CES functions and Dixit-Stiglitz Formulation

Transcription

1 CES functions and Dixit-Stiglitz Formulation Weijie Chen Department of Political and Economic Studies University of Helsinki September, Labour Labour 5 Capital Capital Any suggestion and comment please send to: weijie.chen@helsinki.fi

2 Abstract Constant elasticity of substitution CES) functions are the most extensively used functional form in economics so far, but textbooks seldom give good and enough illustrations on how to use them. The purpose of this note is to show you how it can be derived and its contribution to Dixit-Stiglitz formulation.

3 CES functions We start with the most basic CES function, ux, x ) x + x ) where <. Let us first verify this is a CES function. Take partial derivatives w.r.t. both x i s, ux, x ) x ux, x ) x x + x ) x x + x ) x + x ) x x + x ) x ) x ) Then recall how we define price elasticity of damand, ε p x/x p/p Most of time, we work with continuous case, ε p dx/x dp/p dx p dp x We assume only one good in this case. The price elasticity of substitution looks similar, but a little bit complicated, since it is about substitution, we need at least two goods, say x and x. This is a very important concept in comparative statics, defined as: the relative change of the ratio of the consumption goods x and x over the relative change of the ratio of the according price and p, ε sub dx /x ) x /x dp / ) p / We need to show the case at the optimum, thus the point-elasticity at the optimum is, ε sub dx /x ) x /x dp / ) p / dx /x ) x /x du /u ) u /u d x ) d p ) p x 3)

4 where u u/ x, u u/. At the optimum, Use ) and ), p u u Thus u x + x ) u x + x ) x x ) x x ) x p x x ) p 4) We can use 4) to calculate, d x ) d p ) p ) 5) Then according to 3) we need p / x /x p p ) p ) ) p 6) Multiply 5) and 6), following 3) d x ) d ) p p x ) p p p ) We have verified it is CES function, the elasticity of substitution is / ) which is a constant.. General Case of CES The most general functional form of CES is given by F A α x + α x )

5 where A is a constant or a stochastic process cf. technology parameter A in Cobb-Douglas production function), α denotes distribution parameter as the exponent in C-D function, besides we assume α + α. In this case, is exactly the elasticity of substitution parameter as we have seen in first case as ε sub. The purpose of the outer exponent is designed to render the function as first-order homogeneous linear homogeneous). We can show this as follows: ) F A α tx ) + α tx ) { [ A t α x ta α x + α x + α x ) ]} As the first case, we define the elasticity of substitution at optimum point-elasticity) as ε sub dx /x ) x /x dp / ) p / dx /x ) x /x df /F ) F /F d x ) d p ) p x 7) Take partial derivative w.r.t. x and x, F A [ ] α x + α x α x [ ] Aα x α x + α x Aα x F Aα F x ) 8) Similarly, F Aα α F x ) 9) Divide 9) by 8), F F ) α α x x ) p ) 3

6 Next we shall isolate x x To get p / x, we modify ), on one side at point of optimum, α α x x x ) p p ) α α ) x p p ) α α x p p x p p ) α α ) α α ) From ), we get d x ) d p ) p ) α α 3) According to 7), we multiply 3) and ), p ) α α p ) α α Thus we have confirmed the property of CES function. The larger the the great the substitutability, if, both products have to be in a fixed proportion, such as right and left shoes.. Cobb-Douglas Function As The Special Case We will see in this section that Cobb-Douglas function is simply a limiting version of CES function, which renders C-D function as a special case. Let s go back to the general CES form, F A α x + α x ) 4) 4

7 when, 4) will approaches C-D function. However we cannot simply equal to here, since the denominator of outer exponent is undefined. Thus, the most natural way to proceed is to use L Hôpital s rule. First divide F by A and take natural log, ) ln F ln α A x + α x m) n) 5) To see whether it is form, we set, the denominator obviously equals, and numerator can be shown ln α x + α x ) ln We can perform L Hôpital s rule to 5), but we find it will become extremely messy in notation to handle this problem, since we have to use three times derivative product rule in a nested manner and exponent derivative rule. Actually it is unnecessary to work on such a general function to derive the C-D function, we can simplify 5) a bit and keep its CES features. There will be some slight notation change in this simplified version, F A [ αk + α)l ] 6) as you can see we replace x and x by capital K and labour L. A and α have their corresponding meaning in C-D function, but not yet. Besides, we can notice that we have set implicitly, if then. Thus if, 6) approaches C-D function. Divide 6) by A then take natural log, ln F A ln [αk + α)l ] m) n) To see the, simply plug into numerator, ln [α + α]. Next, use L Hôpital s rule m ) n ) αk + α)l [ αk ln K α)l ln L] 5

8 Then set, we get α ln K + α) ln L α + α α ln K + α) ln L ln K α + ln L α ln K α L α Since we have divided by A and taking natural log, we need to perform the inverse operation to recover the CES form, thus Ae ln Kα L α AK α L α 7) which is the standard form of Cobb-Douglas function..3 CES Production Function We have already seen the CES production function with input capital and labour can be written as: F A [ αk + α)l ] But question is what makes we find the CES production function should look like this? Simply put, why it looks like this? The origin of CES production function comes from Arrow et al 96). In their paper, they tested two models V L c + dw + η ln V L ln a + b ln w + ε where V F K, L), w denotes the nominal wage rate per capita, L the labour input. They found that the log version of the model fit better with data across industries in U.S.. We turn F K, L) into per capita version: y fk), where K L k and V L y. Then MPK and MPL are f k) and fk) kf k) respectively. We will focus on the second model and derive the CES production function here. To start from the function without error 6

