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General Equilibrium and Welfare
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Quiz 12 :
General Equilibrium and Welfare

Quiz 12 :
General Equilibrium and Welfare

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Suppose the production possibility frontier for guns (x) and butter (y) is given by img a. Graph this frontier. b. If individuals always prefer consumption bundles in which y = 2 x , how much x and y will be produced? c. At the point described in part (b), what will be the RPT and hence what price ratio will cause production to take place at that point? (This slope should be approximated by considering small changes in x and y around the optimal point.) d. Show your solution on the figure from part (a).
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The production possibility frontier shows different combinations of two goods which can be produces with given technology and resources.
a.
The production possibility frontier for guns (x) and butter (y) is given below:
img This is an equation of a parabola.
The production possibility frontier for the guns and butter given below:
img Figure 1
On X axis the units of guns is shown and on Y axis the units of butter is shown. PPF curve shows the combination of guns and butter that can be produced with in given resources.
b.
If individuals always choose consumption bundles in which y = 2x so the production should be in this ratio only.
Put value of y in the production possibility equation as shown below:
img Put value of x in the equation of y as shown below:
img If individuals always prefer consumption bundles in which y = 2x, then 10 units of x and 20 units of y will be produced.
The rate of product transformation (RPT) between two outputs is the negative of the slope of the production possibility frontier of those outputs.
The formula for the RPT is given below:
img The production possibility frontier is given below:
img Differentiate both sides with respect to x as shown below:
img RPT is calculated at
img and
img as shown below:
img img At a point where x = 10 and y = 20, the rate of product transformation (RPT) is
img .
At optimality the price ration is same as RPT
Hence, the price ratio that will cause production to take place at this point is
img .
d.
The solution point is shown in the figure below by the point A.
img Figure 2
On X axis the units of guns is shown and on Y axis the units of butter is shown. PPF curve shows the combination of guns and butter that can be produced with in given resources. Point A shows the combination where, x = 10 and y = 20.

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Suppose two individuals (Smith and Jones) each have 10 hours of labor to devote to producing either ice cream (x) or chicken soup (y). Smith's utility function is given by img where as Jones' is given by img The individuals do not care whether they produce x or y, and the production function for each good is given by x = 2l and y = 3l, where l is the total labor devoted to production of each good. a. What must the price ratio, p x /p y , be? b. Given this price ratio, how much x and y will Smith and Jones demand? Hint: Set the wage equal to 1 here. c. How should labor be allocated between x and y to satisfy the demand calculated in part (b)?
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A utility function represents individual's preferences for consuming bundle of goods and services, it calculates individual's preferences in numerical terms. Utility function assign a numerical value to a particular bundle according to its preferences. 
As per the question, Person S and J each have 10 hours of labour for the production of ice-cream or chicken. The quantity of ice-cream and chicken is represented by x and y respectively.
The utility function of Person S is:
img The utility function of Person J is:
img The production function of ice-cream and chicken are:
img where, l is labor employed
a.
In order to calculate the price ratio,
img , first profit is maximized for both ice-cream and chicken.
The profit of ice-cream is maximized. The value of x is put in the given situation as shown below:
img In this the total revenue earned from ice-cream is subtracted from total cost incurred.
Now, it is differentiated with respect to labor, to maximize the profit.
img The value of p x is
img . …… (1)
The profit of chicken is maximized. The value of y is put in the given situation as shown below:
img In this the total revenue earned from chicken is subtracted from total cost incurred.
Now, it is differentiated with respect to labor, to maximize the profit.
img The value of p y is
img . …… (2)
Now, the price ratio is calculated by using values from (1) and (2) as shown below:
img Dividing these two equations, the following is derived:
img Thus, the price ratio is
img .
b.
First, the quantity demanded by Person S is calculated. In order to calculate the quantity demanded of x and y , the utility function of Person S is maximized subject to his budget constraint.
The total number of labor hours is 10. The values of price of ice-cream and chicken are taken from (1) and (2). The budget constraint of Person S is shown below:
img In the above equation, x 1 is ice-cream demanded by Person S
y 1 is chicken demanded by Person S
The condition of utility maximization is shown below:
img Subject to
img .
