Show That Any Strictly Concave Production Function Has Decreasing Returns To Scale

exhibit a non-monotonic behavior when the production function has diminishing returns to scale with respect to capital. But returns to scale is defined as (% change in output) / (% change in all inputs). Let 0 < θ < 1 and let y0 and y1 be real numbers. The size of establishments in terms of employees is n. Returns to Scale. Problem set 9, solutions 1. Our emphasis on dynamics clarifies that the origin of endogenous risk-taking is the convergence of wealth induced by a strictly concave intertemporal production function. Up to now, we have said relatively little about how the cost function C*() depends on y. Explain why 123 2 2 2 1 2 3 { | ( ) 0} aaa S z g z k z z z { t is a superlevel set. - globally constant returns to scale if f(tx)=tf(x) for all t>0 and all x - globally increasing returns to scale f(tx)>tf(x) for all t>1 and all x. returns to scale and no joint production are unlikely to be satisfied in practice. Deadline: 11. plotted against proportionate increases in. The function g is differentiable and weakly concave in n, satisfies g(0,a) = 0 for all a ∈ [a,¯a], and is strictly increasing in both arguments. 1 Model We model a firm as a production function that turns inputs into outputs. , [14, 34]), but limited work has been done to non-parametrically estimate HIV production functions. In that respect, StoNED is similar to DEA, and only imposes free disposability, convexity, and some returns to scale specification. When producing q units of output overall, the firm wants to allocate production at the two loca tions by minimizing. Constant returns to scale For all a ≥ 0, if y∈Y , then ay∈Y. Decreasing returns to scale: , f(tx) f (x0)=x0, decreasing when 0 0) <. Alternatively, decreasing returns could result from a priority-based budgeting rule. We often think of the "typical" production function as having increasing returns to scale initially, then decreasing returns to scale. An output-oriented approach was used for calculating the constant returns to scale (CRS) DEA and variable returns to scale (VRS) DEA measures. with infinite time and space support, and utilitarian objective function with strictly concave preferences, which is admittedly the most natural candidate for spatial Ramsey model, suffers from a structural ill-posedness problem: in contrast to the non-spatial Ramsey model, the initial value of the co-state. In this paper we show that in the setup of Kimbrough (1986a), Faig (1986, 1988), Guidotti and V6gh (1993), and Chari, Christiano, and Kehoe (1993), where the costs of producing real money are assumed to be negligible, the Friedman rule is the optimal policy for any homogeneous transactions costs func-. growth rate is decreasing to zero as time tends to in nity. [35] Bridget’s Brewery production function is given by y ( K, L ) = 2 √ KL , where K is the number of vats she uses and L is the number of labor hours. Finally, household non labor income is denoted by m 2. Suppose that the rm always chooses factors so as to minimize its costs, conditional. (b) Show that the profit function is increasing in p and decreasing in w. The shape of the curve depends on the assumptions made about the opportunity costs. This production function is called Cobb-Douglas function. Here A is a productivity parameter, and fi and 1 ¡ fi denote the capital and. input in the presence of decreasing returns. Likewise, it has diseconomies of scale (is operating in an upward sloping region of the long-run average cost curve) if and only if it has decreasing returns to scale, and has neither economies nor diseconomies of scale if it has constant returns to scale. Each consumer has utility function u(xi) = X2 ℓ,s=1 1 4 lnxi ℓs. Returns to Scale. Returns to Scale. Panagariya (1981: 221) considers a “two-commodity model with increasing returns to scale in one industry and decreasing returns to scale in the other, and discuss … implications of variable returns to scale”. Incentive compatible technology pooling: Improving upon autarky. Particularly, we show how water and capital prices afiect input choices and the important role of soil type via the drainage process. First, we construct the production box. Sure, The PPF is actually all about opportunity cost (in terms of the other option on the chart). A firm has a production function given by f(x 1,x 2)=ln(x 1 +1)+2x 1 2 2, (a) Find the firm’s cost function and the condi-tional input demand functions. In this paper, we compare traditional data envelopment analysis (DEA), three-stage data envelopment analysis (3SDEA) and artificial neural network analysis (ANN) to estimate technical efficiency indices, and to explore the effect of environmental factors (Fried, Lovell, Schmidt and Yaisawarng, 2002) [1] on technical efficiency for policy purposes in the semi-conductor sector. Decreasing Returns to Scale ; If input is doubled, output will less than double ; AC increases at all levels of output; 59 Long-Run Versus Short-Run Cost