The prodution possibility frontier

Transcription

1 The prodution possiility frontier Discussion and derviation firm is descried y its production technologies, methods for transforming input factors such as laor and capital into output commodities such as coffee, computers, or linen. If the firm is operating efficiently, it is intuitive that it will not use any more of its input factors than it must, since ostensily it has to pay for the use of laor and capital and overuse will cut profits. Suppose, however, that the supply of laor and capital in the economy is fixed; the notion of efficient production is then not so intuitively defined, since what is efficiency when the factors to e used are fixed? However, the nature of production in an economy with more than one commodity is such that there is a tradeoff etween production of various goods: resources may e allocated to production of any good, so it makes sense that this allocation of resources will maximize production in some sense. Fix the laor supply at L and the capital stock at K, and let x and y e the goods to e produced. We are given production technologies x = F K x, L x, y = H K y, L y Factor market clearing tells us that laor used in production must equal total laor supply L x +L y = L and capital used in production must equal total capital stock K x + K y = K. The question we now pose is this: given a level of x we would like to produce, what is the maximum amount of y which can e produced? Notice that this is not maximization of production in any proper sense, ut rather the maximization of the production of one good suject to the production level of another; this is not altogether a ridiculous question since, in general, there are many cominations of K x, L x which will otain the same overall level of x in production. In lecture and in the text, a method which seems much different for otaining the production possiility frontier appears. We ll now show that they are equivalent to this more intuitive notion of conditional maximization. To egin, we setup a constrained optimization prolem, From this, we otain a Lagrangian max H K y, L y K y,l y s.t. x = F K x, L x K = K x + K y L = L x + L y L = H K y, L y + λ F K x, L x x + µ K K x + K y K + µ L L x + L y L First-order conditions give us L : K x L : L x L : K y L : L y 0 = λf K K x, L x + µ K 0 = λf L K x, L x + µ L 0 = H K K y, L y + µ K 0 = H L K y, L y + µ L Sustituting out the Lagrange multipliers µ K, µ L we find 0 = H K K y, L y λf K K x, L x 0 = H L K y, L y λf L K y, L y Feruary 8, 011 1

2 which is the same as Dividing the second equation y the first, we get H K K y, L y = λf K K x, L x H L K y, L y = λf L K y, L y H L K y, L y H K K y, L y = F L K x, L x F K K x, L x y the factor market clearing conditions, this ecomes H L K K x, L L x H K K K x, L L x = F L K x, L x F K K x, L x y definition, H L is MPL y and H K is MPK y and similarly for F and x; recall that marginal products are given y the derivative of production with respect to a particular input factor. Then the aove equation is MPL y MPK y = MPL x MPK x The marginal rate of technical sustitution for some good i is RTS i = MPLi MPK i. Then the aove equation is equivalent to RTS y = RTS x This equality is evaluated suject to factor market clearing; this is precisely the formula given in the ook and discussed in lecture! n example onsider a firm which has access to K = units of capital and L = units of laor. It owns the production technologies What is the production possiility frontier? x = K 1 x L 1 x y = K y L 1 y s aove, we eing y finding the marginal rate of technical sustitution for oth goods. We find MPK x = x MPK y = y K x K y = 1 Lx K x = Ly K y MPL x = x L x = 1 Kx L x MPL y = y L y = 1 Ky L y Feruary 8, 011

3 RTS x = MPL x MPK x RTS y = MPL y MPK y = K x L x = 1 Ky L y When we equate RTS x = RTS y, we find K x L x = 1 Ky L y = K x L y = K y L x Factor market clearing conditions tell us K y = K K x = K x and L y = L L x = L x. Plugging in, K x L x = K x L x 4K x K x L x = L x K x L x 4K x = + K x L x L x = 4K x + K x This gives us an explicit form for x, x = K 1 4Kx 4 x = Kx + K x + K x lthough the form for y is slightly more involved, it is still directly computale y = K x 4K x = Kx + K x 6 Kx + K x t this point, we have a parametric form for the production possiility frontier, 4 x, y = Kx, K x 6 Kx + K x + K x To otain the production possiility frontier we can vary K x etween 0 and K = and see what points come out. In general, this should e a sufficient answer and will give more than enough information for plotting the production possiility frontier on a calculator. However, often we have the aility to solve explicity for y in terms of x. While it s not ovious that we can do this here, it s worth the exercise to see how it can e done. From our equation for x, we know 4 x = Kx = 4Kx x K x x = 0 + K x Following the quadratic equation, this gives us values for K x as the roots of the aove polynomial, K x = x ± x x 8 Since x x > x, only the + root of this equation will e valid. We must then have K x = x + x x Feruary 8, 011

