# solving utility function with budget constraint

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The consumer in this case has a utility function expressed as U(XY) = 0.5XY. In other words, people face a budget constraint… 4The budget constraint holds with equality because the utility function is strictly increasing in both arguments (Quiz: Why?). You can use the model of consumer choice and take a look at what a consumer will do to optimize her utility or satisfaction when a constraint exists. This post goes over a question regarding the economics of utility functions and budget constraints: Matt has the utility function U = √XY (where Y represents pears and X represents hamburgers), income of $20, and is deciding how to allocate that income between pears and hamburgers. Will Mainy be better or worse off? maximization problem with an unusual (for us) income. A consumer's budget constraint is used with the utility function to derive the demand function. (2 points) 4. We could have an infinite number of indifference curves. Sign up to join this community. (1 point) 5. 5, Feldman and Serrano Ch 3 2. The budget constraint divides what is feasible from what is not feasible. have a utility function that describes levels of utility for every combination of Apples and Bananas. MRS is the ratio of the marginal utilities i.e. SUBJECT TO the budget constraint • Now it is time to solve it! 3See the following discussion of non-negativity constraints for this utility maximization problem. Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Home Questions Tags Users Unanswered Deriving demand functions given utility. Since, log 0 = – α, the optimal choice of x 1 and x 2 is strictly positive and must satisfy the first order conditions. The price of good is and the price of good is The unusual part is that consumers’ income is given not by a monetary budget but by endowments of the goods. The whole point of having indifference curve (IC) and budget constraint (BC) is to determine the optimal allocation—the feasible bundle that gives the highest utility to the individual. Suggested exercise: Adjust the values of , , , and one at a time, anticipating how the graph will change, and rewriting the Lagrangian and re-solving for the optimal bundle, the value of the Lagrange multiplier, and the resulting optimal utility level; in particular, increase by 1 and note the change in the resulting utility levels. Max U(x), y) = A X α y β. Constraint: Total Budget B = P x X + P y Y . There are relatively few possibly combinations of shoes so we could solve this by brute force to see which combination of goods gives us the most utility. Meanwhile, “happiness” is linked to indifference curves. Budget constraint: graphical and algebraic representation Preferences, indifference curves. 6 Indirect Utility Function De–nition: Plug in the demand functions back into the utility function. Therefore, budget constraint = 4A + 3O = 8 + 12 = 20. P y = Per unit cost of Product 2 MRS = (dU/dC1)/(dU/dC2). Writing the consumption levels simply as x 1 and x 2, we get . It only takes a minute to sign up. The distinction for non-linear functions is not crucial for this class, but it has been tested before. e. = d, but the interest rate is 20%. Moreover, look at Problem 2, where you are actually required to check for corner solutions. Change in budget constraint. This is an example of an optimization problem and there are simple calculus techniques on how to handle this we can exploit. 4. Next we need a set pf prices. υ = V (p, y) The indirect utility function works like an "inverse" to the cost function The cost function maps prices and utility into min budget. (Hint: you can use the answer in point 1) (5 points) 3. 2. Basics . The indirect utility function maps prices and budget into max utility . To do this, you have to take a look at what happens when you put the indifference curves together with the budget constraint. Another method would be to substitute out either C1 or C2 from the utility function using the budget constraint, then solving a one-variable maximisation problem. So the budget constraint will hold with strict equality at any solution. Unlike most utility maximization problems for which you are familiar, you cannot solve this by taking derivatives. $\endgroup$ – worldtea May 18 '15 at 16:35 We can maximize or total utility at all of these other points in between, along our budget line. p 1x 1 +p 2x 2 = M. Write down the Lagrangean function. Can Mark Zuckerberg buy everything? Write down Mainy's marginal rate of substitution. ii) Determine the values of X and Y that will maximise utility in the consumption of X and Y. iii) Determine the total utility that will be generated per unit of time for this individual. sugar tax). The budget constraint is the first piece of the utility maximization framework—or how consumers get the most value out of their money—and it describes all of the combinations of goods and services that the consumer can afford. 2 1 2 1 A B = Well being from consuming (A) Apples and (B) Bananas. Solving Utility Maximization Problem given Cobb-Douglas Utility Function. 2.1. In this subsection, we illustrate the validity of (1) by considering the maximization of the production function f(x,y) = x2/3y1/3, which depends on two inputs x and y, subject to the budget constraint w = g(x,y) = p 1x+p 2y where w is the ﬁxed wealth, and the prices p 1 and p 2 are ﬁxed. See Midterm 1 Spring 2015 #3. Next we have to draw a budget constraint, since the prices are constant (they don’t change) the line will be straight (no weird curvature). This concept can be summed up in the following question: “What affordable set of goods/services will maximize my happiness?” The word “affordable,” here, is linked to budget constraint. However, most people cannot consume as much as they like due to limited income. The goal: maximize total Utility. Use pictures to think heuristically about how to solve the consumer’s problem Varian Ch. We then plot these three points on a graph, and connect the dots so to speak and we will have our have drawn our budget constraint. (1point) 6. c. Suppose that Mainy has the utility function U = c 1 c 2. 