Curves. Consider a production function that exhibits decreasing returns to scale. ) 1 1 2 y x p D x, 2 2 12y p D. Therefore another interpretation of our result is that a higher mobility of unskilled workers are conducive to a higher tax rate if unskilled and skilled workers are substitutes. It is most closely related, however, to Grossman and Lai [2004]. The size of establishments in terms of employees is n. instantaneous utility, which we assume to be non-decreasing, strictly quasi- concave, and homogeneous of degree one in its arguments. Deadline: 11. Cobb-Douglas function represents a convex technology which must line somewhere between perfect substitutes and perfect complements. The domestic production function is F(k t),supposed to be convex for low levels of capital and then concave. (c) What is the cost minimising bundle of capital and labour to produce any xed Q= Q?. For =1show (using a figure) that there exists a unique 0 for any and that is increasing with What is the economic intuition behind these results? (i. Endowment vector ω of the economy: each household has one unit endowment of the commodity with the same index as its index. • We assume that α +η < 1, so there is (still) decreasing returns to the accumulated factors. We find that particular types of homotheticity of technologies, which we refer to here as scale homotheticity, provide necessary and sufficient condition for such equivalence. each has probability of getting the job (1 ) (q)and each –rm has a proba-bility of –lling its vacancy of (1 ) (q) where is strictly decreasing function and is a strictly increasing function. Show that a person’s preferences can be represented by a continuous utility function, if the preferences are complete, transitive, continuous and strictly monotonic. b) If average cost has a unique minimum value at output qe >0,. (a) Prove that the conditional input demand functions can be written in the form x i (w, y) = yz i (w) for all i = 1. 5 so as the amount of labor increases the MPL falls. a) Compute the certainty equivalent of this lottery as a function of η. Decreasing Functions. uk Cobb-Douglas function. We need to show that M is a bijection from \mathscr V to \mathscr P. A production function has decreasing returns to scale if f(tz1;tz2) • tf(z1;z2) for t ‚ 1 (1. Initially, the production function exhibits increasing returns to scale. How do I show that Cobb-Douglas function with decreasing returns to scale is strictly concave using hessian of second derivatives and the defi When the coefficients sum to one, doubling (or tripling, etc) the input will double (or triple, etc) the output, which is defined as constant returns to. The nal good is produced according to a constant elasticity of substituion (CES) production function with constant returns to scale. Show that, if the production function displays decreasing returns to scale, then it is always. These premises turned out to be wrong, and the stationary state has so far been postponed by a steam of highly productive scientific discoveries and investments. each has probability of getting the job (1 ) (q)and each –rm has a proba-bility of –lling its vacancy of (1 ) (q) where is strictly decreasing function and is a strictly increasing function. Let 0 < θ < 1 and let y0 and y1 be real numbers. , linearly homogeneous) is frequently called a neoclassical production function. , an equilibrium. Supply has to be equal to demand. If all contestants have the same production function which has constant elasticity not exceeding one, the proportion of the rent dissipated is equal to the value of this elasticity. A production function that is (twice) differentiable, monotonically increas-ing, concave, and homogeneous of degree one (i. F :3aL,5aK ;min : 3aL 3, 5aK 5 LaQ Constant returntoscale 5. Assume g(N) is strictly increasing, strictly convex, and di erentiable. Box 100131, D-33501 Bielefeld, Germany Abstract The paper analyzes the dynamic properties of the neoclassical one-sector growth model with di!erential savings in the sense of Kaldor}Pasinetti. $\endgroup$ – Alecos Papadopoulos May 26 '14 at 20:24. Decreasing returns to scale C. 0, alpha =. , Sachs, Tsyvinsky, and Werquin, 2016). In this case the amount of G given up to allow additional production of D is the same regardless of the amount of G and D being produced. The production possibilities are common knowledge and exhibit decreasing marginal returns,. Coronation Street has about 5 shows a week. A constant marginal rate of substitution. 