9 term ln y ln a + b ln w 8) Substitute w fk) kf k) into last equation, ln y ln a + b ln [fk) kf k)] Solve for f k), ln y ln a b ln [fk) kf k)] ln ln y a b ln [y kf k)] ) y b a kf k) y ln [y kf k)] kf k) y y b ) y b a a b f k) a b y y b ka b To make further modification in order to simplify the equation, a b y y b a ) b a b y y b y a b y b k k ka b We define α a b, then dy dk y αy k b y αy b ) k y αy ) k where we define b. This is a separable differential differential equation, so separate y and k, dk k dy y αy ) and perform partial fraction decomposition, dk k dy y + αy dy αy 7

10 Take integral on both sides, ln k ln y ln αy ) + ln β where ln β is constant term after indefinite integration. Take antilog on both sides then raise the power of, we get To solve for y, k βy αy k αk y βy k β + αk )y y k β + αk y kβ + αk ) y [k β + αk )] y βk + α) To recover the aggregate form, F K, L) βk + αl) Dixit-Stiglitz Lite In this chapter we present the Dixit and Stiglitz 977) monopolistic competition model, which is currently used in New-Keynesian DSGE model.. Frisch and Marshallian Demand Function There are n + goods, representative household maximises utility function [ n ) U x, x ] i i 8

11 by choosing a consumption plan x R n+. Where is the constant elasticity of substitution between monopolistic goods i,..., n. x is the numeraire good, which can be interpreted as labour or leisure time. With constraint, n x + p i x i I i The representative consumer maximise utility function U ) qω) dω < < 9) There is a continuum of goods, indexed by ω [, n]. ω is the measure of substitutability. With budget constraint, where pω) is the price of good ω. pω)qω) dω I ) In order to maximise utility function easily, we take monotonic transformation to utility function [ U ) ] qω) dω which would yield the same optimisation solution. Form Lagrangian, F.O.C.s are, [ L[qω)] U λ qω) dω λ ] pω)qω) dω I [ L qω) qω) λpω) qω) dω ] pω)qω) dω I We model the numeraire good q) implicitly which was also used in original Dixit- Stiglitz monopolistic competition model. 9

12 the integration sign is gone if we take a derivative w.r.t. a specific ω. Rearrange, qω) λpω) qω) λpω) λpω) qω) ) ) ) is Frisch demand function. Take ratio of Frisch demands for two different goods, ω and ω yields, qω ) qω ) λpω )/ λpω )/ pω ) pω ) ) ) ) Now we define pω )qω ), to simplify notation. Multiply both sides by pω )qω ) qω )pω ) pω ) Integrate both sides out of ω, pω )qω ) dω The left-hand side is I, then I To isolate qω ), I qω ) qω )pω ) pω ) dω qω )pω ) pω ) dω Then we get Marshallian demand function: pω ) pω ) dω qω ) I pω ) pω ) dω Ipω ) n pω ) 3) dω

13 . Indirect Utility Function and CPI Plug 3) into 9) to get optimised indirect utility function, Because { [ U Ipω ) n pω ) dω [ I pω ) ) dω, thus [ U I pω ) ) dω Set utility equal to one and solve for I, [ I pω ) ) dω I n pω ) dω ] dω} ] pω ) dω pω ) dω ] pω ) dω ] pω ) dω ) and we can replace, I pω ) ) dω pω ) dω switch ω to ω, I pω ) dω pω ) ) P 4) dω ) P pω ) dω 5) where P denotes the expenditure to purchase a unit-level utility. 5) is also the consumer price index CPI) in this model..3 Consumption Diversity We assume all varieties have the same price and quantity, thus pω)qω) dω npq I

14 We can see easily, U as you can see U n ) [ ) qω) dω nq ) I ] n I n np P >, utility increases as n goes up..4 Optimum Pricing If you have some experiences of New-Keynesian NK) DSGE models, you will recall that NKs usually employ Calvo pricing mechanism to derive the optimum price and New-Keynesian Phillips curve NKPC). Because the model assume monopolistic competition, every firm and consumer has some monopoly power on its price and wage. And we see the most primitive version of optimum pricing here under Dixit-Stiglitz formulation. This is a static model, no time evolution, we define a firm s profits as π pq wcq + f) pq lq) 6) where c is constant marginal cost, w denotes real wage rate, lq) is the labour demand function with argument q. The firm will choose a p to maximise its profits, set up F.O.C., π p q + p q p wc q p the first two terms are from derivative product rule. Solve for p, p wc q q p 7) Use Marshallian demand function 3) and CPI 5) we can get qω) pω) P I 8) take partial derivative, qω) pω) pω) P I 9)

15 Divide 8) by 9) we can get the second term of 7), Substitute back to 7), pω) P I pω) P I p p wc + p Again, solve for p, + ) wc ) + p wc It is clear that the price should be a proportional mark-up over cost wc. And mark-up level depends on substitutability, if then p wc. End. References [] Arrow K.J., Chenery H.B., Minhas B.S., Solow, R.M.96): Capital- Labor Substitution and Economic Efficiency, the Review of Economics and Statistics, Vol.433), 96 [] Dixit A.K., and Stiglitz J.E. 977): Monopolistic Competition and Optimum Product Diversity, American Economic Review, Vol.673), 977 3