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ECO102 Lecture Notes - Lecture 15: Fxx, Utility, Envelope Theorem

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coppersheep255
3 Nov 2017
School
Bishop's University
Department
Economics
Course
ECO102
Professor
Álvaro

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Document Summary

In a constrained optimization problem we are asked to examine whether a function has a maximum (or minimum) point in a subset of its domain. The subset is identi ed by one or more equations and/or inequalities in the independent variables. These equations and/or inequalities are the con- straints of the problem. Here we will look at the case where the constraints are equations. The following simple constrained maximization problem with one equality constraint is very common in economics. max (x,y) f (x, y) subject to g(x, y) = c where c is a constant. The symbols can be read as follows: choose (x, y) to maximize the value of f subject to the constraint g(x, y) = c". Therefore, we restrict attention only to the (x, y) points in the domain of f which satisfy the equation g(x, y) = c. Remark 1: of course, we assume that the equation has solutions.

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Related Questions

Consider the one period model with endogenous labor supply we studied in class, where workers (consumers) are born with one unit of labor effort and no physical endowments. The total population of consumers is one. Furthermore, assume that workers have the following lifetime utility function: U(c, l) = ln c + ln l Production takes place by a representative firm that has access to a production technology of the Cobb-Douglas form: Y = AK α L 1−α and as usual profits are rebated back to consumers and P = 1. Unlike the model discussed in class, the firm pays (w + τ ) for every unit of labor time used, τ > 0 is a unit tax imposed by the government (or some type of regulation). In this manner, the firm pays τL to the government and wL to workers. Finally, the government does no collect taxes from workers but only from employers as mentioned above. Government revenue is used to finance spending, G where G does not directly affect the economy. The government’s budget constraint is such that: τL = G

A. Write down the individual’s problem of utility maximization. Be sure to derive the budget constraint for the problem. Provide economic interpretation for this model.

2. InvestCo is an investment firm that manages stock portfolios for a number of clients. A new client is requesting that the firm handle his funds. As an initial investment strategy, the client would like to restrict the portfolio to a mix of two stocks: Alaska Oil and Southeast Petroleum. The price of Alaska Oil is $38 per share and the price of Southeast Petroleum is $45 per share. Let x:= number of shares of Alaska Oil y:= number of shares of Southeast Petroleum

a) Write an expression for the total number of shares purchased for the client.

b) Estimated Annual Return is $9 per Alaska Oil share and $7 per Southeast Petroleum share. Develop an objective function to maximize the total annual return.

c) Total investment funds available are $83,000 for the client. Develop a constraint to model this situation.

d) Maximum Alaska Oil investment is $47,000 and maximum Southeast Petroleum investment is $42,000. Develop constraints to model this situation.

e) Of course it is not possible to purchase a negative number of shares. Develop sign restrictions to model this situation.

f) Formulate a Linear Programming model to maximize the annual return for the client (Write the complete model for the problem).

g) Is (x, y) = (1200, 900) a feasible solution for the model you formulated in part f)? Why or Why not?

Need assistance with these problems!


1. The following is a linear programming formulation of a labor planning problem. There are four overlapping shifts, and management must decide how many employees to schedule to start work on each shift. The objective is to minimize the total number of employees required while the constraints stipulate how many employees are required at each time of day. The variables X1 - X4 represent the number of employees starting work on each shift (shift 1 through shift 4).
Minimize X1 + X2 + X3 + X4
Subject to: X1 + X4%u2265 12 (shift 1)
X1 + X2%u2265 15 (shift 2)
X2 + X3%u2265 16 (shift 3)
X3 + X4%u2265 14 (shift 4)
all variables %u2265 0

Find the optimal solution using QM.
How many workers would be assigned to shift 1? (Points : 3) 12
13
0
none of the above

2. When appropriate, the optimal solution to a maximization linear programming problem can be found by graphing the feasible region and: (Points : 3)
finding the profit at every corner point of the feasible region to see which one gives the highest value.
moving the isoprofit lines towards the origin in a parallel fashion until the last point in the feasible region is encountered.
locating the point that is highest on the graph.
none of the above.
all of the above.


3. Multiple optimal solutions can occur when the objective function is __________ a constraint line. (Points : 3)
unequal to
equal to
linear to
parallel to


4. When applying linear programming to diet problems, the objective function is usually designed to: (Points : 3)
maximize profits from blends of nutrients.
maximize ingredient blends.
minimize production losses.
maximize the number of products to be produced.
minimize the costs of nutrient blends.


5. For a maximization problem, assume that a constraint is binding. If the original amount of a resource is 4 lbs., and the range of feasibility (sensitivity range) for this constraint is from 3 lbs. to 6 lbs., increasing the amount of this resource by 1 lb. will result in the: (Points : 3)
same product mix, different total profit.
different product mix, same total profit as before.
same product mix, same total profit.
different product mix, different total profit.


6. The corner point solution method: (Points : 3)
will yield different results from the isoprofit line solution method.
requires that the profit from all corners of the feasible region be compared.
will provide one, and only one, optimum.
requires that all corners created by all constraints be compared.
will not provide a solution at an intersection or corner where a non-negativity constraint is involved.


7. Consider the following linear programming problem.
Maximize 4X + 10Y
Subject to: 3X + 4Y %u2264 480 and 4X + 2Y %u2264 3 360
all variables %u2265 0
The feasible corner points are (48,84), (0,120), (0,0), (90,0).

What is the maximum possible value for the objective function? (Points : 3)
1032
1200
360
none of the above


8. When using Excel's Solver to input and solve a linear programming problem, it is essential that one perform an additional task before submitting the formulation. That important additional function is: (Points : 3)
guessing the values of the variables.
putting in a value for the objective function.
choosing the options for assuming both a linear model and non-negative variables.
resetting the parameters.
none of the above.


9. Consider the following linear programming problem.
Maximize 5X + 6Y
Subject to: 4X + 2Y %u2264 420 and 1X + 2Y %u2264 120
all variables %u2265 0

Which of the following points (X,Y) is not feasible? (Points : 3)
(50,40)
(20,50)
(60,30)
none of the above


10. ____________ is used to analyze changes in model parameters. (Points : 3)
Optimal solution
Feasible solution
Sensitivity analysis
None of the above