Economics 401 Sample questions 2

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1 Economics 401 Sample questions 1. What does it mean to say that preferences fit the Gorman polar form? Do quasilinear preferences fit the Gorman form? Do aggregate demands based on the Gorman form have the property that demands are independent of the distribution of wealth? Justify your answers. The Gorman form means that the indirect utility function can be written as a + bw where a and b are functions of prices alone. Quasi-linear preferences do not necessarily fit the Gorman form. For example, if then u (x 1, x ) = x 1/ 1 + x x 1 x V (, p, w) Range (p / ) w/p p / p / + w/p w p / w/ 0 (w/ ) 1/ w < p / The first row conforms to the Gorman form but the second row doesn t. When wealth is sufficiently low quasilinear preferences may not fit the Gorman form. If preferences do fit the Gorman form and everyone has the same preferences then redistributions of wealth will not affect aggregate demands. To see this in the form of a two-good example write Using Roy s identity for good 1 v j (, p, w j ) = a(, p ) + b(, p )w j, j = A, B. a(,p ) x A 1 (, p, w A ) + x B 1 (, p, w B ) = b(, p ) from which the result is clear. b(,p ) b(, p ) (wa + w B ),. Consider N price-taking consumers in a three-good world; assume all consumers have the same preferences but their incomes may differ. What does it mean to say that their preferences fit the Gorman Polar Form? If their preferences are represented by the following utility function 1

2 u (x 1, x, x 3 ) = x 1 + f (x, x 3 ), f strictly concave, and wealth and prices are such that every consumer consumes a positive amount of good 1 show their preferences fit the Gorman Polar Form and that the total demand for each good is independent of the distribution of income across consumers. With three goods, preferences fit the Gorman Polar Form if the indirect utility function can be written as V (, p, p 3, w) = a (, p, p 3 ) + b (, p, p 3 ) w. With three goods and the assumption of an interior solution for optimal quantities purchased one can solve MRS 1 = /p and MRS 13 = /p 3 together with the budget constraint to obtain x j (, p, p 3, w). With the utility function above, which is quasi-linear in good 1, the marginal utility of good 1 is unity and the two MRS conditions depend only on x, x 3 and prices, not on x 1 or w. In this setting the implicit function theorem allows us to write the demands for goods and 3 as x (, p, p 3 ), x 3 (, p, p 3 ), and then the budget constraint yields Thus x 1 (, p, p 3, w) = w p x (, p, p 3 ) p 3x 3 (, p, p 3 ). which fits the GPF with V (, p, p 3, w) = w p x (, p, p 3 ) p 3x 3 (, p, p 3 ) + f (x (, p, p 3 ), x 3 (, p, p 3 )), a (, p, p 3 ) = p x (, p, p 3 ) p 3x 3 (, p, p 3 ) + b (, p, p 3 ) = 1. f (x (, p, p 3 ), x 3 (, p, p 3 )) Denote total demands for the N consumers with superscript T. From the discussion above we can see x T 1 x T x T 3 ( p1, p, p 3, w 1, w,..., w N) = 1 N j=1 ( p1, p, p 3, w 1, w,..., w N) = Nx (, p, p 3 ) ( p1, p, p 3, w 1, w,..., w N) = Nx 3 (, p, p 3 ), w j Np x (, p, p 3 ) Np 3x 3 (, p, p 3 )

3 so each total demand is either independent of wealth or depends only on total wealth, and not on the distribution of wealth across consumers. 3. Derive the profit function for the following one output, two input technology: y = min (z 1, z ). Do the associated output supply and factor demand functions have the expected properties? Explain. When the firm is maximizing profits with this technology Setting dprofits/dy = 0 obtain y = z 1 = z so Profits py w 1 z 1 w z = py w 1 y w y / And then p w 1 y w y = 0 or p y (p, w 1.w ) = w 1 + w ( ) z 1 (p, w 1.w ) = y (p, w 1.w ) p = w 1 + w z (p, w 1.w ) = 1 y (p, w 1.w ) = 1 ( ) p w 1 + w ( ) p p π (p, w 1.w ) = w 1 w ( p w 1 + w w 1 + w w 1 + w p = 1 ( ) p (w 1 + w ) w 1 + w w 1 + w p = 1 w 1 + w Note that output supply and input demands are homogeneous of degree zero in (p, w 1.w ); the profit function is homogeneous of degree one in (p, w 1.w ); and dy/dp > 0, dz 1 /dw 1 < 0, and dz /dw < Suppose that a price-taking, cost-minimizing firm can produce output q with inputs z 1 and z. Denote the output price by p and the input prices by w 1 and w. (a) Define mathematically what is meant by the term cost function. (b) Prove the cost function is 3 )

