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Unformatted text preview: A (very quick) review of consumer theory Preferences Assume a utility function of U = X°‘5 Y0"5 where X is the quantity of that good consumed ,Y is the quantity of that
good consumed, and U is in units of utility ("satisfaction” ,”happiness”, "hours studying econ ”). Suppose we ask
the question: What combinations of X and Y yield a utility level of 10? That is, what combinations solve this equation 10 = X0'5 Y0'5 ? The table below gives us a few combinations (we call these combinations ”bundles”) that
solve this equation. Bundle X Y
A 1 100
B 2 50
C 3 33.33 If we plot this bundles on a graph we get something called an indifference curve. An indifference curve shows all
bundles that yield a given utility and therefore shows bundles which the consumer is indifferent between.
Y 10 50'
33.3 ' indifference Curveu=10 123 X Suppose we repeat the question, but now for a utility level of 129 12 = XO’SY
We would get a new table and a new graph. 0.5 ? Bundle X Y
A 1 144
B 2 72
C 3 48 Y 100
72 so
ICU=12 Indifference Curveu=1o 48
33.33 ' 1 2 3 X Higher indifference curves correspond to higher levels of utility. The slope of the IC is known as the marginal rate of substitution (MRS). The MRS is defined as the maximum
amount of good Y a consumer is willing to give up to get one more unit of X. For example, referring to the top
graph, if the consumer is initially at bundle X=1 and Y=100, then this particular consumer is willing to give up at
most 50 Y to get one more X. Notice that this trade would leave the consumer on the same IC, hence he or she is
just willing to make this trade. Another important feature of the MRS is that MRS = MUX/MUY = (marginal utility of X)/(marginal utility of Y):
(GU/6X)/(6U/6Y). [ proof not given] We frequently make use a special type of utility function known as the Cobb—Douglas utility function.
The Cobb—Douglas utility function is U = Xa'Yb where a and b are positive constants.
The MRS for CobbDouglas utility functions is MRS = slope of IC = MUx/MUY = (au/ox)/(oU/av) = (ax“‘1 Yb)/(bXaYb'1) = (a/b)(Y/X). Don’t worry about the
derivation of this, just know it. The Budget Line represents the bundles actually obtainable by the consumer and is expressed by the equation I = PXX+PyY where I 2 money income, P, = money price of X and Pv = money price of Y. If, for example, i = $100, PX =
SlO/X and PV = SS/Y then the budget line is: Equation for the budget line: 100 = 10X + 5Y 10 X
A few things to notice about the budget constraint. a. the consumer cannot consume bundles beyond this line
(hence the name ”constraint”). b. the slope of the budget constraint is (PX/FY). This ratio (PX/PY) is, in this
example, ($10/X)/($5/Y) = 2Y/X. This ratio is referred to as the relative price of X since it tells us how many units
of Y the consumer must give up to get one unit of X. NOTICE that (Px/Py) is the slope of the budget line. Consumer Optimization The standard assumption is that consumers desire to choose the bundle the maximizes their utility. Expressed
differently, for a given budget constraint find the bundle on the highest IC—the IC that is tangent to the budget
constraint (we are ignoring corner solutions). It looks like this: 20 10 X Notice two conditions for consumer optimization: a. slope of IC = slope of budget line 9 MRS = (PX/PY) AND
b. we are ON the budget constraint > I: PXX+PVY. A very quick review of production theory. A production function relates output to inputs. Specifically, it tells us the maximum amount of output we can
produce from a given set of inputs and a given technology. A commonly used production function is (yes, you
guessed it) the CobbDouglas production function and is given by (for good X): X = LaKb where L = labor and K: capital. There is usually a technology parameter included in this equation, but for
our purposes we can ignore this (at least for now). The exponents ”a” and "b" are positive constants. We shall be
assuming that a+b = 1. This means we are assuming constant returns to scale—doubling all inputs means exactly
doubling output. Example: Suppose X = LO‘SKO‘5 . What combinations of inputs yield an output level of, say, X =4? The table below gives a few
such combinations. Combination L K A 1 16
B 2 8
C 3 5.33 If we plot these points, we get a curve known as an isoquant. An isoquant shows the combinations of L and K that
yield a given (maximum) output level. In this case, an output level of 4. Higher isoquants correspond to higher
levels of output (not shown on graph). 5.33. . lsoquantx=4 123 L Suppose we start at point c (L=3 and K =5.33). If we reduce L by one unit (so AL = 1 in absolute value terms), then
how many units of K would the firm need to add to keep output at the same level of 4? As you can see from both
the table and graph, it would be 2.66 (so AK = 2.66). For ”small” changes we can see that amount of K that must be