Now, using Lagrangian, the following is obtained as shown below:
img img …… (3)
img …… (4)
img Now, dividing equations (3) and (4), the following is derived:
img Therefore, the value of y 1 is
img . …… (5)
The value obtained of y 1 in (5) is substituted in the budget constraint in following manner as shown:
img The value of y 1 is calculated as shown below:
img Thus, for Person S x 1 is
img and y 1 is
img .
Similarly, the quantity demanded for Person J is calculated. The budget constraint for Person J is shown below:
img In the above equation, x 1 is ice-cream demanded by Person J
y 1 is chicken demanded by Person J
The condition of utility maximization is shown below:
img Subject to
img .
Now, using Lagrangian, the following is obtained as shown below:
img img …… (6)
img …… (7)
img Now, dividing equations (6) and (7), the following is derived:
img Therefore, the value of y 1 is
img . …… (8)
The value obtained of y 1 in (8) is substituted in the budget constraint in following manner as shown:
img The value of y 1 is calculated as shown below:
img Thus, for Person S x 1 is
img and y 1 is
img .
c.
It is given that the production function of ice-cream and chicken are:
img where, l = labor employed
To calculate the number of hours spent on the production of ice-cream, first total number of ice-cream produced is calculated.
The total number of ice-cream produced is computed as follows:
img Now, the labor allocated to production of ice-cream is calculated as follows:
img Thus, the labor devoted to the production of ice-cream is
img .
To calculate the number of hours spent on the production of chicken, first total number of chicken produced is calculated.
The total number of chicken produced is computed as follows:
img Now, the labor allocated to production of Chicken is calculated as follows:
img Thus, the labor devoted to the production of Chicken is
img .

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Consider an economy with just one technique available for the production of each good. img a. Suppose land is unlimited but labor equals 100. Write and sketch the production possibility frontier. b. Suppose labor is unlimited but land equals 150. Write and sketch the production possibility frontier. c. Suppose labor equals 100 and land equals 150. Write and sketch the production possibility frontier. Hint: What are the intercepts of the production possibility frontier? When is land fully employed? Labor? Both? d. Explain why the production possibility frontier of part (c) is concave. e. Sketch the relative price of food as a function of its output in part (c). f. If consumers insist on trading 4 units of food for 5 units of cloth, what is the relative price of food? Why? g. Explain why production is exactly the same at a price ratio of img . h. Suppose that capital is also required for producing food and clothing and that capital requirements per unit of food and per unit of clothing are 0.8 and 0.9, respectively. There are 100 units of capital available. What is the production possibility curve in this case? Answer part (e) for this case.
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a) In this case, labor has a limit of 100. The PPF would look like below.
img The corresponding PPF graph is given below in Figure 1.
img Figure 1
b) In this case, land has a limit of 150. The PPF would look like below.
img The corresponding PPF graph is given below in Figure 2.
img Figure 2
c) The combined PPF (ABC) is shown below in Figure 3.
img Figure 3
d) The combined PPF is concave because of increasing rate of product transformation. In the case of labor, RPT is 1 and in the case of land, RPT is 2.
e) The relative price is shown by curve DBE in Figure 4. The slope of curve represents the relative price. Before B, slope is 1 and after B, slope is 2.
img Figure 4
f) The individual can buy 4 units of food by paying 5 units of cloth. So, the relative price is 5/4.
g) We have seen in Figure 4, the way the relative price changes at point B. The relative price rises instantaneously from 1 to 2. So, the production would remain the same whatever it is at point B.
h) The PPF for capital is given in equation below and sketched in Figure 5.
img img Figure 5

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Suppose that Robinson Crusoe produces and consumes fish (F) and coconuts (C). Assume that, during a certain period, he has decided to work 200 hours and is indifferent as to whether he spends this time fishing or gathering coconuts. Robinson's production for fish is given by img and for coconuts by img where lF and lC are the number of hours spent fishing or gathering coconuts. Consequently, img Robinson Crusoe's utility for fish and coconuts is given by img a. If Robinson cannot trade with the rest of the world, how will he choose to allocate his labor? What will the optimal levels of F and C be? What will his utility be? What will be the RPT (of fish for coconuts)? b. Suppose now that trade is opened and Robinson can trade fish and coconuts at a price ratio of p F /p C = 2/1. If Robinson continues to produce the quantities of F and C from part (a), what will he choose to consume once given the opportunity to trade? What will his new level of utility be? c. How would your answer to part (b) change if Robinson adjusts his production to take advantage of the world prices? d. Graph your results for parts (a), (b), and (c).