3: Possible Shapes of the Total Cost Curve * Cost Curves Panels b and c reflect the cases of decreasing and increasing returns to scale, respectively. The cost of posting a vacancy is and a job of specialization level of produces output g( ) where g is an increasing and concave function. ), which is strictly increasing and strictly concave. Returns to Scale. In fact, the function appears to be strictly quasiconcave. find that uncertainty may reduce free-riding if utility is concave or there are decreasing returns to scale in the production of public goods. Suppose a firm’s production function, y = f(x), is homogeneous of degree one. Production processes having this property exhibit decreasing returns as one of the inputs is increased while holding other inputs constant. that if a homogeneous production function which displays decreasing returns to scale (DRS) is also quasiconcave, then the function is strictly concave. any concave production function of several variables. To produce each unit of output, a firm must use exactly 3 units of L and 5 units of K. (2008) provide a meta-analysis of empirical studies. A recent literature shows how an increase in volatility reduces leverage. [email protected] This is because, for our hypotheses, we estimated technical efficiency in milk production in terms of farmers’ capacity to maximize production given a certain bundle of inputs. (Note: “log” denotes the natural loga-rithm, with base e. Moreover, for fixed v \in \mathscr V, the derivative v' is a continuous, strictly. • We preserve the assumption of constant returns to scale (the exponents sum to 1). The assumption that real balances is the only factor of production is made for analytical convenience. Homogeneity of degree one is constant returns to scale. (In equilibrium, the profits are zero, so we will ignore them. Are short run production functions always straight lines? Properties of Production Functions: - they pass through the origin. 22 Let F be fixed costs. (b) For each of the following hypothetical production functions, determine whether the function has constant, increasing, or decreasing. When k < 1 the production function exhibits decreasing returns to scale. This paper examines theoretically the effects of the program set up under AMFA. the production function and study implications for input choices, i. , an equilibrium. Show that if a rm has xed proportion technology (assume a xed propor-tion production function where inputs are used in the proportion : ) then the cost function will be linear, and if it has linear technology then the cost function will be xed coe cient type. This implication of pin-making technology may be another reason why the distinction is most often fudged, particularly in the neoclassical literature. in [a,a¯] ⊂ (0,+∞). R&D spending in agricultural, medical and biological sciences has increased from $26 billion in 1980 to $97 billion in 2002, with an average annual growth rate of 6% (Fig. instantaneous utility, which we assume to be non-decreasing, strictly quasi- concave, and homogeneous of degree one in its arguments. • We assume that α +η < 1, so there is (still) decreasing returns to the accumulated factors. Business; Economics; My (very rough) notes - Agricultural & Applied Economics. Constant returns to scale implies that we have a homogeneous of degree 1 function: $F(K,L)=KF_k+LF_l \implies F_l=KF_{kl}+LF_{ll}+F_l~~ \&~~ F_k=LF_{kl}+KF_{kk}+F_k$. IP AND MARKET SIZE 2 Our model is related to a series of papers by Grossman and Helpman [1991, 1994, 1995] studying innovation in a Dixit-Stiglitz framework. Repeat using the strictly concave production function, Q = f(K,L,D). Thus, the production possibilities curve is concave. Moreover, estimated returns to scale are roughly constant. input in the presence of decreasing returns. We need to obtain the value of the Lagrange multiplier l. [Hint: For the second part let x2 = x1 +¢x with ¢x ‚ 0. NA = y tx c. Assume that the individual is an expected utility maximizer with a preference scaling function over consumption of u(:), where u is increasing and strictly concave. Also φ(0, z e) = 0, ∀z e ≥ 0. But returns to scale is defined as (% change in output) / (% change in all inputs). In practice, the shape of the production functions can be determined from available data on program scale up (e. • You have to check whether it’s appropriate to apply the Lagrangean method • You may need to use other ways of finding an optimum. The endowments are ω1 = (1,2. Constant Returns to Scale. Increasing returns to scale lead to decreasing cost. For =1find an implicit function defining as a function of b. By employing ko. This paper deals with this issue by focusing on the interaction between decreasing returns to labour (which imply firms' convex production costs) and centralised unionisation in a differentiated duopoly model. (a) Find the MRTS. When this decreases, investment and growth slow down delivering convergence toward a steady state. We show that the optimization problem admits a unique solution that can be characterized by the Euler equation. ), which is strictly increasing and strictly concave. A production function f(x) has globally returns to scale. Marginal productivity: The partial derivative of F with respect to v i represents the marginal product of factor i, which indicates how. See also: comparative advantage. This is because, for our hypotheses, we estimated technical efficiency in milk production in terms of farmers’ capacity to maximize production given a certain bundle of