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Decision-Making in Agribusiness: Quantitative Applications

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Problem 1. (3×7 = 21 points)

Are the following statements about indifference curves (ICs) correct or incorrect?

If you answer is incorrect, give your suggestion(s) to correct the statement.

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(a) IC shows all combinations of goods that give a consumer the same level of utility.

(b) The slope of IC shows that change in utility from an additional unit of good.

(c) Along an IC, the ratio of marginal utility is constant.

(d) The slope of IC is the rate at which the consumer is able to exchange one good in the

market.

(e) IC shows that, as a consumer has more of a good, he/she is less willing to exchange it for

one unit of another good.

(f) IC shifts out if income increases.

(g) The slope of IC is equal to the price ratio at all points.

Problem 2. (6 points)

If marginal rate of substitution (MRS) of X for Y is 2, the price of X is $3 and the price of Y is $1, a

utility maximization consumer should (choose one or more of the following that are correct):

(a) be indifferent between one unit of X and two units of Y.

(b) prefer three units of Y to one unit of X.

(c) choose less X and more Y.

(d) choose more X and less Y.

Explain why you make your choice and show your work.

Decision-Making in Agribusiness: Quantitative Applications (AGSC5300) Lan Li

Tennessee State University

2

Problem 3. (2+3+3=8 points)

Sybil has a limited income and consumes only sausage and cheese. Her current consumption is 4

pounds of sausage and 10 pounds of cheese. The price of sausage is $10 per pound, and the price of

cheese is $4 per pound. The last pound of sausage added 50 units to Sybil’s utility, while the last

pound of cheese added 40 units to her utility.

Is Sybil making the utility maximization choice? If yes, WHY? If not, WHY not? What should she

do instead? Explain why she should do what you suggest?

Problem 4. (4×4 = 16 points)

In terms of optimization and the consumer theory that we have covered, EXPLAIN the meaning of

the following statements.

(a) “I think you get more for your money from Nike shoes than from Reebok shoes.”

(b) “I wanted to buy a Mercedes-Benz rather than a Toyota-Corolla, but it just wasn’t worth it.”

(c) “I’d like to go to Mexico over spring break, but I just can’t afford it,” said Edith.

Cora then asked, “Don’t you have enough money in your bank account to go?”

Edith replied, “Yeah, but I can’t afford to go.”

(d) “I will have to flip a coin to decide whether to buy chocolate or vanilla ice cream.”

Problem 5. (25 points)

Please refer to the following graph to answer the questions. Suppose the graph illustrates Mary’s

utility maximization (U-Max) problem. There are two goods, X and Y. The price of Y is $10.

Y

60

X

0 300

B

A

U1

U 2

200

26

20 U3

C

Units of good X

Units of good Y

120

Decision-Making in Agribusiness: Quantitative Applications (AGSC5300) Lan Li

Tennessee State University

3

(a) (2) What is Mary’s income/budget in $10?

(b) (1) What could have caused Mary’s budget line to rotate from left to right?

(c) (5) What are the prices of X? (X has more than one price values in this graph.)

(d) (6) What are MRSs of X for Y at points A, B, and C, respectively?

(e) (5) If the price of X is $5, what combination of X and Y will Mary choose optimally?

(f) (6) List three points on Mary’s individual demand curve for X. (Note: A point on a demand

curve is a combination of price and quantity demanded).

Problem 6. (18 points)

The following graph illustrates Isobel’s U-Max problem. Use it to answer the questions. The price

for X is $20 and the price for Y is $40.

(a) (3) To achieve the utility level at U1, what is Isobel’s budget?

(3) To achieve the utility level at U2, what is Isobel’s budget?

(b) (3) How many units of X and Y are, if A is Isobel’s U-Max optimal choice?

(3) How many units of X and Y are, if B is Isobel’s U-Max optimal choice?

(c) (3+3) What are MRSs of X for Y at points A and B, respectively?

Y

30

X

40

B

A

U1

U2

0

14

24

Units of good X

Units of good Y

Decision-Making in Agribusiness: Quantitative Applications (AGSC5300) Lan Li

Tennessee State University

4

Problem 7. (32 points)

Assume Violet’s preference is represented by the following utility function:

=U(X,) = 20− X2 +802

The budget for Violet is M=$80. The price for X is $1, and the price of Y is $2.

(a) (2) What is Violet’s budget constraint?

