Econ Homework 4 - Answers ECONOMIC APPLICATIONS OF CONSTRAINED OPTIMIZATION. 1. Assume that a rm produces product x using k and l, where

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1 Econ Homework 4 - Answers ECONOMIC APPLICATIONS OF CONSTRAINED OPTIMIZATION Graded questions: : A points; B - point; C - point : B points : B points. Assume that a rm produces product x using k and l, where x = f (k; l) = 0k :5 l :5 Further assume that the rm purchases labor and capital at prices w and r. A) Derive the rm s conditional demand function for labor; lc d = l c (x; w; r). B) Then derive the rm s conditional demand function for capital, kc d = k c (x; w; r). C) What happens to the conditional demand for capital if the price of labor increases marginally. D) What is the rms cost function? Answer: A) This is the production manager s problem: subject to mine (l; k) = wl + rk l;k x = 0k :5 l :5 Note that there constant returns to scale in production. Turn the constraint minimization problem into unconstraint problem by solving the costraint for one of the endogenous variables, say k: x = 00kl k = x 00l and substituting the result in the objective function: Look for critical point(s): x mine (l) = wl + r l 00l

2 Solution (l) = w + ( ) r 00 l = 0 Discard the negative one. So the solution is: w = r x 00 l l = rx 00w l = x r r 0 w l = x 0 r r w Check the second order conditions for a e = ( ) r x 00 ( ) l > 0 It is positive for any positive l, including l = x p r w. So, e (l) is in fact minimized at l = x p r w, and therefore, l is the conditional deman function: l c (x; w; r) = x 0 r r w B) The conditional demand function for capital is: k c = x l = x x p r w = x 0 Note that both of these are linear in output. This follows from the fact that we assumed a r w r production function that exhibits constant returns to scale. C) To determine what happens to the conditional demand for capital when the wage rate increases take the partial = x 0 p wr > 0 So, if w marginally increases, holding x and r constant, the rm s demand for capital will x increase by 0 p wr. D) Determining the cost function. Expenditures on inputs are, by de nition the amount spent on labor and the amount spent on capital, that is, e = wl + rk. Cost-minimizing expenditures to produce x given w and r are therefore

3 e = wl c + rk r c r r w = w x 0 w + r x 0 r = x p x p rw + rw 0 0 = 5x p rw = c (x; r; w). Assume that an individual s preferences can be described by the following utility function u (x ; x ) = x + 5x where a and b are exogenous parameters. Assume that the consumer s income is y, and it is an exogenous variable. Assume that the prices of goods and are and p, and these are exogenous variables. A) Is the common "more is always preferred to less" assumption satis ed for this utility function? B) Derive the individual s demand functions for goods and. Answer: A) More is preferred to less requires that the marginal utility of each good is always positive. That is, (x ; x = > (x ; x = 5 > 0 So, the "more is always preferred to less" assumption satis ed for this utility function. B) The individual s demand functions for goods and are the solutions to the consumer s problem of maximizing utility subject to budget constraint: subject to max x ;x u (x ; x ) = x + 5x where y denotes the individual s income. y = x + p x

4 First, turn the problem into an unconstrained maximization problem in one variable, x, by solving the budget constraint for x and substituting the result into the objective function: Then, look for critical point(s): max x u (x ) = y p (x = p x Note that in this case the derivative is not a function of x and depending on the values of and p it will be positive or negative or zero. It means that utility is either always increasing, always decreasing in x ; or always constant. If p + 5 < 0, then the individual should consume no x and spend all of her income on x. That is, if p + 5 < 0, x = y and x = 0. If p + 5 > 0, then the individual should consume no x and spend all of their income on x. That is, if p + 5 > 0, x = 0 and x = y p. Finally, if p + 5 = 0, then the individual is indi erent i how much of x she consumes, and so she can spend any amount in the interval h0; y, and spend the rest of his income, y p p x, on x.. Assume that an individual s preferences can be described by the following utility function u (x ; x ; x ) = x + x + x Let y denote the consumer s income and let ; p and p denote the prices of goods and, where y, ; p and p are exogenous variables. A) Is the "more is always preferred to less" assumption satis ed for this utility function? B) Derive the individual s demand functions for the three goods. C) What happens to the demand for good if the price of good increases? D) Are goods and are substitutes or complements? Answer: A) More is preferred to less requires that the marginal utility of each good is always positive. That (x ; x ; x = > (x ; x ; x ) x > (x ; x ; x ) x > 0 4

5 The three marginal derivatives are in fact positive, and so the "more is always preferred to less" assumption is satis ed for this utility function. B) The individual s demand functions for the three goods are the solutions to the consumer s problem of maximizing utility subject to the budget constraint: subject to max x ;x u (x ; x ; x ) = x + x + x y = x + p x + p x Solve the budget constraint for one of the endogenous variables, say x. Solution is: Substitue for x in the utility function to get maxu (x ; x ) = x ;x x = y p x p x y p x p x + x + x Notice that we reduced the constraint optimization problem in three endogenus variables (x ; x and x ) to an unconstraint optimization problem in only two endogenous variables (x and x ). Look for critical (x ; x (x ; x ) p p + x + x Set each of these partials to zero and solve for x and x. With this simple utility function each can be solved separately. p + x = 0 p = p = p x p x = x = p x p p p Similarly, the solution for x is: 5

6 x p = p These are possibly the demand functions for goods and. To see if these critical values in fact maximize utility u (x ; x ), check if the second derivatives are u (x ; u (x ; x = = x < 0 x < 0 They are negative and so the critical values x and x in fact maximize u (x ; x ), and so they are the demand functions for goods and : x d = x d = p p p p And by substituting with x and x in the budget constraint, we obtain the demand function for good : x d = y p x p x y p p p = = y p p p p p C, D) To determine what happens to x d when p increases, we have to look at the sign of the partial derivative of x d with respect to p = p ( ) p > 0 That is, demand for x increases by p p p when p marginally increases. This means that goods and are substitutes. 6