inputs. For some ranges of parameter and input values the function will exhibit decreasing returns to scale, for some others, increasing returns to scale. The nal good is produced according to a constant elasticity of substituion (CES) production function with constant returns to scale. Each individual faces the time constraint. Under these assumptions, firms using the traditional technology have cost functions for production levels , and firms using the modern technology have costs for. Comparing rows 1 and 4 shows increasing returns to scale. We show that under standard assumptions, each player has a well-defined share function which also satisfies other useful properties; notably it is strictly decreasing where positive, which rules out multiple equilibria. We show that the amount of contribution towards the provision (in the case of common property resources this has to be interpreted as extraction) of the collective input is a concave function of the private input endowment for most well-known production functions (e. Households begin life with equal claims to the profits of these firms. The typical mathematical expression linking the input variable is (1) Y x, t = A x, t F K x, t, L x, t, where F is a production function which is continuous and twice differentiable. tainty, and current resources simply have to be divided into current consumption and savings. An isoquant map can also indicate decreasing or increasing returns to scale based on increasing or decreasing distances between the isoquants on the map as you increase output. Decreasing returns could result from short-run rigidities in the production function. input in the presence of decreasing returns. L If this function is to exhibit constant returns to scale, what restrictions should be placed on parameters?. Which of the following production functions exhibits constant returns to scale? and B have increasing returns to scale, and D has decreasing returns to scale. utility function, we maintain the standard assumptions that it be intertemporally separable, quasi-concave, non-decreasing, and strictly increasing in at least one argument. The price of the bought-in good is unity. Consider a production function that exhibits decreasing returns to scale. In the latter case, we show that the transitional dynamics depends on the values of both capital and the reference level of consumption, whereas the transition only depends on the ratio of capital to the reference level of. Thus, the production possibilities curve is concave. We consider a one-sector growth model with homogeneous consumers who derive utility from consumption and wealth. period 1970-1998, using a translog function. This production function is called Cobb-Douglas function. 1361-1373 1. n, where z i (w) is a function that depends on the price vector w = (w. A local measure of returns to scale is: f tx lim f x t 1. Output in the traditional sector is produced with labour alone under constant returns to scale. The only reason loans are taken in this example is to smooth consumption over time. Unlike Wicksteed, who saw constant- and nonconstant-returns production functions as mutually exclusive phenomena, Wicksell (1901, 1902) argued that a firm's production function might exhibit successive stages of increasing, constant, and decreasing returns to scale. This implies that the. 2 Suppose that the economy's supply side can be described by the following Production function: where A-10 (a) Graph the production function for N-5, 10, 15,20, and 25. This constructive proof is presented on pages 14 and 15 of Jehle & Reny. In addition, Noulas et al. To see this, first observe that, in view of our assumptions above, u' is a strictly decreasing continuous bijection from (0,\infty) to itself. One example of this type of function is Q=K 0. ) Households also own their time endowment, k0 units of capital, and M0 units of money. All Docsity's contents are fully available from any version English Español Italiano Srpski Polski Русский Português Français. Agents in a population choose the level of input. The technology choice problem is identical for all types of firms because of constant returns to scale, so all firms use the modern technology if and the traditional technology otherwise. Advanced Microeconomics: Problems Atsushi Kajii Institute of Economic Research, Kyoto University September 1, 2013 Abstract This is a master copy - do not think my students. Clearly, f(0) = g(0) = 0. production function of transactions are convex and that the demand for money depends negatively on the nominal interest rate. A production function has decreasing returns to scale if f(tz1;tz2) • tf(z1;z2) for t ‚ 1 (1. We assume that the utility function in the problem (1) is a-concave with large positive a in the relevant sub-domain. advertisement. Comparative statics analysis shows that increases in uncertainty will increase donations. This describes a firm. Firm j may make investment Ij t to increase its existing capital stock K j t. $\endgroup$ – Alecos Papadopoulos