(2) In addition, present the budget constraint in the format of Y as a function of X.

(2) What is the intercept and what is the slope of the budget line?

(b) (2) The First Step: State Violet’s U-Max problem in the mathematical format.

(3) What is the objective function? What is the constraint? What are the choice variables?

(2) State Violet’s U-Max problem in the Lagrangian functional form.

(c) (3) The Second Step: Derive the F.O.Cs for Violet’s U-Max problem.

(d) (3) The Third Step: Solve for the optimal consumption bundle (X*, Y*) that maximizes

Violet’s utility.

(e) (5) Derive and present marginal utility of X (MUX) and marginal utility of Y (MUY) for

Violet, respectively. Derive the marginal rate of substitution (MRS) of X for Y.

(f) (3+3) What is the Tangent condition for U-Max problem? Explain in words and in math

equation(s).

(2) From the Tangent condition, write Y as a function of X.

Decision-Making in Agribusiness: Quantitative Applications (AGSC5300) Lan Li

Tennessee State University

5

Problem 8. (14 points)

A consumer has the indifference map shown below. The market price of X is $24 and the price of Y

is $8, respectively. The consumer has $120 to spend on good X and Y.

(a) (2) Construct the consumer’s budget line and find the utility-maximizing consumption

bundle. Label this bundle “E.” Bundle E is composed of ______units of X and ______

units of Y.

(b) (3) For bundle E, the marginal rate of substitution is, in absolute value, _______ (greater

than, less than, equal to) the slope of the budget line. The ratio MU/P for good X is

_______ (greater than, less than, equal to) the ratio MU/P for good Y.

(c) (2) Bundle E _________ (is, is not) a corner solution.

Now suppose the consumer’s income and the price of Y remain the same, but the price of X

decreases to $8.

(d) (2) Construct the new budget line and find the new utility-maximizing consumption bundle.

Label this bundle “N.” Bundle N is composed of _____ units of X and _____ units of Y.

(e) (3) For bundle N, the marginal rate of substitution is _________ (greater than, less than,

equal to) the slope of the budget line (in absolute value). The ratio MU/P for good X is

________(greater than, less than, equal to) the ratio MU/P for good Y.

(f) (2) Bundle N _________ (is, is not) a corner solution.

Decision-Making in Agribusiness: Quantitative Applications (AGSC5300) Lan Li

Tennessee State University

6

Problem 9. (10 points)

The following figure shows a portion of Lavinia’ indifference map and budget lines. The price of

good Y is $17 and Lavinia’ budget is $7,650. Let Lavinia begin in U-Max point A on the indifferent

curve II. Next, the price of good X changes so that Lavinia moves to a new U-Max point B on

indifference curve I. Use the graph to approximate the quantities.

(a) At pint A, what are the price and consumption quantity of good X?

At point B, what are the price and consumption quantity of good X?

(b) The substitution effect of the change in the price X has caused ______(an increase or a

decrease) in consumption of X by ______ units.

The income effect of the change in the price of X has caused _______(an increase or a

decrease) in consumption of X by ______ units.

The total effect of the change in the price of X on the consumption of X is_______(an

increase or a decrease) by ______ units.

(c) Is good X a normal or an inferior good? Why?

(d) What is the Law of Demand? Explain why the Law of Demand holds in terms of income

and substitution effect of a price change on quantity demanded.

II

200

Quantity of good Y

Quantity of good X

A

300 400 500 600 700 800

400

0 100

300

500

600

700

100

B

I

200

X

Y.

Problem 1. (10 points)

(1) Construct a general demand function for tea or tomatoes or a bioenergy crop. Pick one, if

you have more time, do the exercise for two or all.

(2) Find a product (a good or service) that you are keen or just feel fun to estimate its demand.

Then next construct a general demand function for this product.

List a set of factors (or variables) that you consider to be important factors that affect quantity

demanded for this product. You can either use f(…) as a general functional form, or choose some

specific functional form.

Suggestions: Have a little fun with this. It is just a small exercise for the mind, not a serious project.

• I encourage you use your imagination to select the product. The set of variables in

textbook/lecture are fundamental. Variables may change depending on the product, and you

may find some interesting factors just for the product you analyse.

• A note on personal preferences: We can use personal experience/preference to think about

what factors may influence the demand for a product. At the same time, take care that you

are estimating the demand function at the aggregate or market level, and therefore do NOT

let personal or idiosyncratic preference interfere with what factors you pick. Step back,

observe the overall or the general or the average.

