a consumption bundle that lies on both the Budget Constraint and the highest possible indifference curve. By this reckoning, she cannot choose any bundle on I2 even though she would like to, since these bundles lie beyond her Budget Constraint and are, therefore, unaffordable. She can choose bundles on I0 , but she can do better. The best possible bundle that she can choose is the one that lies exactly at the point where the indifference curve I1

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  Economics for Investment Decision Makers

  Micro, Macro, and International Economics

  • Christopher D. Piros, Jerald E. Pinto(Authors)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  Exhibit 2-10 . The investor’s choice of a portfolio on the frontier will depend on her level of risk aversion.

  EXHIBIT 2-10 The Investment Opportunity Frontier

  Note: The investment opportunity frontier shows that as the investor chooses to invest a greater proportion of assets in the market portfolio, she can expect a higher return but also higher risk.

  5. CONSUMER EQUILIBRIUM: MAXIMIZING UTILITY SUBJECT TO THE Budget Constraint

  It would be wonderful if we could all consume as much of everything as we wanted, but unfortunately, most of us are constrained by income and prices. We now superimpose the Budget Constraint onto the preference map to model the actual choice of our consumer. This is a constrained (by the resources available to pay for consumption) optimization problem that every consumer must solve: choose the bundle of goods and services that gets us as high on our ranking as possible, while not exceeding our budget.

  5.1. Determining the Consumer’s Equilibrium Bundle of Goods

  In general, the consumer’s constrained optimization problem consists of maximizing utility, subject to the Budget Constraint. If, for simplicity, we assume there are only two goods, wine and bread, then the problem appears graphically as in Exhibit 2-11 .

  EXHIBIT 2-11 Consumer Equilibrium

  Note: Consumer equilibrium is achieved at point a , where the highest indifference curve is attained while not violating the Budget Constraint.

  The consumer desires to reach the indifference curve that is farthest from the origin while not violating the Budget Constraint. In this case, that pursuit ends at point a , where the consumer is purchasing

  Wa

  ounces of wine along with

  Ba

  slices of bread per month. It is important to note that this equilibrium point represents the tangency between the highest indifference curve and the Budget Constraint. At a tangency point, the two curves have the same slope, meaning that the MRSBW must be equal to the price ratio,

  PB

  /

  PW

  . Recall that the marginal rate of substitution is the rate at which the consumer is just willing to sacrifice wine for bread. Additionally, the price ratio is the rate at which the consumer must

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