A Utility Maximization Example


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1 A Utilit Maximization Example Charlie Gibbons Universit of California, Berkele September 17, 2007 Since we couldn t finish the utilit maximization problem in section, here it is solved from the beginning. 1 The ingredients First, we start with a budget constraint: p x x + p = M = 10x + 0 = 60 (we are assuming that p x = 10, p = 0, and M = 60). Next, we are given a utilit function: U = U(x, ) = x / 1/. An important concept captured b the utilit function is that of marginal utilit. The idea here is, if we get one more unit of a good (i.e., a marginal unit), how much does m utilit go up? To find the marginal utilit of x, MU x, we find du dx : d ( ) x 1 dx = x 1 1 = Similarl, to find MU = du d : d ( ) x 1 d = 1 x = 1 ( x cgibbons@econ.berkele.edu 1
2 Next, we wish to calculate the marginal rate of substitution, MRS x,. The MRS x, states how much ou would give up for another unit of x without reducing our utilit. As shown in the book, U = MU x x + MU = 0 = x = MU x MU The amount of ou would give up for another unit of x is the ratio of the marginal utilities. 1 So if x is much more valuable to ou than (i.e., MU x is much greater than MU ), ou would give up a lot of to get more x. In our case, MRS x, MU x MU = 1 = ( = x x ( 1 x ) ) 2 Maximization We wish to maximize utilit subject to the budget constraint. To do this, we use the Lagrangian equation. The generic Lagrangian equation is L = objective function + λ(constraint = 0) Think of this equation as taking the function that we wish to maximize and adding 0 to it a perfectl acceptable mathematical trick that doesn t change the function at all, since the constraint must be satisfied and equal to 0. The smbol λ is called a Lagrange multiplier 1 The give up is wh there is a negative sign in there. The negative sign also reflects the fact that the slope of the indifference curve is the negative MRS. 2
3 and, of course, multipling 0 b an number will return 0, so the Lagrangian is still equated to the objective function. For us, utilit is our objective function and the budget constraint equated to 0 is our constraint. Now, maximizing the Lagrangian, we have max L = max x, x, x/ 1/ + λ(m p x x p ) The x, below max indicates that we are choosing both x and to maximize the function. All other variables are beond our control. 2 We take the following derivatives and set them equal to 0. These are known as first order conditions. ) 1 dx = x = 1 ( x d dλ = M p xx p = 0 λp x = MU x λp x = 0 λp = MU λp = 0 Note that the derivative with respect to λ simpl returns the budget constraint. One of the functions of the Lagrange multiplier is to ensure that we don t lose our constraint. We now have three equations defining our optimal bundle: MU x = λp x (1) MU = λp (2) M = p x x + p () First see that MRS x, = MU x MU = λp x λp = p x p 2 Since x and are in our control, the are called control variables. This procedure onl works if there is an interior solution that is, the individual consumes some x and some. If he does not consume one of these goods, there will be a corner solution. You know that there will be an interior solution if each marginal utilit is a function of the quantit of the good and thus the first order conditions will be solvable. For example, MU x = 7 is not a function of x and thus MU x = 7 λp x in general. See LearningbDoing exercises. and. for examples.
4 Recall the interpretation of the marginal rate of substitution: it is how much ou are willing to give up to get one more unit of x, holding utilit constant. The equation above states that, at the optimum, our willingness to exchange the goods must be equal to the ratio of the prices of the goods. So, if jeans cost three times more than shirts, when I am optimizing, m marginal utilit of jeans must be three times that of shirts and m marginal rate of substitution of jeans for shirts must be three. Here s another wa to think of equations (1) and (2). Solve them this wa: λ = MU x p x = MU p This tells us that the utilit that we get from an additional unit (i.e., a marginal unit) of a good per dollar spent must be the same across ever good that we consume. Let s sa that this wasn t true; if I receive more utilit per dollar that I spent on jeans than per dollar that I spend on shirts, then I should spend more on jeans to get higher utilit. If marginal utilit per dollar is equal across all goods, then I can t change m consumption in an wa to get higher utilit that is, I am at a maximum of utilit. Now, let s solve the equations with our example utilit function and prices. First solve equations (1) and (2) for λ: MU x = λ = 0 MU = 1 ( x λ = = λp x = 10λ = λp = 0λ ( x
5 Setting these two equations equal to one another and solving for in terms of x: 0 x ) 1 = 1 ( x 120 = 1 ( x 9 ( x x = 1 9 ) = 1 x 9 x = 1 9 = x 9 Taking this result and entering it into the budget constraint: M = p x x + p 60 = 10x + 0 x 9 x = 27 = x 9 = At the optimum, MU x p x = 1 p x MU = 1 1 p p x ( x ) 1 = = As required, these values are the same, validating our results. Tell me more about the Lagrange multiplier! (Note: This section isn t required, but ma help ou better understand what λ is.) 5
6 As alread mentioned, one of the important functions of the Lagrangian multiplier is to ensure that we don t lose our constraint during maximization (see page ). Let s see what else we can learn from the Lagrangian equation. Take the derivative with respect to income, M: dm = d dm (U(x, ) + λ (M p xx p )) = λ What does this mean? The Lagrangian multiplier tells us the increase in utilit (that s what the Legrangian function is counting utilit) when we get an extra dollar of income. This has two interpretations. First, we could see it in the literal sense of the value of increasing our income. Alternativel, we could view it more generall as the value of relaxing our constraint. In other words, how much can we increase our objective function b making our constraint less restrictive? This interpretation can be used for an Lagrange function. It is the use in the context of utilit maximization that gives the Lagrange multiplier its alternate name a shadow price, the value (or price) of the budget constraint relaxed (i.e., increased) b a dollar. 6
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