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Sample NIH non-modular Budget Justification

By Mario Rogers,2014-11-11 22:30
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Sample NIH non-modular Budget Justification

Topic #3 General Equilibrium

    Our consumer theory will have a framing premise and three assumptions. A framing premise is something that can’t be proven – it’s either accepted or rejected. Our framing

    premise is that individuals are self-interested. That is, they act to be made as well off as possible under the circumstances. Further, our consumers are rational and goal-directed. We assume that they act in a purposeful manner. This does not mean that our consumers are selfish. Being self-interested does not mean you benefit from depriving others.

For example, let Carol’s ordinal utility be represented by the following equation,

     U = F + B,

    where F is footballs and B is books. An indifference curve connects all the consumption

    bundles that generate the same level of utility. Thus, a consumer is indifferent among the consumption bundles on a particular indifference curve. For example, the following indifference curves represents X units of utility. Higher indifference curves yield more utility. Thus, consumption bundles to the north-east are superior bundles and consumption bundles to the south-west are inferior bundles.

     Footballs Figure 1

     SUPERIOR BUNDLES

     36 a d

     b

     16 INFERIOR c

     BUNDLES

     1

     4 16 49 Books

In addition, our theory has three assumptions:

1. Completeness - the consumer has and can rank in order of preference all

    consumption items. No flipping of coins or enemeneminemo.

    2. Transitivity - preferences are rational and consistent. No rock, paper, scissors. The following indifference map violates the assumption of transitivity. For example, try to rank consumption bundles a, b, and c.

    Consumption Good Y

     a

     c

     b

    Consumption Good X

    It appears that a>b, a=c, and b=c, which does not make sense. So, we conclude that indifference curves can't intersect.

    3. Nonsatiation - more is preferred to less. This means that indifference curves can’t be positively sloped and that indifference curves can’t be fat.

The Marginal Rate of Substitution (MRS) is the amount of the consumption good YX

    measured on the Y axis that an individual would be willing to give up in order to obtain another unit of the consumption good measured on the X axis. We assume that the marginal rate of substitution is diminishing as we move from left to right along an indifference curve. This makes indifference curves convex.

For example MRSis the maximum amount of footballs the consumer is willing to give FB

    up to obtain another book.

Find the MRS between bundles “a” and “b” and “b” and “c”.

Does the MRS increase or decreases as we move from left to right?

Suppose consumption bundle “e” has 49 footballs and 1 book. Calculate the MRS

    between “e” and “a”.

     Footballs Figure 1

     36 a

     b

     16 c

     1

     16 49 Books

Formally, ?U = MU(?F) + MU(?B). Since utility is constant along an indifference FB

    curve, 0 = MU(?F) + MU(?B) when calculating MRS. Thus, MU/MU = ?F/?B. FBBF

    To see why the MRS is diminishing, suppose I give you 10 large pizzas (each with 10 slices) and 1 six-pack of coke. What is the maximum number of slices of pizza would you be willing to give up in order to obtain one more six-pack of coke?

    Suppose instead that I give you 1 large pizza (with 10 slices) and 10 six-packs of coke (60 cokes). What is the maximum number of slices of pizza would you be willing to give up in order to obtain one more six-pack of coke?

    Now, is your answer to the first question larger than to the second? If so, then you exhibit a diminishing MRS. Again, this is why indifference curves are convex to the origin.

     2

    Suppose Bill really likes footballs relative to books and that a nerd really likes books relative to footballs. Draw a set of indifference curves for Bill and a set for the nerd on the same graph that represent the above information about their preferences. If they intersect each other, have we violated an axiom?

Footballs

     Nerd

     Bill

     Books

    Thus far, our theory of consumer choice has suggested that to maximize utility, individuals will consume an infinite amount of consumption goods. However, this does not represent the behavior of consumers in the real world because consumers are constrained by scarcity. In particular, consumption goods cost money and consumers have limited resources. Therefore, we must derive a budget constraint that illustrates the consumption bundles that are attainable given a consumer’s scarce resources.

Suppose that a consumer has income of I, consumption good 1 costs p per unit, and 1

    consumption good 2 costs p per unit. The following budget constraint defines what 2

    combinations of consumption goods 1 and 2 that are attainable.