4 When we sustitute this into our equation for y, we get y = This form simplifies dramatically to y = x + x x x x x x +x x x +x x x x x x + x x + 48 which in turn is 1 y = 4 x x x x + x 1 x + 48 Production possiility frontier y x Figure 1: production possiility frontier; the lack curve is a straight line, and serves to show that the PPF in this setup ows out slightly. Feruary 8, 011 4

5 n exchange economy efore discussing general equilirium, we consider a simpler setup in which agents trade goods to one another and there is no production; this is referred to as an exchange economy. The most asic case of such an economy has two consumers and two goods: with fewer consumers there is no one to trade with, and with fewer goods there are no goods to trade for! In these setups, we are given agent utility functions and endowments, and are asked to find final consumption and prices. s an example, consider two consumers, and, and two goods, x and y; utility functions are given y u x, y = x y 1 u x, y = x 1 y Endowments are x e, y e, x e, y e ; this leads to udget constraints for the consumers, p x x + p y y = p x x e + p y y e p x x + p y y = p x x e + p y y e To otain prices and consumption, we setup the agents maximization prolems and solve; we will then need to apply market clearing conditions to make sure consumption is in order. While maximizing these prolems straight-up is not an issue, it would e simpler to do so without the exponents. We can ause a neat mathematical fact to do this: maximizing any monotonic increasing function of utility is identical to maximizing utility. That is, rather than maximize utility itself, we can apply maximization to an increasing transformation of utility; if this transformation makes the maximization simpler, we re doing alright. The standard transformation with o-douglas utility is to take logarithms. We have ln u x, y = ln x + 1 ln y ln u x, y = 1 ln x + ln y To solve the consumers prolems, we setup a Lagrangian we solve only s prolem, since s solution will follow y rough symmetry, L : 0 = λp x x x L : 0 = 1 λp y y y Sustituting through for λ, we find L = ln x + 1 ln y + λ p x x e + p y y e p x x p y y y x = p x p y = x = p yy p x Similar logic will otain, for consumer, y x = p x p y = x = p yy p x ppealing to the agents udget constraints, for we find p x py y p x + p y y = p x x e + p y y e = y = p xx e + p y y e p y Similarly, for py y p x p x + p y y = p x x e + p y ye = y = p x x e + p y y e p y Feruary 8, 011

6 In turn, this gives us consumption of x as x = p x x e + p y ye x = p xx e + p y ye p x p x This is a general feature of o-douglas utility: equilirium udget shares are proportional to the exponent associated with the good in utility. s we see here, of agent s spending power is dedicated to consumption of good x, and 1 is dedicated to good y. Lastly, we use market clearing conditions to pin down prices in terms of one another. x e + x e = x + x = p x x e + p y ye + p xx e + p y ye p x p y = p x x e + x e + py y e + ye p x p x x e + p x x e = p x x e + x e + py y e + ye p x x e + p x x e = p y y e + p y y e p x = y e + ye p y x e + x e Sustituting in to our demand equations, we find x = x py e + ye p x x = x e + e + x e ye + ye y e = 4x e ye + x e ye + x e ye + 4x e ye 6ye + ye Descriptions of other consumption variales will follow analogously. Since there is always one price free or normalizale in the description of equilirium, we now have a full characterization: x = 6x e ye + x e ye + 4x e ye 6ye + ye x = x e ye + x e ye + x e ye 6ye + ye y = y e x e + ye x e + ye x e x e + 6x e y = 4y e x e + y e x e + 6y e x e x e + 6x e p x = y e + ye p y x e + x e When calculating equilirium parameters, it is often helpful to leave things a little simpler than this; generally if you can phrase everything in terms of the price ratio you are well-set to otain numerical answers to a question. s a follow-up question, what happens to prices, allocations, and utility and what is the intuition when: x e = x e = y e = y e = 1 x e = x e = 1, y e = y e = x e = y e = 1, x e = y e = 4 Feruary 8, 011 6

