Economics 386-A1 Practice Assignment 3 S Landon Fall 003 This assignment will not be graded. Answers will be made available on the Economics 386 web page: http://www.arts.ualberta.ca/~econweb/landon/e38603.html. The goal of this assignment is to help you review the course material and to expose you to the types of questions that I may ask on exams. The assignment is not intended to be comprehensive (as not all the questions asked on the exams will necessarily relate directly to the questions on the assignments). In particular, the exams may include more questions that ask you to explain concepts or answers in words than does this assignment. The material covered by this assignment generally corresponds to the material in Chapters 10-1 of Chiang. I am happy to discuss any of the questions and answers with you (or any other questions you may have about the course material). The questions are loosely organized by chapter. Chapter 10 1. Consider the utility function: U = x α y β where x and y are the quantities of the two goods consumed, and α and β are parameters with 0<α<1, 0<β<1 and 0<α+β<1. Show that the slopes U of the indifference curves associated with the three utility functions: U, V=log(U), and Z=e are all the same.. Consider the function: y = f(x) = α x where α is a constant parameter and 0<α. Is this function concave, convex or neither? Explain. 3. Consider the function: y = A x α x where A and α are both constants (0<A, 0<x). Find d y. dx Chapter 11 4. Consider the following function: y = f ( x, w) = g( x) + h( w). Find conditions that must be imposed on the derivatives of the g(x) and h(w) functions to ensure that the f(x,w) function is strictly concave? Explain. 1
5. Suppose the production function of the firm is the following: α Q = L + K β firm's output. The firm takes the price of its output (P), the price of labour (w) and the price of capital (r) as exogenous. The costs of the firm are wl+rk while revenues are PQ. Assume w, r and P are all positive and that L and K must also be positive. a) Set up the firm's unconstrained profit maximization problem with L and K as the firm's choice variables. b) Derive the first order conditions for a profit maximum. c) What are the sufficient second order conditions for a unique profit maximum? d) For the sufficient second order conditions to be satisfied, what restrictions must be placed on α and β? Explain briefly. e) Derive the firm's optimal choices of L and K as functions of exogenous variables only. f) Find the impact of an increase in r on the firm's optimal choice of K. g) Find the impact of an increase in r on the firm's optimal choice of L. h) Find the impact of an increase in r on the firm's profit (assume the firm is maximizing profit). 6. Suppose the production function of the firm is the following: Q = F( L, K) F ( ) > 0, F ( ) > 0, F ( ) < 0, F ( ) < 0, F ( ) > 0, L e) Find the impact of an increase in r on the firm's optimal choice of L. K firm's output. The firm takes the price of its output (P), the price of labour (w) and the price of capital (r) as exogenous. The costs of the firm are wl+rk while revenues are PQ. a) Set up the firm's unconstrained profit maximization problem with L and K as the firm's choice variables. b) Derive the first order conditions for a profit maximum. c) What are the sufficient second order conditions for a unique profit maximum? d) Find the impact of an increase in r on the firm's optimal choice of K. LL KK LK
f) Find the impact of an increase in r on the firm's profit (assume it is maximizing profit). 7. Suppose the production function of the firm is the following: Q = L α K β α>0, β>0, α+β<1. firm's output. The firm takes the price of its output (P), the price of labour (w) and the price of capital (r) as exogenous. The costs of the firm are wl+rk while revenues are PQ. a) Set up the firm's unconstrained profit maximization problem with L and K as the firm's choice variables. b) Derive the first order conditions for a profit maximum. c) What are the sufficient second order conditions for a unique profit maximum? d) Derive the firm's optimal choices of L and K as functions of exogenous variables only. All exponents in these solution equations should be positive. e) Find the impact of an increase in r on the firm's optimal choice of K. f) Find the impact of an increase in r on the firm's optimal choice of L. 8. A firm producing good X faces the following demand function for its output: P = α βx + γa where P is the price of X, A is the firm's expenditure on advertising, and α, β and γ are positive parameters. The firm's total cost of production is: X TC = F + + A, where F represents an exogenous fixed cost and F>0. The firm chooses X and A to maximize profits taking F and all the parameters as given. Assume that the firm always makes positive profits. The firm pays a tax equal to a fraction τ of its revenues from selling X. a) Set up the firm's profit maximization problem for the case in which the firm chooses both the quantity of X to produce and the quantity of advertising (A). b) Find the necessary first order conditions. 3
c) Find the second order sufficient conditions for a unique profit maximum. (Assume they hold.) d) What is the impact of an increase in F on the firm's optimal choice of A and X? e) Find the effect of an increase in the tax rate, τ, on the optimal quantity of advertising by the firm. 9. Consider the production function: β / α [ L y = F( L, K) = + K ] 1, 0<α<1, 0<β<1, where y is units of output produced, and L and K are units of labour and capital, respectively. a) What is the marginal product of capital? Show your work. b) Find the derivative F LK ( ) and sign it. Show your work. c) Find the slope of an isoquant associated with this production function. Assume K is on the vertical axis and L is on the horizontal axis. Simplify your answer as much as possible. d) Now assume that α=1. Is the function strictly concave or strictly convex? Explain and show your work. Chapter 1 10. Suppose the production function of the firm is the following: Q = + 1/ 1/ L K firm's output. The firm takes the wage (w) and the price of capital (r) as exogenous. The total per unit price of labour to the firm is: v = (1+t)w where t is a payroll tax (such as EI or health insurance premiums). Thus, the total costs of the firm are vl+rk. The goal of the firm is to choose the quantity of labour and capital that will minimize the costs of production subject to producing a given level of output Q (the production function is the constraint). 4