     Good 2

     I/p2

    I/p Good 1 1

    Note that the slope of the budget constraint is p/p. In other words, the opportunity cost 12

    in terms of good 2 of consuming 1 more unit of good 1 is p/p. The slope of the budget 12

     3

    constraint is always given by the price of the good measured on the X axis divided by the price of the good measured on the Y axis.

    Footballs Nice, But Unattainable

     b a

     c

     d

     Attainable,

     But Inferior

     Books

    We can rank the consumption bundles as follows: a > c > b = d. But, consumption bundle a is unattainable. So, consumption bundle c maximizes utility.

    Now consider an economy with only two consumers. Let the consumers have endowment bundles of two goods. Further, let the two consumers trade to achieve the highest possible utility. We depict this with an Edgeworth box.

    Suppose Mr. Smith and Ms. Jones live on an island with no contact with the outside world. Each owns a part of the island and, with no labor expended, each obtains a positive quantity of both apples and bananas from his/her part of the island. No production is possible.

     SSSJ JJLet endowments be = (, ) >> 0 and = (, ) >> 0. abab

    Apples Jones

     Bananas

     W

     W*

     Bananas

     Smith Apples

     4

    Mr. Smith and Mrs. Jones will engage in trade only if their MRS are not equal at the initial endowment when that endowment lies in the interior.

     SSJJAs drawn, U/U > U/U at W*. For trade to take place, neither trader can be made abab

    worse off. Therefore, only points in the eye between the two indifference curves are

    acceptable. Mutually advantageous trade is possible because Smith can move to a higher indifference curve by trading bananas for apples, and Jones can move to a higher SSindifference curve by trading apples for bananas: (U/U) decreases until it equals abJJS(U/U), which correspondingly increases. Trade will take place only if MRS doesn’t abJequal MRS. If they are equal, such as at point W, then no gains can be had by trading.

    Suppose one or both of the 2 people have a positive endowment of only one good. The endowment must lie on the boarder of the Edgeworth box. As before, agents will trade if their MRS are not equal. However, we must consider the feasibility of the trades. In certain instances, Pareto improvements will not be feasible.

    Apples Jones

     W 2

     Bananas

     W W13

     Bananas 4 W

     Smith Apples

     SSJJSSJJAt W and W: U/U > U/U. At W and W: U/U < U/U. 12abab34abab

     SSW, W: Smith trades b for a and Jones trades a for b: (U/U) decreases until it equals 12abJJ(U/U), which increases. ab

     34SSW, W: Smith trades a for b and Jones trades b for a: (U/U) increases until it equals abJJ(U/U), which decreases. ab

     5

All of the above are feasible, so trade can take place.

    Apples Jones

     W 6

     Bananas

     W W57

     Bananas 8 W

     Smith Apples

     SSJJSSJJAt W and W, U/U < U/U. At W and W, U/U> U/U. 56abab78ab ab

     SAt W and W, Smith wants to trade a for b but W = 0. Jones wants to trade b for a, but 56aJW = 0. bSAt W and W, Smith wants to trade b for a but W = 0. Jones wants to trade a for b, but 78bJW = 0. a

Since these are not feasible, trade cannot take place.

    If the two agents trade, the trade must be feasible and acceptable (neither trader is worse off and the trades are Pareto optimal: within the eye and on the contract curve). The first part says trade to a feasible allocation must make neither agent worse off. The second part says that the trade is Pareto optimal since nobody can be made better off at any other feasible allocation without making the other agent worse off. These conditions narrow the set of trades to the section of the contract curve that lies within the eye.

     SJSJFeasible can be expressed as: x+ x = + for i = a and b. i iii

     SsSSJJJJAcceptable can be expressed as: U(x) U(), U(x) U(), and for any y x, if SSSSJJJJU(y) > U(x) then U(y) < U(x).

    Now suppose that a ship comes and from then on they are connected to world markets. In world markets, the price of apples is p and the price of bananas is p. They can trade ab

    freely at world prices. Once connected to world markets, the agents face the price ratio p/p. They are now competitive agents unable to affect the terms of the trade between abSJiiithe two goods. The trades will be the selection of x and x that maximize U(x,x) such ab

     6

    iiiithat p + p = px + pxfor i = S and J. The agents are no longer constrained by aabbaabb

    the feasibility condition defined by the dimensions of the Edgeworth Box. Thus, it is not necessary for trades to take place such that each agents’ indifference curve is tangent to

    the budget line at the same point. The first order condition to the maximization problem iiis U/U = p/p for i = S and J. Again, neither will make a trade that makes them worse ababiioff. Note also that the budget line must pass through the endowment. Since U/U = ab

    p/p for i = S and J, no further gains can be made by trading between themselves. ab