• You do not need to pick ALL the factors that may affect demand, focus on important

factors, although I encourage you to think as creatively, broadly, and deeply as you like. In

practice, we generally pick out important factors because we cannot model with too many

variables, infeasible to collect the data and/or statistically inefficient. This may not be a

problem if you have big data.

Problem 2. (72/2 points)

The general demand function for product A is

€

Q=160 −12+ 0.01−10P+ 8P+12+ 2P+ 3N

where

€

Q=quantity demanded of product each month in units; =price of product in $/unit;

= average household income in $;

€

P=price of related product in $/unit

€

P=price of related

product in $/unit; T=a consumer taste index ranging in value from 0 to 10 (the higher the value,

Decision-Making in Agribusiness: Quantitative Applications (AGSC5300) Lan Li

Tennessee State University

2

the more consumers like product A);

€

P= the price that consumers expect to pay next month for

product in $/unit; and = the number of buyers in the market for product A.

(a) (3 points) Interpret the intercept parameter in the general demand function.

(b) (3 points) Interpret the coefficient (or called slope parameter) for P. Does it have the

correct algebraic sign? Why?

(c) (3 points) Interpret the coefficient for income. Is product a normal or inferior good?

Why?

(d) (3 points) Are products and B, and products and substitutes or complements? Why?

(e) (5 points) Are the algebraic signs on the coefficients for T,

€

P, and correct? Follow your

answer “yes” or “no” to the sign of each coefficient, explain why?

(f) (5 points) Derive the (direct) demand function for product A, when

€

P= $20,

€

P= $40,

€

= $40,000,

€

= 8,

€

P= $2, and

€

=1,000.

(g) (2+2+3 points) Interpret the intercept and slope parameters of the demand function

derived in part (f). How many units of product are demeaned when its price is $100/unit

according to the demand function in part (f).

(h) (5 points) Derive the inverse demand function from the demand function in in part (f).

(i) (6 points)) Sketch a demand curve based on the inverse demand function derived from part

(h). Where does the demand curve intersect the price axis, and where does it intersect the

quantity-demanded axis?

(Hint: It is easier to draw the graph if you scale the axis measurements. For example, you

may let one unit on the quantity-demanded axis represent 500 units of product A.)

(j) (6 points)) The demand curve in part (i) passes through the point P=$265/unit and

€

Qd

=600. Give two interpretations of this point on the demand curve. (Note: One

interpretation is from the angle of quantity and the other is from the point of price.)

(k) (6 points) Using the graph in part (i), explain the difference between a movement along

demand curve and a shift in demand curve.

(l) (10 points) What happens to demand for product when the following changes occur?

(increases or decreases?) If a shift in the demand curve for product occurs as a result of

the change, in which direction does the demand curve shift?

i). The price of product rises?

ii). Income increases?

iii). The price of product increases?

iv). The number of consumers increases due to a seasonal demand peak for product A?

v). A number of retail stores advertised product in their store flyers?

(m) (10 points) The price of product decreases from $40/unit to $10/unit, all else constant.

Calculate the (direct) demand function when

€

P= $10, while values of other variables are

the same as specified in part (f). Does the demand curve shift to the right or left, and by

how many units? Illustrate with a graph.

Decision-Making in Agribusiness: Quantitative Applications (AGSC5300) Lan Li

Tennessee State University

3

Problem 3. (5X5=25/2 points)

Construct a graph showing equilibrium in the market for movie tickets. Label both axes and denote

the initial equilibrium price and quantity as

€

P0 and

€

Q0 . Draw one graph for each of the following

events. In your graph, include the initial demand and supply and equilibrium, and draw an

appropriate new supply and demand curve for movies, and predict the impact of the event on the

market price of movie ticket and the number of tickets sold in the new equilibrium situation.

(a) Movie theatres double the price of soft drinks and popcorn.

(b) A national video rental chain cuts its rental rate by 25 percent.

(c) Cable television begins offering pay-per-view movies.

(d) The screenwriters end a 10-month strike.

(e) Kodak reduces the price it charges Hollywood producers for motion picture film.

Problem 4. (5X6 = 30/2 points)

Florida Citrus Mutual, an agricultural cooperative association for citrus growers in Florida, needs to

predict what will happen to the price and output of Florida oranges under the conditions below.

