Explicit solutions of some utility maximization problems. Intermediate microeconomics solutions to problem set 2. Step 2 evaluate the function you are maximizing at all candidate solutions. Econ 101a solution to problem set 2 no late problem sets. Lagrange multipliers and constrained optimization a constrained optimization problem is a problem of the form maximize or minimize the function fx,y subject to the condition gx,y 0. Indirect utility and the marshallian demand correspondence are formally defined and the fundamental implications for them of the pricetaking utility maximization behavioral postulate are established. It turns out that this is general to all utility maximization problems not specic to. If done correctly with the logic stated clearly, this gets full credit. Dual to the utility maximizationproblem is the costminimizationproblem min x. This problem set tests the knowledge that you accumulated in the lectures 5 to 8. Problem 2, where you are actually required to check for corner solutions. Pdf explicit solutions to utility maximization problems.
However, for problems involving more than two variables or problems involving a large number of constraints, it is better to use solution methods that are adaptable to computers. It is focused on preferences, utility functions, and utility maximization. However, no argument can be provided to exclude corner solutions to this utility maximization problem. Utility maximizationconsumer behaviorutility maximizationindirect utility functionthe expenditure functiondualitycomparative statics 5 we will often need to assume that the solution to the utility maximization problem ump is unique. This is other form of saying that the solution to this problem is homogeneous of degree 0. The results presented here extend the results of zariphopoulou,14 and of henderson and hobson 6. That is, the utility function is the sum of a standard ces constant elasticity of substitution utility. Notice that the utility function which is a standard cobbdouglas, is a monotonic transformation of the utility function with exp hence, the two utility functions represent the same preferences, and thus the two utility maximization problems are identical. To discuss the problem of profit maximisation we shall consider here a simple production process where the firm uses two variable inputs x and y to produce a single output q and where the firm buys the inputs at fixed prices r x and r y and sells the output also at a fixed price p. Microeconomics practice problem utility maximization. Performance limits of solutions to network utility.
It makes sense to think that people are generally motivated to do what is best for them, to purchase the most satisfying goods, to make the. In most problems you will see, there will be no more than one such candidate solution. These turn out to be the trickiest utility functions to be confronted with. The solution of the first order condition, or equation 4, is y. Utility maximization problem questions and answers test your understanding with practice problems and stepbystep solutions. We will now explore these functions in more detail. Of course you could solve the usual utility maximization problem subject to the budget constraint. Problem set 2 solutions university of colorado boulder. The price of good xis pxand the price of good yis py.
Solving for the consumers utility maximizing consumption bundle. Their values will thus be determined inside the modeland they are, accordingly, called endogenous variables. Utility maximization can be achieved at the peak of the total utility curve. Relationship between total utility and marginal utility including a brief digression using calculus d. This approach is taken to satisfy the need for a simple objective for the. First, we transform this problem into an unconstrained optimization by replacing. Then any convex combination of c1 and c2 is also weakly preferred to c3. Because you have convex preferences, there will be a tangency between the ic and the.
Of course, if g is the exponential utility function gx. Maximization for linear programming problems involving two variables, the graphical solution method introduced in section 9. The condition for utility maximization the rational spending rule iv. Chapter 9 profit maximization done university of tennessee. Her utility function is u xy3 find her utility maximizing x and y as well as the value of. C2 and c3 be commodity bundles such that c1 c3 and c2 c3. The solution to utility maximization problem 3 gives the consumers choice of x and y, as a function of prices and income, which we denote by xpx. We make the same assumptions on the commodities as the last example, but assume the utility is given by u xa 1 1 x a 2 2 x a 3 3 with a 1 0, a 2 0, and a 3 0. A version of the envelope theorem yields roys identity relating indirect utility and marshallian demand. The hicksiandemand equations are sometimescalled compensateddemand equationsbecause they hold u constant. To solve the utility maximization problem, begin by setting the mrs price ratio. Problems with solutions, intermediate microeconomics. The solutions to the primal and dual problems coincide in the.
Suppose marias preferences for eggs and milk can be described by the following marginal utility schedules. The missing sentence should be \how are the indirect utility functions related. The rational choice assumption, that consumers pick the best a. Problems with solutions, intermediate microeconomics, lecture 12. It turns out that strict convexity ensures uniqueness. The values of x1and x2are to be chosen to solve the maximization problem. Mandy, in producers, consumers, and partial equilibrium, 2017. Our consumer, skippy, wishes to maximize utility, denoted ux,y. Utility maximization an overview sciencedirect topics. Of course everybody who handed in the homework gets full credit for the question 1. The mrs and the cobbdouglas consider a twogood world, xand y. Utility maximization given the consumers income, m, and prices, p x. The solutions to the problems are my own work and not necessarily the only way to solve the problems.
With quasilinear utility functions, indifference curves can cross the axes, so we do need to worry about corner solutions. Using the same reasoning again, the problem does not ask for this, the total amount spent on nuts, x 1, will be p. As you will see in the next lecture, the utility maximization problem for a consumer and lots of other problems that you will see in this class can be stated as a problem on the form max a. Utility maximization is the guiding notion underlying consumer choices analyzed with consumer demand theory and utility analysis. In this section we solve three utility maximization problems. Performance limits of solutions to network utility maximization problems r. Chapter 9 profit maximization economic theory normally uses the profit maximization assumption in studying the firm just as it uses the utility maximization assumption for the individual consumer. Assume x is a solution to the problem with prices and income p. Sally jones gets utility from consuming two goods, spaghetti and. Steele and are reprinted with his generous permission. Exponential and power utility maximization problems under.
This video shows how to use marginal utility and prices to maximize utility. If you cannot solve a problem fully, write down a partial solution. Mathematical economics practice problems and solutions. You can show that the utility maximization of uleads to the same demand xp. Substitute the budget constaint into the utility function so the utility function is a. Opmt 5701 optimization with constraints the lagrange. Since more is better, inequality 1 must hold with equality at the solution to the consumers problem. Eco 305 fall 2003 range number princeton university.
To show the effects on consumer choice of changes in prices and income. The standard optimization problem is to maximize a utility function subject to a budget constraint. Frittelli and others published optimal solutions to utility maximization and to the dual problem find, read and cite all the research you need on researchgate. Learn to identify basic variables, read feasible solutions from a tableau, and pivot to manipulate your. The mrs and the cobbdouglas simon fraser university. Explicit solutions to utility maximization problems in a regimeswitching market model via laplace transforms article pdf available april 2018 with 110 reads how we measure reads. Utility maximization problem questions and answers. Variation in how quickly marginal utility declines e.
Derive the demand of good x and y in terms of the parameters. Test your understanding with practice problems and stepbystep solutions. Because at the utility maximizing solution to this problem, x. You clearly want to spend all your income, so the bc is binding. Pdf optimal solutions to utility maximization and to the. These are known as the generalized demand functions.
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