    Given these conditions (trade with the outside world), the two consumers will not trade with the outside world only if world price ratio p/p is such that the utility maximizing ab

    trade with the outside world are feasible between themselves. Each would equate MRS to p/p and the outcome would be a tangency between the agents’ indifference curves. ab

    Pareto Optimal Allocation: an allocation (x*) for which no other allocation (x’) exists with the property that the other allocation (x’) is preferred to x* by at least one consumer with the other consumer(s) being indifferent between x* and x’.

    Core or Contract Curve: the set of Pareto Optimal Allocations that dominate the initial endowment.

Walrasian Equilibrium: an allocation (x*) and a price vector (p*) such that (i) x*

    maximizes each consumer’s utility given each consumer’s budget constraint and (ii)

    demand equals supply in all markets with x*.

Excess demand function: (denoted by z) is the difference in the amount of a good

    demanded (x*) and the endowment of that good : z = x* - . xx

    Example 1: Consider the following pure exchange, Edgeworth box economy. There are 2 consumers and 2 goods. Consumer 1 has an endowment of 7 units of good 1 and 3 units of good 2 (i.e. = (7,3)), while consumer 2 has an endowment of 3 units of good 1 and 1

    7 units of good 2 ( = (3,7)). The consumers’ utility functions are given by: 2

    U = x + x and U = min{xx} 11112221,22

     where xis consumption of good 1 by consumer i. i1

    a) Find the set of Pareto optimal allocations of this economy.

    b) Find the core of the economy.

    c) Find the Walrasian equilibrium.

    The indifference curves of consumer 1 are straight lines and the indifference curves of consumer 2 are Leontief.

    a) The set of Pareto optimal allocations is the 45 degrees line in the Edgeworth Box.

    That is, the set {x,x,x,x} such that {x=x, x = x, x + x = 10, x+x = 111221221112212211211222

    10}.

    b) The core is the dark dotted line in the Edgeworth Box. It is the subset of Pareto

    optimal allocations such that consumer 1 has at least 5 units of each good and

     7

    consumer 2 has at least 3 units of each good. That is, the set {x, x, x, x} such 11211222

    that x = x 5, x=x 3, where x + x = 10 and x + x = 10}. 1112212211211222

    c) There is a unique Walresian equilibrium: x* = x*= x* = x* = 5, p*/p* = 1. 1121 122212

    This is found by maximizing utility for consumer 1: max U such that px + px = 1111212

    p7 + p3, or max(x, x): x + x + { px + px - p7 - p3} which gives the 121112111211121212

    following three first order conditions:

     1 - p = 0 1st 1 - p = 0 (which with the 1 first order condition implies that p/p = 1) 212

     px + px - p7 - p3 = 0 11121212

so, x = (7p + 3p px)/p. We know that with Leontief preferences, x = x at the 111221212122

    optimum. Therefore, (p + p)x = 7p + 3p , x = x = (7p+3p)/(p+p) = (7+3)/2 = 12111212111212

    5. So, x= x = 5 and x = x = 5. 12 112221

     10 Good 1 5 3 Consumer 2

     Good 2

     5 Core 5

     3 7

     Good 2

     Consumer 1 Good 1 5 7 10

Example 2: Consider the following pure exchange, Edgeworth Box economy. There are

    2 consumers and 2 goods. Consumer 1 has an endowment of 8 units of good 1 and 30

    units of good 2 ( = (8,30)), while consumer 2 has an endowment of 10 units of each 1

    good ( = (10,10)). The consumers’ utility functions are given by: 2

    U = xx + 12x + 3x and U = xx + 8x + 9x. 1 11121112221222122

    a) Determine the excess demand functions for the two consumers.

    b) Find an equilibrium price ratio for this economy. Sketch the situation in an

    Edgeworth Box.

