ECON Answers Homework #3

Transcription

1 ECON Answers Homework #3 Exercise 1: (a) First, I calculate the derivative of y with respect to t. Then, to get the growth rate, I calculate the ratio of this derive and the function: (b) dy dt = arert and dy/dt y = r (c) d(ln(y)) dt = = r = 1 y d[ln(a) + rt] dt dy dt where the last equality follows from (a). This result means that the growth rate of the function y can be obtained as the derivative (or instantaneous rate of change) of the log-function. 1

2 Exercise 2: (a) The price elasticity of demand is dened as: El p (q) = dq/q dp/p = p dq q dp = p ( ) p 1.5 q = 0.5 (b) Starting from the demand function, I take the logarithms on both sides to get: ln(q) = ln(16) 0.5 ln(p) And it is obvious to conclude that the log-quantity log(q) is obtained as a linear function of the log-price log(p). This is the characterization of a log-log linear function. (c) I now dierentiate the above expression found in (b) to get: d(ln(q)) d(ln(p)) = 0.5 = El p (q) where the last equality follows from (a). As a result, the price elasticity of demand is equal to the slope of our log-log linear function. (d) 2

3 Exercise 3: (a)(i) To nd the equation for the isoquant at Q = 100, I set Q = 100 in the production function and solve to get K as a function of L (or L as a function of K): 100 = 100K 0.25 L = K 0.25 L 0.75 K = L 3 (ii) To show that the isoquant is negatively sloped and convex, I calculate the rst-order and second-order derivatives: (iii) Graph of the isoquant: dl = 3L 4 < 0 for any L > 0. d 2 K dl 2 = 12L 5 > 0 for any L > 0. (iv) - The negative slope of the isoquant (negative rst-order derivative) results from a neoclassical assumption stating that, when both K and L are positive, an increase in either K or L will always increase Q. Since we are considering an isoquant, that is a curve where Q remains constant, when K increases, L has to decrease and vice-versa. - Convexity of the isoquant (positive second-order derivative) results from decreasing marginal rate of substitution between K and L. (v) Other isoquants (that is for other xed values of Q, say Q > 0) have the following equation: Q 100 = K0.25 L 0.75 K = Q 100 L 3 3

4 with rst- and second-order derivatives equal to dl = 3Q 100 L 4 < 0 for any L > 0. d 2 K = 12Q dl L 5 > 0 for any L > 0. Hence, for this production function, all the isoquants have a similar shape. (b) (i) The marginal products of labour and capital are: ( ) 0.25 dq K dl = 75 L ( ) 0.75 dq L = 25 K (ii) Since K > 0 an L > 0, it is easy to check that the above functions are positive at every level of output. (iii) Yes, we expect this to hold for any production function. This comes from the neoclassical production theory that assumes that provided both K and L are positive, an increase in either L or K will always increase Q. (c) To show that the marginal product of labour diminishes as the labour input increases with the capital input held constant, I calculate its derivative with respect to L: [ ] [ d dq = d ( ) ] 0.25 K 75 dl dl dl L = K0.25 L 1.25 < 0 Hence, I conclude that MPL diminishes as the labour input increases. Similarly, to show that the marginal product of capital diminishes as the capital input increases with the labour input held constant, I calculate its derivative with respect to K: d [ ] dq [ = d ( ) ] 0.75 L 25 K = L0.75 K 1.75 < 0 4

5 (d) The 'law' of diminishing marginal productivity (DMP): as we repeatedly increase the labour input (or capital input) by one small unit with the capital (respectively the labour) held constant, then the successive increases in output become smaller and smaller. Thus we have diminishing (or decreasing) marginal product of labour (respectively capital). - No, we do not expect this to hold for every production function at every level of output. A productive process in which the marginal product of labour (MPL) is constant or increasing over a certain range of output, but eventually decreases is perfectly possible. However, the Cobb-Douglass function is too simple to accommodate this subtlety. One reason for that could be that, for technological reasons, the xed stock of capital equipment cannot operate eciently when the labour is very small. A machine may have been designed to be operated by a crew of three, say, and with only two operators it is very slow. Hiring extra workers then raises marginal productivity instead of lowering it. As more and more workers are hired, marginal product of labour continues to rise until it reaches a maximum. Beyond this point, diminishing marginal product of labour begins take eect and the slope starts to decrease. (e) (i) To show that an increase in capital input increases the marginal product of labour, I calculate its derivative with respect to K (holding L constant): [ ] [ d dq = d ( ) ] 0.25 K 75 dl L 1 = (KL) > 0.25 To show that an increase in labour input increases the marginal product of capital, I calculate its derivative with respect to L (holding K constant): [ ] [ d dq = d ( ) ] 0.75 L 75 dl dl K 1 = (LK) > (ii) - The MPL is increasing in the capital. This means that a small increase in K (with L constant) increases the MPL. Economic theory does not gives us strong reasons for 5

6 expecting this sign to be invariably positive at any point on any short-run production function (that is for xed K). However, this property must hold at some points at least. Otherwise, the short-run production function (for xed K say K 1 ) could not lie above the one for xed K, say K 2 > K 1. - The MPK is increasing in the labour. This means that a small increase in L (with K constant) increases the MPK. Again, this property does not have to hold at any point, but must hold at some points at least. 6