7 General equilirium n equilirium is descried y prices p 0 1 and allocations x such that a llocations are feasile only positive quantities of goods and factors are consumed or utilised Markets clear for oth consumption goods and productive factors c gents are maximizing utility suject to their udget constraints d Firms are maximizing profits With this in mind, when we are asked to find equilirium in an economy we are eing asked to list the prices and allocations which support an equilirium. Solving for equilirium involves a lot of algera, a it of calculus, and patience. There are tricks to solving particular models, ut they are ad hoc and not generally applicale so we won t discuss them here. The only real way to get comfortale with general equilirium is practice, so let s proceed with an example prolem. There are capitalists and W workers in an economy. ll agents are endowed with 1 unit of laor, and capitalists also have 1 unit of capital apiece. We have production functions for goods and g, gent utility is given y g = K g + L g, = K 1 L 1 u c c, g c = 1 c g c, u w w, g w, l w = w g 1 w So capitalists enjoy g slightly more than, and workers enjoy slightly more than g relatively speaking; in an asolute sense, this depends on where they are in their margins. Equilirium is descried y prices and quantities. We therefore work to find prices on all goods p K, p L, p, p g and consumption/production levels, c, w, g, g c, g w of commodities and K, K, K g, k c, L,, L g, l c, l w of productive factors that solve the definition of equilirium listed aove. We egin looking for equilirium y applying a few intuitive arguments. Since within each type all agents are identical, this equilirium should have a symmetric structure; all workers have the same consumption and laor supply, and all capitalists have the same consumption and laor supply. With this in mind, the factor market clearing conditions are W l w + l c = + L g, k c = K + K g and commodity market clearing conditions are = c + W w, g = g c + W g w Notice that laor and capital are oth supplied inelastically in this model! That is, agents realize some income from wages and, if they are capitalists, from rents ut are otherwise unaffected; utility does not decrease when working or renting out capital. Since a greater udget is etter and allows more purchasing options, agents will provide as much of each productive factor as they have at their disposal. That is, l c = k c = 1 and l w = 1; so we can now restate the factor market clearing conditions as + L g = W +, K + K g = 1 side: there are some fairly general conditions under which all prices must e strictly greater than 0; although I can y no means guarantee this, my intuition is that all questions we will see in this class will have this feature. Feruary 8, 011 7

8 The worker s prolem We need to start somewhere, so we egin with the worker s prolem; we turn to calculus to help uncover relationships etween variales. The prolem of an individual worker is max w,g w,l w w g 1 w, s.t. p w + p g g w = p L l w We have already reasoned that l w = 1, so we can rephrase the optimization as max w,g w w g 1 w, s.t. p w + p g g w = p L Now, we could solve this prolem using a Lagrangian or through direct sustitution; ut, since we know that utility in o-douglas form has consumption s share of udget proportional to the exponent see the previous exercise with an exchange economy we know w = p L p g w = p L p g The capitalist s prolem The capitalist s prolem is given y max c,g c,k c,l c 1 c g c, s.t. p c + p g g c = p K k c + p L l c We have already reasoned through k c = 1 and l c = 1, so we can simplify the optimization to max c,g c 1 c g c, s.t. p c + p g g c = p K + p L s is the case in the worker s prolem, this is a o-douglas form so we know that consumption levels will e c = p K + p L g c = p K + p L p p g The firm s prolem With access to two production technologies, the firm solves max p K 1 L 1 + p g K g + L g p K K + K g p L + L g We can apply some intuition to a few price relations: Suppose p g > p K. Then for every unit of g produced using K g as input, positive profit is otained; the firm s profit-maximizing consumption of K g is then infinite! This will certainly violate the capital market clearing condition, so we cannot have p g > p K ; it must e that p g p K. Suppose p g > p L. Similar logic to the aove will hold; it must e that p g p L. Suppose p g < p K and p g < p L. Then the firm realizes a loss for every unit of g produced; it will then choose g = 0. ut y the equation for demand, g = 0 only if p w = + ; this contradicts the notion that p g < p K and p g < P L! So we cannot have oth p g < p K and p g < p L : at least one of p K or p L equals p g. Feruary 8, 011 8

9 This method of otaining restrictions on prices works only ecause we have a production function for which permits a neat, linear comparison. With regards to the production of, we cannot apply any such logic; it is necessary to use calculus to determine tradeoffs etween production and expenditure. We see K x : L x : 0 = p 0 = p L 4K K 4 pk pl From these equations, we otain p L = K p K p L p K = K 1 Putting it all together From the demand conditions for the agents together with the commodity market clearing conditions, we know W w = W p L p c = p L + p K p W g w = W p L p g g c = p L + p K p g ppealing to our earlier discussion of the firm s prolem, we reduce prices to three cases: p g < p L. Then p g = p K. Further, L g = 0 so = W +. Solving through aggregate demand for, pl pl W w + c = W + + p K p p p [ K ] K L + + = W = W K W From market clearing, we know = K 1 L 1. Then we have [ K W + K 1 W + W K = + W + 6 K W + = W W W W + K ] [ K W + + W + K W + 6K W 1 = W + + 6W + + 6K W + = 1 6K 6W + 4W + = 6K 6W + 6 W + K = 4W + ] Feruary 8, 011 9