Consumer 1 maximizes xx + 12x + 3x such that px + px = 8p + 30p. 1112111211121212

    The first order conditions are:

    x + 12 - p = 0, 121

    x + 3 - p = 0, 112

    and px + px - 8p - 30p = 0 11121212

     8

Solve for x and x: x = 5/2 + 21p/p (demand function of good 1 by consumer 1) 11121121

    and x = 9 + 11p/2p (demand function for good 2 by consumer 1). Thus, the excess 1212

    demand functions of consumer 1 are:

    z = 5/2 + 21p/p 8 = 21p/p 11/2 and 112121

    z = 9 + 11p/2p 30 = 11p/2p 21. 121212

    Consumer 2 maximizes xx + 8x + 9x such that px + px = 10p + 10p. 2122212212122212The first order conditions are:

    x + 8 - p = 0, 221

    x + 9 - p = 0, 212

    and px + px - 10p - 10p= 0 12122212

Solve for x and x to get x = ? + 9p/p (demand function for good 1 by consumer 2) 21222121

    and x = 1 + 19p/2p (demand function for good 2 by consumer 2). 2212

    The excess demand functions are:

    z = ? + 9p/p 10 = 9p/p 19/2 and 212121

    z = 1 + 19p/2p 10 = 19p/2p-9. 221212

At equilibrium, z + z = z + z = 0. So, z + z = 21p/p 11/2 + 9p/p 19/2 = 2111211222112121

    30 p/p 15. Hence, p/p = ? or p/p = 2 is an equilibrium price ratio and it is unique. 212112

    Therefore, x = 13, x = 20, x = 5, and x= 20. 11 12 2122

     10 5 Good 1 Consumer 2

     Good 2

     30 10

     W*

     20 20

     Good 2

     Consumer 1 8 13 Good 1

Example 3: Consider the following pure exchange, Edgeworth box economy. There are 2

    consumers and 2 goods. Consumer 1 has an endowment of 1 unit of good 1 and 2 units of

    good 2 (i.e. = (1,2)), while consumer 2 has an endowment of 2 units of good 1 and 1 1

    unit of good 2 ( = (2,1)). The consumers’ utility functions are given by: 2

    U = ;ln(x) + (1-;)ln(x) and U = ;ln(x) + (1-;)ln(x) 1111222122

     where xis consumption of good 1 by consumer i, i = 1, 2. i1

    Find the Walrasian equilibrium.

     9

The Legrangian for consumer 1 is

Max: ;ln(x) + (1-;)ln(x) + (px+px-p-2p). x11,x12111211121212

The first order conditions are: ;(1/x) + p = 0 111

    (1-;)(1/x) + p = 0 122

    and px + px p 2p = 0 11121212

From the first order conditions, x = ;(p+2p)/p and x = (1-;)(p+2p)/p. 1112112122

We can do the same for consumer 2 to get x = ;(2p+p)/p and x = (1-;)(2p+p)/p2112122122.

To find the Walrasian equilibrium, set supply equal to demand: x + x = + , or 11211121

;(p + 2p)/p + ;(2p+ p)/p = 1 + 2 121121

3;p + 3;p = 3p 121

;p = (1-;)p so p/p = ;/(1-;). 2112

Now, plug the price vector back into the demand functions to get x = 2-;, x= 2-;, 1112

    x=1+;, and x = 1+;. 2122

    Example 4: Consider a two-period economy with n individuals and a single good in each period. The n individuals have identical preferences given by the utility function U(c, c) 01

    = ln(c) + αln(c) where c denotes the individual’s period 0 consumption and c denotes 0101

    period 1 consumption. Each individual has an endowment of one unit of period 0 good and zero units of period 1 good. Each individual has an equal share in the economy’s 1/2only firm, which has production function x = β(x), where x denotes production of 101

    period 1 good and x denotes the input of period 0 good. In period 0, the firm purchases 0

    the period 0 good to use as an input in the production of the period 1 good which it sells in period 1. In period 0, each individual decides how much of the endowment to consume and how much to sell to the firm. In period 1, the individual purchases period 1 good from the firm.

    a) Let p and p denote the period 0 and period 1 prices. Set up and solve the 01

    firm’s profit maximization problem taking prices as given.

    b) First, find a representative individual’s demand functions for period 0 and

    period 1 consumption, taking prices as given. Then, find the market

    demand functions.

    c) Define Walrasian equilibrium for this economy. Normalize prices by

    taking p = 1. Solve for the equilibrium price in period 1. 0

     1/2a) The profit function is Π = pβ(x) px. The first order condition is 1000-1/2220.5pβ(x) = p. So, x = (0.5pβ/p) and x = 0.5pβ/p. Corresponding 10010110

    profits are

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