May 26 '14 at 20:24. c(p; y) > c(p;y) for any >1 2. We show that any homogeneous production function can be written as a CD function 2 For a formal proof in the case of the Translog function see e. We are interested 4. This follows from the definition, with some algebra. This production function is called Cobb-Douglas function. It will be assumed that the production function displays constant returns to scale and that the standard Inada conditions are fulfilled. The postulate of cost minimization does not restrict the shape of the cost function C*(y) Suppose now the production function is strictly concave (decreasing returns). Population and employment (L) are con-stant, so-given constant returns to scale-we might as well nor-malize to L = 1, after which Q, K, C, and R can be thought of as. 1 Production functions with a single output 0. This function has constant returns to scale, with TFP given by a function of the elasticity of substitu-tion. (c) This technology has locally increasing returns to scale if x 1. , the uniqueness of ,that is strictly positive, and that it is increasing in ) c. A function g : R 7!R is -concave if g(x) + ( =2)jxj2 is concave. Each consumer has utility function u(xi) = X2 ℓ,s=1 1 4 lnxi ℓs. Greater than 1, you have increasing RTS. 5 Fsa N K b a a N H b b112a2, which is a similar expression to the one derived for a perfect competition version of the theory. If that expression is less than 1, you have diminishing RTS. Let the production be defined as f(z1,z2)=za1z b 2 If a+b<1, thenY satisfies NI returns to scale. Production The maximum amount of output that can be produced depends on K according to an aggregate production function: Y = F(K) A crucial property of the aggregate production function is diminishing returns to capital. But the converse is not true. Since a nonnegative linear combination of concave functions is quasiconcave, it suffices to prove that ln(x+1) is concave, but this is straightforward from the properties of the logarithmic function, i. Economics Bulletin, 2013, Vol. on the other, if K λ(KL), the production function has increasing returns to scale. Cobb-Douglas function represents a convex technology which must line somewhere between perfect substitutes and perfect complements. Check it out at Concave and convex technology, www-users. In this paper, we compare traditional data envelopment analysis (DEA), three-stage data envelopment analysis (3SDEA) and artificial neural network analysis (ANN) to estimate technical efficiency indices, and to explore the effect of environmental factors (Fried, Lovell, Schmidt and Yaisawarng, 2002) [1] on technical efficiency for policy purposes in the semi-conductor sector. at low output levels. Solution: Q = F(L,K) = min : P 7, O 9 ; Think along the efficient production only. , the Cobb-Douglas andthe CES) and also that the equilibrium level of joint surplus (of bothcontributing. 2 +1): As this is the case for any x ˛ 0, and we know the production function is continuous on R2 +; it must be f(x) = x 1(x 2 +1) on R2 +: (This production function is easily veri–ed to be correct by solving the cost minimization problem for it to show that the resulting cost function is indeed the one given in the problem. When the production function is continuous, strictly increasing and strictly quasiconcave on n \+, and f (0) 0G= and it is homothetic (a) the cost function is multiplicatively separable in input prices and output and can be written cw(, ) ()(,1)y =h y cw, where h(y) is strictly increasing function and cw(,1) is the. Conversely, if inputs were to be increased by 100% but output were to increase by less than this, then the production function would exhibit decreasing return to scale. This is equivalent to the cost function being strictly concave in output. We address the issue of equivalence of primal and dual measures of scale efficiency in general production theory framework. Generally, by equilibrium. For example, the input might. This material, like any other properly issued regulation, has the force of law. • If k = 1 have constant returns to scale and the production function is linear homogeneous. (Chp18) Consider the production function Q = B 0 + B 1 (K*L) 0. Returns to Scale Figure 7. This implies that the CES function exhibits constant returns to scale and hence has an elasticity of scale of 1. Local indeterminacy has been shown to arise both in endogenous growth models with constant returns to scale in reproducible factors (Bond, Wang, and. An output-oriented approach was used for calculating the constant returns to scale (CRS) DEA and variable returns to scale (VRS) DEA measures. Assume that F is strictly increasing, strictly concave, continuously difierentiable, and satisfles the usual Inada conditions. (a) Define the profit function of the firm. ) c) (12 points) Here is another production function. 