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Smith and Jones are stranded on a desert island. Each has in his possession some slices of ham ( H ) and cheese ( C ). Smith is a choosy eater and will eat ham and cheese only in the fixed proportions of 2 slices of cheese to 1 slice of ham. His utility function is given by U S = min( H , C/ 2). Jones is more flexible in his dietary tastes and has a utility function given by U J = 4 H+ 3 C. Total endowments are 100 slices of ham and 200 slices of cheese. a. Draw the Edgeworth box diagram that represents the possibilities for exchange in this situation. What is the only exchange ratio that can prevail in any equilibrium? b. Suppose Smith initially had 40 H and 80 C. What would the equilibrium position be? c. Suppose Smith initially had 60 H and 80 C. What would the equilibrium position be? d. Suppose Smith (much the stronger of the two) decides not to play by the rules of the game. Then what could the final equilibrium position be?
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In the country of Ruritania there are two regions, A and B. Two goods (x and y) are produced in both regions. Production functions for region A are given by img here l x and l y are the quantities of labor devoted to x and y production, respectively. Total labor available in region A is 100 units; that is, img Using a similar notation for region B, production functions are given by img There are also 100 units of labor available in region B: img a. Calculate the production possibility curves for regions A and B. b. What condition must hold if production in Ruritania is to be allocated efficiently between regions A and B (assuming labor cannot move from one region to the other)? c. Calculate the production possibility curve for Ruritania (again assuming labor is immobile between regions). How much total y can Ruritania produce if total x output is 12? Hint: A graphical analysis may be of some help here.
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Use the computer algorithm discussed in footnote 24 to examine the consequences of the following changes to the model in Example 13.4. For each change, describe the final results of the modeling and offer some intuition about why the results worked as they did. a. Change the preferences of household 1 to img : b. Reverse the production functions in Equation 13.58 so that x becomes the capital-intensive good. c. Increase the importance of leisure in each household's utility function. Reference: Equation 13.58 img
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Tax equivalence theorem Use the computer algorithm discussed in footnote 24 to show that a uniform ad valorem tax of both goods yields the same equilibrium as does a uniform tax on both inputs that collects the same revenue. Note: This tax equivalence theorem from the theory of public finance shows that taxation may be done on either the output or input sides of the economy with identical results.
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Returns to scale and the production possibility frontier The purpose of this problem is to examine the relationships among returns to scale, factor intensity, and the shape of the production possibility frontier. Suppose there are fixed supplies of capital and labor to be allocated between the production of good x and good y. The production functions for x and y are given (respectively) by img where the parameters ?, ?, ?, and ?, will take on different values throughout this problem. Using either intuition, a computer, or a formal mathematical approach, derive the production possibility frontier for x and y in the following cases. img Do increasing returns to scale always lead to a convex production possibility frontier? Explain.
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The trade theorems The construction of the production possibility curve shown in Figures 13.2 and 13.3 can be used to illustrate three important ''theorems'' in international trade theory. To get started, notice in Figure 13.2 that the efficiency line O x , O y is bowed above the main diagonal of the Edgeworth box. This shows that the production of good x is always ''capital intensive'' relative to the production of good y. That is, when production is efficient img no matter how much of the goods are produced. Demonstration of the trade theorems assumes that the price ratio, p = p x / p y , is determined in international markets-the domestic economy must adjust to this ratio (in trade jargon, the country under examination is assumed to be ''a small country in a large world''). a. Factor price equalization theorem: Use Figure 13.4 to show how the international price ratio, p, determines the point in the Edgeworth box at which domestic production will take place. Show how this determines the factor price ratio, w / v. If production functions are the same throughout the world, what will this imply about relative factor prices throughout the world? b. Stolper-Samuelson theorem: An increase in p will cause the production to move clockwise along the production possibility frontier- x production will increase and y production will decrease. Use the Edgeworth box diagram to show that such a move will decrease k/l in the production of both goods. Explain why this will cause w / v to decrease. What are the implications of this for the opening of trade relations (which typically increases the price of the good produced intensively with a country's most abundant input). c. Rybczynski theorem: Suppose again that p is set by external markets and does not change. Show that an increase in k will increase the output of x (the capital-intensive good) and reduce the output of y (the labor-intensive good). Reference: img img img
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An example of Walras' law Suppose there are only three goods ( x 1 , x 2 , x 3 ) in an economy and that the excess demand functions for x 2 and x 3 are given by img a. Show that these functions are homogeneous of degree 0 in p 1 , p 2 , and p 3. b. Use Walras' law to show that, if ED 2 = ED 3 = 0, then ED 1 must also be 0. Can you also use Walras' law to calculate ED 1 ? c. Solve this system of equations for the equilibrium relative prices p 2 / p 1 and p 3 / p 1. What is the equilibrium value for p 3 / p 2 ?