substituted for one unit of labor, holding output constant, is given by AK/AL = 2.66/1 = 2.66 K per L. This is the
slope of the isoquant. The slope of the isoquant is known as the marginal rate of technical substitution (MRTS) and,
as already mentioned, MRTS = the amount of K that must be substituted for one unit of labor, holding output
constant. IMPORTANT: MRTSx = MPLx/MPKX = (ox/au/(ox/ak) = (aLa'l Kb)/(bLaKb'1) = (a/b)(K/L) The other aspect of production theory is the isocost curve. It is given by the equation TC = wL + rK where TC is total
cost of production, w is the money wage rate (price per unit of labor) and r is the rental rate of capital (price per
unit of capital). For example, if TC = $100, w = $10/L and r = $5/K then the isocost curve is: 100 = 10L + 5K
20 lsocost curveTc=1oo 10 L
The isocost curve shows the various combinations on inputs, L and K, a firm can use for a given TC. If TC was instead at TC = 50, then the curve would like
K 100 = 10L + 5K
20
10 isocost curveTc=50 lsocost curveTc=1oo Notice that other things constant, lower isocost curves correspond to lower total costs. Also NOTICE that the slope
of the isocost curve = w/r. In this case, the slope of the above isocost curves = w/r = ($10/L)/($5/K) = 2K per L. In
other words, for every unit of L used by the firm it must give up 2 units of capital. This ratio,w/r, is known as the
relative price of labor since it tells us how many units of capital must be given up to use one more unit of labor. Producer optimization If we combine the isoquant and isocost curves we can talk about producer optimization. There are two ways to
think about the optimization problem. First, for given TC, find the input combination that maximizes output (i.e.,
for a given isocost curve find the highest possible isoquant—the isoquant tangent to the isocost). Alternatively, we
can pose the question as for a given output level, choose the input combination of L and K that minimizes TC. This
means for a given isoquant find the lowest possible isocost curve—the isocost curve that is tangent to the
isoquant. K
For given TC, find L and K that maximizes X. L
K For given X, find L and K that minimizes TC. TC* < TC
TC L
EITHER WAY, the condition for producer optimization is that the slope of isoquant = slope of the isocost —> MRTS
: w/r 9 MPL/MPK = w/r END OF REVIEW TWO GOODS (X&Y), TWO INPUTS (L & K) AND TWO COUNTRIES (A & B) I. We need to first know what is meant by "capital intensive” and ”labor intensive”. These terms refer to
comparisons across goods. We say that X is ”capital intensive" if (Kx/Lx) > (Ky/Ly) at any given set of input prices. In
words, if the amount of capital per worker in industry X exceeds the amount of capital per worker in industry Y,
then X is said to be capital intensive. For example, the manufacturing sector will likely be capital intensive relative
to the service sector. If we interpret K as land, then agriculture would be land intensive relative to manufacturing.
We say that X is "labor intensive” if (LX/Kx )> (Ly/Ky) at any given set of input prices. Obviously, if one industry is
capital intensive, then the other industry must be labor intensive. Here is an example. Assume the following production functions for goods X and Y. x = L,”3 K,“ and Y = L,”3 K,”3 1. In order to find the optimal combination of L and K for a given TC, we want to make sure the slope of the
isoquant (MRTS) equals to the slope of the isocost (w/r). So naturally we would first find the respective MRTS.
MRTSX = slope of isoquant X = MPLX/MPKX = (1/3)/(2/3) KX/Lx = 1/2 KX/LX MRTSV = slope of isoquant Y = MPLy/MPKy = (2/3)/(1/3) Ky/Ly = 2Ky/Lv 2. Now that we have MRTS for each good, and by costminimization, we can set MRTS = slope of isoquant = w/r =
slope of isocost and solve for the respective K. For purposes of this example, I shall be assuming w/r = 2/1 = 2 and
TC = $12. If we produce X: MRTSx = 1/2 KX/LX = w/r = 2 9 K,< = 4Lx If we produce Y: MRTSV = 2Ky/Lv = w/r =2 9 Ky = LV 3. The next step is to substitute the result from step (2) into the isocost curve equation (recall that we are
assuming TC is fixed at 12) and solve for respective L: For X: 12 = ZLX + KX which by substitution gives us 12 = ZLX + 4LX = 6LX or L)( = 2. For Y: 12 = 2Ly + Ky which by substitution gives us 12 = 2Lv + LV = 3Lv or LV = 4. 4. We can now solve for the respective values of K ForX: by K,,=4Lx =8 and ForY: by Ky: Ly =4 SUPPOSE TC was instead 24. If we repeat the above steps we get: For X: 1/2 KX/LX = 2 9 KX = 4Lx 9 24 = 2LX + KX which by substitution gives us
24=2LX+4LX=6oner=4. 9 bny=4Lx =16 For Y: 2Kv/Ly = 2 9 Ky = LV 9 24 = 2Lv + Ky which by substitution gives us
24=2Ly+Ly=3LyorLy=89 byKy=Lv :3. Two things to notice: a. As jump from TC =12 to TC =24, the optimal combination of L and K for each industry exhibits the m K/L ratio.