3 Today’s Aims 1. Suppose that Apples cost $4 apiece and Bananas cost $2 apiece. Consider now the maximization subject to a budget constraint. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function. Solving Utility Functions Utility function Marginal rate of substitution (MRS), diminishing MRS algebraic formulation of MRS in terms of the utility function Utility maximization: Tangency, corner, and kink optima Demand functions, their homogeneity property Homothetic preferences. Will she borrow or save in the first period. This is obtained by solving the original equation for a and setting it equal to u. Consumers maximize their utility subject to many constraints, and one significant constraint is their budget constraint. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Write down the Lagrangean function. Use maths to turn this intuition into a solution method It will be useful to review the materiel on first order conditions, Lagrangians etc From your calculus class Varian Ch. The utility function describes the amount of satisfaction a consumer gets … To start, I'm going to assume the utility function is U(x,y) = (x^1/3)(y^2/3), as that is your typical CRTS Cobb-Douglas production/utility function (the assumption doesn't change the fundamental approach to solving the problem, just the algebra). Budget constraint constitutes the primary part of the concept of utility maximization. Further, assume that this consumer has $120 available to spend. So to actually maximize our total utility what we want to do is find a point on our budget line that is just tangent, that exactly touches at exactly one point one of our indifference curves. Solving for Optimal Bundle . So, the goal is to determine the demand of x and y given U(x,y) and the budget constraint, B(x,y): 4x + 5y = 60. Where: P x = Per unit cost of Product 1 . (20) In consumption of two goods at levels (x,y), a consumer has the following utility function and budget constraint u(x, y) B (3.x2 + 2)(y +1) P. x + Pyy Given that B = $100, P: = $5, Py = $2 (ii) Form the Lagrange function for maximization of the utility subject to the budget constraint, and its 1st order conditions. in ﬁnding the consumption bundle that gives the highest value of the utility function of all bundles in the budget set. 1. y = C (p, υ) The two solution functions have to be consistent with each other. Write down the ﬁrst order conditions for this problem with respect to x 1, x 2,and λ. The utility function is ( )= log( )+(1− )log( ) This function is well-deﬁned for 0 and for 0 From now on, assume 0 and 0 unless otherwise stated. Using the Slutsky equation, determine the size of income effect and substitution effect of $1 unit sales tax on the consumption of Donuts (e.g. By now you should be very familiar with where the optimal allocation is graphically; in this section we shall work it out mathematically. U=(xy) du/dx=y du/dy=x Mux=y Muy=x Then Budget constraints =Px.X+Py.Y=M Price ratios px/py=y/x 1000/500=y/x Cross multiply when you get y, put it in the budget constraints and solve for y and use it to solve for x When you finish, equate x and y values to the price ratio to … commodity vectors x = (x1,x2), and seeks to maximize this utility function subject to the budget constraint P1x1 + P2x2 I, where I is income and P = (P1,P2) is the vector of commodity prices. The consumer maximizes utility subject to a budget constraint. i) Express the budget constraint mathematically. Consider the following utility function and budget constraint U(Donuts, Yogurt) = Day P.D+PyY=M Currently, a = 0.3,8 = 0.6, P, = $1,Py = $1 and M = $20. In reality, there are many goods and services to choose from, but economists limit the discussion to two goods at a time for graphical simplicity. d. Set this slope equal to the slope of the budget line and solve for the consumption in period 1 and 2. Even Bill Gates cannot consume everything in the world and everything he wants. Thus our budget constraint looks like: $40 = $10L + $10R. Consider ﬁrst the case where there are no constraints on x,so that xcan take on any The third first-order condition is the budget constraint. I. The Intertemporal Budget Constraint: Rational individuals always prefer to increase the quantity or quality of the goods and services they consume. Define u = V(I,P) to be the value of utility attained by solving this problem; V is called the indirect utility function. Write down the maximization problem of the consumer with respect to xand y.Explain brieﬂy why the budget constraint is satisﬁed with equality. Thus, budget constraint is obtained by grouping the purchases such that the total cost equals the cash in hand. the budget constraint and the perfect substitutes utility function (see below). The agent maximizes max x 1,x 2 u(x 1,x 2)= (x 1 −r 1) αx 2 β s.t. Note: interpreting the slope like this technically only works for linear functions, e.g. Is the ratio of the consumer in this case has a utility function to derive the demand function that consumer... Varian Ch this slope equal to the budget constraint: Rational individuals always to... Algebraic representation Preferences, indifference curves and 2 the consumer with respect to x and... Cost equals the cash in hand part of the goods and services they consume,. 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An infinite number of indifference curves which you are actually required to check for corner solutions example of optimization... Original equation for a and setting it equal to U marginal utilities i.e for linear functions, e.g the and...

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