What are your predictions? Sketch separate graph for each question showing the appropriate

demand and supply analysis.

(a) A major freeze destroys a large number of the orange trees in Florida.

(b) Two new companies producing orange juices began to operate.

(c) The scientists in the agricultural extension service of the University of Florida discover a way

to double the number of oranges produced by each orange tree.

(d) The American Medical Association announces that drinking orange juice can reduce the risk

of heart attack.

(e) The price of Florida grapefruit falls.

(f) The price of Californian oranges increases.

Decision-Making in Agribusiness: Quantitative Applications (AGSC5300) Lan Li

Tennessee State University

4

Problem 5. (63/2 points)

Consider the following demand and supply functions for tomatoes:

Demand function:

€

Q= 60 − 40P;

Supply function:

€

Q= −10 +100P;

where P is the market price for tomatoes in $/pound; Quantity demanded and supplied are

measured in 1000 pounds per unit (e.g., one unit of

€

Qor

€

Qcorresponds to 1000 pounds.).

(a) (8 points) Plot demand and supply functions in a demand-supply panel. Calculate the slope

coefficients for demand and supply curves, respectively.

(Note: Remember price is on the vertical axis).

(b) (8 points) The demand curve passes through the point P=$1/pound and

€

Qd=20 units. Give

two interpretations of this point on the demand curve.

The supply curve passes through the point P=$0.6/pound and

€

Qs=50 units. Give two

interpretations of this point on the supply curve.

(Note: One interpretation is from the angle of quantity and the other is from the point of

price.)

(c) (6 points) Calculate the market equilibrium (also called market clearing) price and quantity of

tomatoes.

(d) (10 points) Calculate the producer surplus, consumer surplus, social surplus in the market

equilibrium in part (c).

(e) Suppose there is a tomato grower association in Tennessee. The association plans to begin a

regional advertising campaign in the Southeast US to promote tomatoes by informing

consumers of the nutritional benefits of tomatoes, and the benefits of eating healthy food

like fresh tomatoes and buying local.

The association expects that the advertising will make consumers want to eat more

tomatoes, and hence tomatoes growers will benefit by higher prices and more sales.

Suppose you are a consultant commissioned by the association to predict the effect of the

advertising. First, suppose you estimate that the demand will increase to

€

Qd

1 = 70 − 35(We

will learn how to estimate demand later in the course). Assume that the supply is unaffected

by the advertising.

Decision-Making in Agribusiness: Quantitative Applications (AGSC5300) Lan Li

Tennessee State University

5

Now answer the following questions:

(5 points) First, would you agree with the association’s expectation? EXPLAIN WHY (you

may use a graph to illustrate)?

(5 points) Second, calculate the new market clearing price and quantity. Do your numbers

confirm with your answers to the first question.

(f) The advertising by the association is funded by collecting money from tomato growers. The

association demonstrates to growers the advertising will be effective and beneficial to

growers.

• First, the advertising will lead to higher price and more sales as demonstrated in part

(e). This means higher revenue (price times quantity). This does not indicate profits

(revenue minus costs).

• Second, the adverting will increase profits (or producer surplus) for tomato

growers.

(5 points) Calculate the increase in producer surplus due to the effect of the advertising.

(Hint: the difference in producer surplus between the initial and the new market

equilibrium).

(6+4 points) Calculate the change in social surplus due to the effect of the advertising. The

association wants to make sure that the advertising is also sound to the society (consumers

and producers) as a whole. (Hint: The social surplus is the sum of consumer and producer

surplus).

(g) The association reports to growers a measurement called benefit-cost ratio, which measures

the effectiveness of the advertising campaign:

€

Benefit – Cost Ratio =

Benefit of the advertising

Cost of the advertising .

(6 points) Calculate two benefit-cost ratios of the advertising campaign. Suppose the

advertising campaign costs $2500. One benefit-cost ratio is calculated based on the benefit

represented by changes in producer surplus due to the advertising from part (f). The other

benefit-cost ratio is calculated based on the benefit represented by changes in social surplus

due to the advertising from part (f).

You’ve just learned and practiced how to carry out program evaluation in this simple example.

This is a very representative example in agriculture industry: many commodity groups conduct

this type of evaluations. Many businesses also apply this analysis for their promotions,

investment in research and development, and various types of programs. The framework and

method are also applicable for evaluating impacts of certain programs, projects, and policies