For X: KX/LX = 8/2 = 16/4 (or 4) and for Y: Ky/Ly = 4/4 = 8/8 (which is 1). This can also be seen in the graph below.
The slope of the ray through the origin and through the optimal tangency points (this line is called the expansion
path) gives the K to L ratio for that good. b. The other thing to notice is that X is the capital intensive good. For this given w/r ratio it always the case that
(Kx/Lx) > (Ky/Ly) [4 >1]. This can be seen by comparing the expansion paths in the graph below. Again, the slope of
the expansion path is the K to L ratio for the respective good. The slope of the X expansion path is greater than the
slope of the Y expansion path. Even if we used a different w/r ratio we still get (Kx/Lx) > (Ky/Ly) , although at values
different from 4 and 1. [Trick: to determine which good is capital intensive, look at the exponent in the production function for each good.
The good with the larger exponent for K will be the capital intensive goodl] Capital (K) fExpansion Path for X (slope = Kx/Lx)
24 _." X*
Expansion Path for Y (slope = Ky/Ly) 2 4 6 8 12 Labor (L) II. The next step in our analysis is to understand what is meant by ”labor abundance" and ”capital abundance”.
These terms refer to comparisons across countries. We say that country A is capital abundant if (K/L)A > (K/L)B. In
words, if country A has a larger capital to labor ratio than B, then A is said to be capital abundant. For example, we
would say the US. is capital abundant relative to India because the US. has more capital per worker than does India. Likewise, a country is labor abundant if it has a higher labor to capital ratio than another country. Thus,
India is labor abundant relative to the us. To understand the implications of capital or labor abundance, we need to see what it implies about the PPF of each
country. We will again be assuming X = Lxl/3 KX 2” and Y = LVZ/3 Ky”3 for both countries. This means that they have
the same technology. [NOTE: The Ricardian model explains comparative advantage by technology differences —
different MPLs across countries. Here we see how factor endowments can possibly explain the source of
comparative advantage, so to isolate on this explanation we assume technology is the same] In country A, assume L = 27 and K = 1 and in country B, L = 1, K = 27. We would say that B is capital abundant since
(K/L) is higher in B than in A (27/1 > 1/27). Likewise, we would say A is labor abundant since (UK) is greater in A
than in B. What does this imply about the shape of the PPF? I will employ the following ”trick” to answer this
question: find the respective intercepts of the PPF for each country. For example, suppose country A only produces good X (so Y = 0). Then, X = L,“3 me = (27)1/3(1)2/3 = 3. Now suppose A produces only good Y (so X=0). Then, Y = LYN3 Ky“3 = (27)”3 (1)“3 = 9. Using this information I can
draw in the following PPF. [NOTE: Although I am not offering a proof, it won’t be linear. See below.] Y
9 Country A 3 X Notice that the curve is ”stretched out” along the vertical axis. Why? Because A is labor abundant and Y is the labor
intensive good. That is, Y is good that tends ”to use lots of labor” and country A has "lots of labor”! Let us now do the same thing for B. Suppose country B only produces good X (so Y = O). Then,X = L,”3 me: (1)“3
2 /3 3
(27) 2/3 = 9. Now suppose B produces only good Y (50 X=0). Then, Y = Ly B Kyl/3 = (1)2 (27)” = 3.
