Utility Maximisation ProblemSimon Board∗This Version: September 20, 2009First Version: October, 2008.The utility maximisation problem (UMP) considers an agent with income m who wishes tomaximise her utility. Among others, we are interested in the following questions:• How do we determine an agent’s optimal bundle of goods?• How do we derive an agent’s demand curve for a particular good?• What is the effect of an increase in income on an agent’s consumption?1 ModelWe make several assumptions:1. There are N goods. For much of the analysis we assume N = 2, but nothing depends onthis.2. The agent takes prices as exogenous. We normally assume prices are linear and denotethem by {p1, . . . , pN}.3. Preferences satisfy completeness, transitivity and continuity. As a result, a utility func-tion exists. We normally assume preferences also satisfy monotonicity (so indifferencecurves are well behaved) and convexity (so the optima can be characterised by tangencyconditions).∗Department of Economics, UCLA. http://www.econ.ucla.edu/sboard/. Please email suggestions and typosto [email protected], Fall 2009 Simon Board4. The consumer is endowed with income m.The utility maximisation problem is:maxx1,...,xNu(x1, . . . , xN) subject toNXi=1pixi≤ m (1.1)xi≥ 0 for all iThe idea is that the agent is trying to spend her income in order to maximise her utility. Thesolution to this problem is called the Marshallian demand or uncompensated demand. It isdenoted byx∗i(p1, . . . , pN, m)The most utility the agent can attain is given by her indirect utility function. It is definedbyv(p1, . . . , pN, m) = maxx1,...,xNu(x1, . . . , xN) subject toNXi=1pixi≤ m (1.2)xi≥ 0 for all iEquivalently, the indirect utility function equals the utility the agent gains from her optimalbundle,v(p1, . . . , pN, m) = u(x∗1, . . . , x∗N).1.1 Example: One GoodTo illustrate the problem, suppose N = 1. For example, the agent has income m and is choosinghow many cookies to consume. The agent’s utilities are given by table 1.In general, we solve the problem in two steps. First, we determine which bundles of goods areaffordable. The collection of these bundles is called the budget set. Second, we find whichbundle in the budget set the agent most prefers. That is, which bundle gives the agent mostutility.Suppose the price of the good is p1= 1 and the agent has income m = 4. Then the agent canafford up to 4 units of x1. Given this budget set, the agent’s utility is maximised by choosingx∗1= 4, yielding utility v = 28.2Eco11, Fall 2009 Simon BoardUnits of x1Utility1 102 183 244 285 306 297 268 21Table 1: Utilities from different bundles. Observe that this agent is satiated at 5 units.Next, suppose the price of the good is p1= 1 and the agent has income m = 8. Then theagent can afford up to 8 units of x1. Given this budget set, the agent’s utility is maximised bychoosing x∗1= 5, yielding utility v = 30. In this example, the consumer can afford 8 units butchooses to consume 5. If the agent’s preferences are monotone, then she will always spend herentire budget.Finally, suppose the price of the good is p1= 2 and the agent has income m = 8. Then theagent can afford up to 4 units of x1, as in the original case. This illustrates that the budgetset is determined jointly by the prices and income: doubling both does not change the agent’sbudget set. When maximising her utility, the agent once again chooses x∗1= 4.2 Budget SetsAs in Section 1.1, we will solve the agent’s problem in two steps. First, we determine whichbundles of goods are affordable. Second, we find which of these bundles yields the agent thehighest utility. In this section we look at the first step.2.1 Standard Budget SetsIn the standard model, we assume there are unit prices {p1, p2} for the 2 goods. The budgetset is the collection of bundles (x1, x2) such that (a) the quantities are positive; and (b) thebundle is affordable. Mathematically, the budget set is{(x1, x2) ∈ <2+: p1x1+ p2x2≤ m}3Eco11, Fall 2009 Simon BoardFigure 1: Budget Set.where <+is the positive part of the real line, and <2+is the positive orthont in <2.Figure 1 illustrates such a budget set. The equation where the budget binds is given byp1x1+ p2x2= m (2.1)We can rearrange this to be in the form of a standard linear equationx2=mp2−p1p2x1(2.2)Hence the budget line is linear with intercept m/p2and slope −p1/p2. Crucially, the slope onlydepends on the relative prices.The two endpoints are easy to calculate. If the agent spends all her money on x1she can affordx1=mp1and x2= 0If the agent spends all her money on x2she can affordx1= 0 and x2=mp2Figure 2 shows that an increase in the agent’s income leads the budget line to make a parallel4Eco11, Fall 2009 Simon BoardFigure 2: An Increase in Income.shift outwards. Mathematically, this can be seen from equation (2.2). Intuitively, if the agent’sbudget doubles then she can double her consumption of both goods. Since relative prices donot change, the new budget line is parallel to the old one.Figure 3 shows that an increase in p1leads the budget curve to pivot around it’s left endpoint.Mathematically, this can be seen from equation (2.2). Intuitively, if the agent only buys x2,then her purchasing power is unaffected by the increase in p1. As a result, the left endpoint doesnot move. If the agent only buys x1, then the increase in p1reduces the amount she can buy,forcing the right endpoint to shift in. As a result, the budget line become steeper, reflectingthe change in the relative prices.2.2 Nonlinear Budget SetsWhile we focus on linear budget constraints, agents often face nonlinear prices. Here we presentsome examples.Figure 4 shows an example of quantity discounts. In this example, the agent has income m = 30.Good 1 has per–unit price p1= 2 for x1< 10, and per–unit price p1= 1 for x1≥ 10. Good 2has a constant price, p2= 2. Lets consider 2 cases. First, when the agent buys x1< 10, theprice of good 1 is p1= 2 and the equation of the budget line is therefore 2x1+ 2x2= 30 or5Eco11, Fall 2009 Simon BoardFigure 3: An Increase in the Price of Good 1.x2= 15 − x1. For example, when the agent spends all her money on good 2, she can affordx2= 15. Second, when x1≥ 10 the agent spends $20 on the first 10 units of x1and $1 per unitthereafter. Hence her budget constraint is20 + (x1− 10) + 2x2= 30Figure 5 shows an example of rationing. In this example, the agent has income m = 30. Good1 has