1 Model We model a firm as a production function that turns inputs into outputs. (2008) provide a meta-analysis of empirical studies. This means that there is a maximal sustainable output Y: for all k > Y, f(k) < k (the production function crosses the 450 line). with infinite time and space support, and utilitarian objective function with strictly concave preferences, which is admittedly the most natural candidate for spatial Ramsey model, suffers from a structural ill-posedness problem: in contrast to the non-spatial Ramsey model, the initial value of the co-state. The purpose of this study is to study an economic growth model with heterogeneous households for providing insights into relations between economic growth and income. A cost function of this kind can be derived from the firm’s production function, prices of variable inputs, and employment of fixed assets, on the assumption that the firm is a cost minimizer. We need to show that M is a bijection from \mathscr V to \mathscr P. Each individual faces the time constraint. Suppose now the production function is strictly concave (decreasing returns). The first posibilty is decreasing returns of scale on the production function, that means when we add one additional unit of both inputs (labor and capital) to the production proccess the production will growth in less than one unit. Any Pareto-Optimal solution solves the following constrained maximization program P1, namely it maximizes a linear social welfare function Max λU1(u. Here A is a productivity parameter, and fi and 1 ¡ fi denote the capital and. Show that every cost function is superadditive in input prices. Fully label all graphs. Let xi ℓs denote consumer i’s consumption of good ℓ in state s. Promising adaptations of the model are the models of Ehrlich and Chuma (1990) and Galama (2011), who have extended the Grossman model to include a health production process that is characterized by decreasing returns to scale (DRTS), whereas the standard model assumes a linear health production function with constant returns to scale (CRTS). We characterize the class of output-sharing rules for which the labor-supply game has a unique Nash equilibrium. Show all the calculations. This function has constant returns to scale, with TFP given by a function of the elasticity of substitu-tion. The temperature distribution function over that plate is T(x,y)=x2+2y2-x. • Technology: There is a constant returns to scale technology over capital and labor such that per capita output is f(k),where kis capital input per unit of labor. A production function which is strictly concave but intersects the horizontal axis at a positive level Consequently, whether we have increasing, constant or decreasing returns to scale depends upon Now, earlier on we showed that the average product of labor is higher than marginal product only. It is twice continuously differentiable in K,N and it is strictly increasing in K,N, F K ,F N > 0 , concave in K,N, F KK ,F NN 0 , and F (0 , , ) = 0. decreasing returns. Returns to Scale. Daron Acemoglu (MIT) Economic Growth. Homogeneity of degree one is constant returns to scale. the nature of these functions. The motivation for this method is that in some problems, especially large-scale problems where the number of DMUs is very much greater than the total number of inputs and outputs, the efficient frontier tends to a nonlinear surface and, therefore, a nonlinear formulation better fits the production function. Fixed and Variable Costs: Since variable costs depend on the level of production, they can be represented as a function of output. Also φ(0, z e) = 0, ∀z e ≥ 0. A utility function that has the desired property is (any monotone transformation of) cv(l); assume also that v is not only strictly increasing but that cv(l) is strictly quasiconcave. This assumption implies that the average cost function is U-shaped, although the production function does not have to be concave. The MPL =. Deadline: 11. technology is the source of rents for a competitive firm with a constant-returns-to-scale production function. The purpose of this study is to study an economic growth model with heterogeneous households for providing insights into relations between economic growth and income. With decreasing returns to scale, a proportional increase in all inputs will increase output by less than the proportional constant. (c) This technology has locally increasing returns to scale if x 1. • Preferences: The instantaneous household utility function over consumption, c, is u(. In the case of a constant returns to scale firm, the net supply function becomes a correspondence and differs from the net supply functions of the preceding chapters on two accounts:. Then, the neoclassical. 1 Introduction The objective of this chapter is to estimate alternative specifications of production functions in the banking industty in India over the period 1985 to 1995-96. If a firm employs a mass n of workers at the time of production, then its output is g(n,a). We assume that the utility function in the problem (1) is a-concave with large positive a in the relevant sub-domain. To produce each unit of output, a firm must use exactly 3 units of L and 5 units of K. The combined assumptions of increasing returns to scale (IRS) in production and sticky prices identify this approach. To see this, first observe that, in view of our assumptions above, $ u' $ is a strictly decreasing continuous bijection from $ (0,\infty) $ to itself. 