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Productive efficiency with calculus In Example 13.3 we showed how a Pareto efficiency exchange equilibrium can be described as the solution to a constrained maximum problem. In this problem we provide a similar illustration for an economy involving production. Suppose that there is only one person in a two-good economy and that his or her utility function is given by U ( x , y ). Suppose also that this economy's production possibility frontier can be written in implicit form as = 0. a. What is the constrained optimization problem that this economy will seek to solve if it wishes to make the best use of its available resources? b. What are the first-order conditions for a maximum in this situation? c. How would the efficient situation described in part (b) be brought about by a perfectly competitive system in which this individual maximizes utility and the firms underlying the production possibility frontier maximize profits. d. Under what situations might the first-order conditions described in part (b) not yield a utility maximum?
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Initial endowments, equilibrium prices, and the first theorem of welfare economics In Example 13.3 we computed an efficient allocation of the available goods and then found the price ratio consistent with this allocation. That then allowed us to find initial endowments that would support this equilibrium. In that way the example demonstrates the second theorem of welfare economics. We can use the same approach to illustrate the first theorem. Assume again that the utility functions for persons A and B are those given in the example. a. For each individual, show how his or her demand for x and y depends on the relative prices of these two goods and on the initial endowment that each person has. To simplify the notation here, set p y = 1 and let p represent the price of x (relative to that of y). Hence the value of, say, A's initial endowment can be written as img b. Use the equilibrium conditions that total quantity demanded of goods x and y must equal the total quantities of these two goods available (assumed to be 1,000 units each) to solve for the equilibrium price ratio as a function of the initial endowments of the goods held by each person (remember that initial endowments must also total 1,000 for each good). c. For the case x A = y A = 500, compute the resulting market equilibrium and show that it is Pareto efficient. d. Describe in general terms how changes in the initial endowments would affect the resulting equilibrium prices in this model. Illustrate your conclusions with a few numerical examples.
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Social welfare functions and income taxation The relationship between social welfare functions and the optimal distribution of individual tax burdens is a complex one in welfare economics. In this problem, we look at a few elements of this theory. Throughout we assume that there are m individuals in the economy and that each individual is characterized by a skill level, a i , which indicates his or her ability to earn income. Without loss of generality suppose also that individuals are ordered by increasing ability. Pretax income itself is determined by skill level and effort, ci, which may or may not be sensitive to taxation. That is, I i = I ( a i , c i ). Suppose also that the utility cost of effort is given by img : Finally, the government wishes to choose a schedule of income taxes and transfers, T(I i ), which maximizes social welfare subject to a government budget constraint satisfying img (where R is the amount needed to finance public goods). a. Suppose that each individual's income is unaffected by effort and that each person's utility is given by img Show that maximization of a CES social welfare function requires perfect equality of income no matter what the precise form of that function. (Note: for some individuals T(I i ) may be negative.) b. Suppose now that individuals' incomes are affected by effort. Show that the results of part (a) still hold if the government based income taxation on ai rather than on Ii. c. In general show that if income taxation is based on observed income, this will affect the level of effort individuals undertake. d. Characterization of the optimal tax structure when income is affected by effort is difficult and often counterintuitive. Diamond shows that the optimal marginal rate schedule may be U-shaped, with the highest rates for both low- and high income people. He shows that the optimal top rate marginal rate is given by img where img is the top income person's relative weight in the social welfare function and e L,w is the elasticity of labor supply with respect to the after-tax wage rate. Try a few simulations of possible values for these two parameters, and describe what the top marginal rate should be. Give an intuitive discussion of these results.
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