Y Country B 9 X
Notice that the curve is ”stretched out” along the horizontal axis. Why? Because B is capital abundant and X is the
capital intensive good. That is, X is a good that tends ”to use lots of capital” and country B has ”lots of capital”. See if this helps: suppose we think of K as land. Agriculture (X) is very land Intensive. So a country with lots of land
will tend to be able to produce (relatively) large amounts of X. Ill. Equations for the PPF curves. An equation that would describe A’s PPF is: 81 = 9X2 + Y2 (don’t worry about the
derivation—too hard—l will give you this equation). For B, PPF would be 81 = X2 + 9Y2. [We interrupt this story for a brief review from Econ 11 or 101. The above equations are what are called implicit
functions. An implicit function f(X, Y) = 0. If we take the total differential of this function we get 0 =fXdX +fde (fX is the partial derivative of the function with respect to X). Rearranging this (ignoring the negative
sign) we get dY/dX =fX/fy which is the slope of the PPF! The slope of PPF, also called the marginal rate of
transformation(MRT), was discussed in an earlier lecture and as we explained at that time it represents the
opportunity cost of X in terms of Y. 50 ifl ask to find the MRT for country A, you would proceed as follows: For 81 = 9x2 + y2 the MRT =fx /fy = 18X/2Y = 9X/Y. For B, 81 = x2 + 9y2 9 MRT = 2X/18Y : X/9Y.] IV. We will now bring preferences (utility functions) into the model. Suppose each country has the same
preferences and is given by the utility function U =XY. The reason for assuming identical preferences is to isolate
on the role of factor endowments. Obviously, countries may not have the same preference and we can address
that later. Pre—trade. Country A wants to be on the highest possible IC. This means the IC that is tangent to the PPF. To solve
for the U max combination ofX and Y we do the following. a. U maximization requires that the slope of lC = slope of PPF 9 MRS = MRT. For this problem, Y/X = 9X/Y which equals Y2 = 9X2 or Y = 3X. b. Plug this result into A's PPF and solve for x. 81 = 9x2 + v2 = 9x2 + 9x2 =18x2 9 x = 3/v2 c. Since Y = 3X, Y equals 9/\/2. d. The implicit equilibrium price (see earlier lecture) is equal to slope of the PPF or, as we now see, the slope of the
IC. So we can use either MRT or MRS to find (PX/PY). (PX/PY) = MRS = Y/X = (9/v2)/(3/v2) = 3 Y per X. e. U = XY = 27/\/2. Pre—trade situation in Country A (the labor abundant country) Y /slope=MRT=MRS=(Px/PY)=3Y perX 3N2 3 X Pre—trade. Country B wants to be on the highest possible lC. This means the IC that is tangent to the PPF. To solve
for the U max combination of X and Y we do the following.
a. U max requires that the slope of IC = slope of PPF ) MRS = MRT. For this problem, Y/X = X/9Y which equals
ex2 = x2 or Y = (1/3)x.
b. Plug this result into B’s PPF and solve for X. 81 = X2 + 9Y2 = X2 + X2 =2X2 ) X = 9/\/2
c. Since Y = (1/3)X, Y is 3/V2.
d. The implicit equilibrium price (see earlier lecture) is equal to slope of the PPF or, as we now see, the slope of the
IC. 50 we can use either MRT or MRS to find (PX/PY). (PX/Py) = MRS = Y/X = (3/\/2)/(9/\I2) = 1/3 Y per X.
e. U = XY = 27/\I2. Y Pretrade situation in country B ICU= 27/V2 slope = MRT = MRS = (PX/PY) = (1/3)Y perX V. Trade. The price of good X is relatively high in country A compared to country B9 3Y/X > (1/3)Y/X. Country A would
therefore like to import good X from country B. Likewise, Y is relatively expensive in country B, so B would like to
import good Y from country A. As good X ”floods into” country A this will drive down its price in that country. In country B, the increased
competition from A’s consumers will drive the price of X up in B. The new price will be somewhere between 3Y and
1/3Y per X. In other words, 1/3Y per X < (PX/Py)wo..d < 3Y/X and this means that the slope of the isorevenue
curve or budget line (recall that is simply (PX/Py) ) will get flatter in country A and steeper in country B with the
following consequences: Country A after—trade (slope is something less than 3Y per X)
Pre—trade (slope is 3Y per X) Xp 3/v2 3 X
Country A will decrease its production of X and increase its production of Y. it will export good Y and import X (not
shown on the above graph). Y Country B Pretrade (slope if 1/3 Y per X)
X Country B will increase its production of X and decrease its production of Y. It will import good Y and export X (not
shown on the above graph). Conclusion: One of impacts of moving from no trade to trade is that the country that is labor abundant will
increase its production of the labor intensive good (thus, country A increased its production of Y). This country will then export that good for the other good. The country that is capital abundant will increase its
production of the capital intensive good (thus, country B increased its production of X). This argument implies that we should observe countries that are abundant in particular inputs exporting those
goods that are intensive in the input. If agriculture is intensive in land, then we should observe countries that are
land abundant exporting agriculture goods. If high—tech goods are intensive in skilled, trained labor, then we
should observe countries that are skill labor intensive exporting those goods. This will have further implications as we shall see in a later lecture. X67: WRODVMTLUW 0 )L X q nua Maﬁa
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 Spring '10
 McDevitt

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