10 This compensation can come in the form of transfers of current wealth (dividing up marriage 16 At a level utility of marriage that previously would not have resulted in divorce, under no-fault However, the supply curve (the production function of human capital) is concave up and the demand … Read Article. It also is subject to constant returns to scale. 6 This is half true: if g0 (x) > 0, then the function must be strictly increasing, but the converse is not true. 0 (trivial aggregation): Suppose that (i) every firm has the same productivity and the same CRS production function ,where is strictly increasing and strictly concave; (ii) consumers have the same initial endowments, and same preferences, and their utility function is strictly increasing and strictly concave. During the so-called \Dot-Com boom" in the late nineties, many software companies located. An extreme example of the latter is that of a monopolist hiring workers. The two sector model growth model with constant returns to scale in production has frequently been utilized to show how dynamic indeterminacy can arise in the presence of factor market distortions. 5 < m, our new production has increased by less than m, so we have decreasing returns to scale. In Figure 1, we show the production side of the economy,6 with the two sectors as X and Y. The usual production function satisfies constant returns to scale or linear homogeneity, which means that the marginal products depend only on the ratio K / L. We aim to show that AFC is strictly convex in two different ways. It will be assumed that the production function displays constant returns to scale and that the standard Inada conditions are fulfilled. ) Households also own their time endowment, k0 units of capital, and M0 units of money. 1 Model We model a firm as a production function that turns inputs into outputs. Both they and we show that as the total monopoly revenue function has increasing (de-. t ≥ T , population grows at constant rate n, and the production function Fˆ K (t) , L (t) , Aˆ (t) exhibits constant returns to scale in K (t) and L (t), then: g K = g Y = g C. Indifference curves that are convex to the origin reflect: An increasing marginal rate of substitution. Production side same as before: competitive firms, constant returns to scale aggregate production function, satisfying Assumptions 1 and 2: Y (t) = F (K (t) , L (t)). This paper extends the analysis to a model with production and capital ac-cumulation. Equal to 1, you have constant RTS. Supply has to be equal to demand. (Chp18) Consider the production function Q = B 0 + B 1 (K*L) 0. Assumptions about the production function • The function F(;) is assumed to be • quasi-concave, • strictly increasing in both arguments, • homogeneous of degree one or constant-returns-to-scale, • and twice differentiable. In section 5, we show that the results of this section also hold under a more general production. • If k > 1 have increasing returns to scale. Returns to Scale. First, we construct the production box. The y-value decreases as the x-value increases Strictly Increasing (and Strictly Decreasing) functions have a special property called "injective" or "one-to-one" which simply means we never get the same "y" value twice. High type workers have a reservation return u h and low type workers have a reservation return u l 0 and all non-negative input combinations (L;K), f(tL;tK) = tf(L;K):. This paper surveys the 20:th century discussion on resource scarcity. Thus, C(y) is both concave and convex. 6 This is half true: if g0 (x) > 0, then the function must be strictly increasing, but the converse is not true. Comparative statics analysis shows that increases in uncertainty will increase donations. Consider a two-input production function that employs capital (K) and labor (L) as the two inputs. A contribution of this paper is to show. , irriga-tion water demand and technology adoption. (a) Does the production function exhibit increasing, decreasing or constant returns to scale? Explain your answer. Does it have diminishing Y- A [(12)N (1/4)N2] returns? Explain. Let k¯ >0 be some reference capital-labor ratio, and let ¯ >0 and A >0 be two constants. Appendix: Properties of an Expenditure Function Suppose that a consumer’s utility function is continuous, increasing, strictly quasi-concave. In the current paper, which has no convex adjustment costs, rents are earned as a result of monopoly power or as a result of decreasing returns to scale in the production function. The nal good is produced according to a constant elasticity of substituion (CES) production function with constant returns to scale. In fact, the function appears to be strictly quasiconcave. returns to scale. The MPL =.