a) Set up the firm's constrained cost minimization problem with L and K as the firm's choice variables (there may be more choice variables). b) Derive the first order conditions for a cost minimum. c) What are the sufficient second order conditions for a unique cost minimum? Are they satisfied? d) Find the optimal choices of K and L as a function of non-choice variables only. e) Find the impact of an increase in t on the firm's optimal choice of K. 11. Suppose and individual has the following utility function: U = x y where U represents the individual's utility level, and x and y are the quantities of two goods consumed by the individual. The budget constraint of the individual is: M = P x x + P y, y where M is the individual's income, and P x and P y are the prices of x and y, respectively. The goal of the individual is to pick the values of x and y that maximize utility subject to the budget constraint. a) Set up the Lagrangean to maximize utility subject to the budget constraint. b) Find the necessary first order conditions. c) Find the sufficient second order conditions for a unique utility maximum and determine whether they are satisfied. d) Find the solution equation that describes the optimal choice of x as a function of nonchoice variables only. e) Show how the optimal choice of x changes as P y changes. f) What is the own price elasticity of demand for x? The income elasticity? 1. Suppose a benevolent social planner has the utiltiy function: X U = + G N 1/ 5
where U is the planner s utility, X represents the total units of private consumption in the economy, N is the exogenous size of the population and G represents the units provided of a public good. The economy has an endowment of goods equal to E that is exogenous. The goal of the social planner is to maximize utility by choosing X and G subject to the constraint that this choice cannot exceed the economy s endowment of E. That is, it must always be the case that E = X + G. a) Find an expression for the slope of an indifference curve associated with the planner s utility function and try to sign it. Assume G is on the vertical axis. b) Is the planner s utility function strictly concave? Show why or why not. c) Set up the social planner s utility maximization problem as a Lagrangean with one constraint (E = X + G) and identify the choice variables. d) Find the necessary first order conditions for a utility maximum to the problem given in (c). e) Find the sufficient second order conditions for a unique utility maximum. Are they satisfied? f) Using calculus show how the optimal level of X changes as N changes. Find the sign of this effect. 13. Consider the production function: Y = LK where Y is the output produced by a firm, and L and K are the quantities of labour and capital used by the firm to produce Y. a) What is the slope (in the standard (L,K) diagram) of an isoquant associated with this production function? Suppose the goal of the firm is to minimize the cost (wl+rk) of producing a given level of output ( Y ) by choosing L an K. In other words, it faces the constraint Y = LK. b) Write down the Lagrangean function associated with this cost minimization problem. c) Find the first order conditions for a cost minimum. d) Check the second order conditions. e) Find the derivative that shows how the optimal choice of L changes as a result of a change in w. 6
14. Consider the production function: Y = LK, L>0, K>0. a) Find the slope of an isoquant associated with this production function in the usual diagram with L on the horizontal axis and K on the vertical axis. b) Find the derivative that shows how the slope of the isoquant changes following a movement down the isoquant, sign this derivative, and explain what the derivative implies about the slope. c) Suppose this is the production function of a firm that is trying to choose the quantity of labour (L) and capital (K) that will minimize the cost of producing a given level of output, Y 0. That is, the firm wants to choose L and K to minimize wl +rk subject to Y 0 = LK. i) Set up the cost minimization problem of the firm. ii) iii) iv) Find the necessary first order conditions. Find the sufficient second order conditions for a minimum. Are they satisfied? Explain. Solve for the firm's cost minimizing choice of capital. 15. Suppose an individual has the following utility function: U = g(x)h(y) = g(x)(y-α) where x and y are the quantities consumed of two goods, h(y) = y-α, 0<α<y, and 0<g (x) and g (x)<0. Suppose the individual has a given income (M) and faces the exogenous prices for x and y of P x and P y, respectively. Thus, the individual faces the budget constraint: M = P x x + P y y. For simplicity, assume that P x = P y = 1 so that the budget constraint is M = x + y. a) Set up the utility maximization problem of the individual as a Lagrangean with one constraint and find the first order conditions for a utility maximum. Show your work. b) Find the second order conditions for a utility maximum and explain whether they hold. c) Find the derivative that gives the impact of an increase in α on the demand for x. Show your work and sign the derivative. 7