10 From factor market clearing, we know then that W + K g = 4W + 4W + W + = 4W + W + 4 = 4W + Since L g = 0, this gives us g directly. s a final check, we have assumed that p K < p L. Then y equation 1, we need K > 1 Verifying against our determined levels of capital and laor, K W + = 4W + W + = 4W + < 1 So the prices arising in this equilirium contradict our assumptions; this cannot e an equilirium, and we will never see p g < p L! Why might this e the case? onsider the fact that while only one type of agent is supplied with capital, oth types of agents are supplied with laor. lthough this is y no means a mathematical argument, it follows that laor will, in general, e the less dear factor of production as it is supplied inelastically y all memers of the economy. Since p g < p L implies p K < p L, relative dearness of capital violates the pricing assumption aove. p g = p K, p g = p L. From equation 1 it follows that K = ; the total production of is then = K =. We know that aggregate demand for is pl pl + p K W w + c = W + p p = W + pl = W + 1 = 1 W + Then K = = 1 W +. Since = K g + K, we have then that K g = W It follows that for this equilirium to e valid, we must have W < otherwise K g is negative! s we found K g aove, we find L g according to clearing constraints, p L g = + W = W + Total g production is then g = 4 + W Feruary 8,

11 p g < p K. Then p g = p L. Further, K g = 0 so K =. Turning again to aggregate demand for, we find pl pl W w + c = W + + p K p p p = W [ K ] K L = W [ + 6 K + L ] From market clearing, we know = K 1 L 1. Then we have It follows that K 1 L 1 = W 1 L = W 1 Since K g = 0, we know g = W +4. Having assumed p K > p L, we need K = W = W + L g = L [ = W + W + W + 4 = < 1. This is checked as K = W + = W + + L ] K < 1 < W + W > Then this equilirium is valid when W >. That is, when the numer of workers is sufficiently large relative to the numer of capitalists, we will see the price of capital rise aove the price of laor. How can we summarize these results? We taulate what we ve found on the asis of the necessary relation- Feruary 8,

12 ships etween W and to support different equiliria. W < < W p K p K p K p L p K W + p p K W + p g p K W + p K p K p K L + W +W + W W + +W W +4 L g K K +W W 0 K g W < < W +W W c w W + g 4+W W +4 g 4 W +6 c g 1 1 w u 16 c a mess u 1 1 w W + W +6 W + That is a more-than-thorough answer to the question. In general, it will e clear what is eing asked for in the question prices, quantities, etc. so everything should not need to e taulated. ut a full description of the economy requires all variales to e accounted for. It is oviously tough to keep track of all cases, equations, and variales. The est tip here is to keep organized and consider cases one at a time. s far as I can tell, there are no real tricks to solving general equilirium prolems, only a willingness to work through the algera. If you work ack through the question aove, you can see that if we try to use the market clearing condition for good g we wind up mired in a morass of square roots; we can get the same answer y going this direction, ut it is not as clean as using the market clearing condition for clean here in the relative sense. How can you recognize that this is the case? Frankly, y solving the question. If you get stuck or start getting forms that look unholy, try approaching a different clearing condition. s I was working through this prolem, I initially solved for g until it proved too tedious. Solving for was much simpler. The advice for solving general equilirium prolems is then twofold: e patient, and don t e so attached to your approach that you re unwilling to try a different tack. note on maximization It was mentioned in section that maximizing a function f is the same as maximizing some strictly increasing function g of f. To make this look a little more concrete, suppose x is such that That is, x maximizes f. f x = max fx x Let g e a strictly increasing function, and suppose that there is some x such that g f x > g f x Now, since g is strictly increasing it is uniquely-valued and hence invertile. Moreover, this inverse is also strictly increasing for an illustration of this ut not a proof think of the graphs of y = x which is its own inverse or of y = x and y = x. If a > and h is a strictly increasing function, then ha > h; we apply this logic to see that g 1 g f x > g 1 g f x Which will hold if and only if f x > f x Feruary 8, 011 1

13 ut this contradicts the statement that x solves max x fx! So we cannot have that g fx > g fx, and x must also maximize g fx. Where is this useful? Often it is helpful to transform utility functions to find optimum consumption think aout why we, in general, cannot do this with production functions in any regular way. Suppose we are given a o-douglas utility function, ux, y = x α y 1 α. lthough y now we are all familiar with solving this optimization, it s kind of messy; this is even more true as we add consumption goods. In this case, we note that ln is an increasing function, so we can just as well optimize ln ux, y; that is, we can solve max x,y ln ux, y = max α ln x + 1 α ln y x,y This prolem is much simpler to optimize on paper, since consumption goods no longer affect each other s optimization. Further, we can use this trick to see that some utility functions are more familiar than we might think. Suppose ux, y = x m y n for some m, n N. gain, this optimization is a little messy on paper; we could apply the logarithm trick from aove, ut to appeal to intuition we ll do something slightly different. Notice that fu = u 1 m+n is a strictly increasing function. Then maximizing the utility aove is equivalent to solving max x,y f ux, y = max x,y x m m+n y n m+n = max x m x,y m+n y 1 m m+n So solving this utility form is equivalent to solving a o-douglas utility form! Since we know that o- Douglas consumption expenditure is proportional to the exponent on the good, we don t even need to perform any real calculus here and can safely plug into a known formula. Feruary 8, 011 1