4 concave in input prices. Given (a) and (b) what are the properties of the input demand functions and the marginal cost function? (a) Denote the production function by q = f (z 1, z ). The cost function is the value function for the cost-minimization problem. C (w 1, w, q) = Min z 1, z, µ w 1z 1 + w z + µ (q f (z 1, z )) (b) For some t (0, 1) and two vectors of input prices (w 1 1, w 1 ) and (w 1, w ) define We need to prove that w t 1 = tw (1 t) w 1 w t = tw 1 + (1 t) w C ( w t 1, w t, q ) tc ( w 1 1, w 1, q ) + (1 t) C ( w 1, w, q ) Proof: Let (z t 1, z t ) be the cost-minimizing input vector for (w t 1, w t ). Then simple algebra yields C ( w1, t w, t q ) w1z t 1 t + wz t t = t ( ( ) w1z 1 1 t + wz) 1 t + (1 t) w 1 z1 t + wz t Now note that this number of dollars w 1 1z t 1 + w 1 z t is enough to produce output q at (w 1 1, w 1 ) but probably this is not the minimum number of dollars to produce output q at (w 1 1, w 1 ) so Likewise, w 1 1z t 1 + w 1 z t C ( w 1 1, w 1, q ). w 1z t 1 + w z t C ( w 1, w, q ). This completes the proof. Looking back at the definition of the cost function we see that the envelope theorem tells us that Marginal cost dc dq = µ (w 1, w, q) In addition, the cost minimization problem yields input demands for a cost minimizing firm: 4

5 z 1 (w 1, w, q) z (w 1, w, q) These input demands have to be homogeneous of degree 0 in (w 1, w ) which then tells us that total cost and marginal cost have to be homogeneous of degree 1 in (w 1, w ). That the cost function is concave in (w 1, w ) tells us that ] H [ C w1 C w 1 w C w w 1 C w is negative semi-definite. Among other things this means that the main diagonal is non-positive. Applying the envelope theorem again z 1 (w 1, w, q) = C (w 1, w, q) 0 w 1 w1 z (w 1, w, q) = C (w 1, w, q) 0, w w that is, input demands are not upward-sloping in their own prices. 5. The AP mill in Dryden Ontario produces wood pulp with labour, N, capital, K, and wood, W. Assume AP acts to maximize its profits and is a price-taker in all input and output markets. Denote the input prices by P N, P K, P W and the output price by P. Using data on profits, π (which are always positive) and prices Andy has estimated AP s profit function using the following functional form: ln π = α 0 + α 1 ln P N + α ln P K + α 3 ln P W + α 4 ln P. Since AP is a major employer in Dryden and there is a lot of unemployment in the area the Ontario Ministry of Finance has decided to subsidize at a proportional rate, λ, the price of labour to the firm. Assume AP s data fit the above profit function perfectly and no price changes as a consequence of the wage subsidy. Find a formula for λ in terms of the parameters of the profit function if the government s objective is to increase AP s use of labour by 50 percent. Since π(p N, P K, P W, P ) = Max N, K, W P f(n, K, W ) (P NN + P K K + P W W ), where f is the production function, we can use the envelope theorem to obtain the input demand for N as a function of the input prices and the output price. 5

6 N(P N, P K, P W, P ) = π(p N, P K, P W, P ) P N Taking the derivative of the profit function with respect to P N we obtain Since 1 π π P N = α 1 P N or N(P N, P K, P W, P ) = α 1π P N. we can see that ln π = α 0 + α 1 ln P N + α ln P K + α 3 ln P W + α 4 ln P exp (ln π) = exp (α 0 + α 1 ln P N + α ln P K + α 3 ln P W + α 4 ln P ) exp (ln π) = exp (α 0 + ln P α 1 N + ln P α K + ln P α 3 W + ln P α 4 ) π = e α 0 P α 1 N P α K P α 3 W P α 4 N(P N, P K, P W, P ) = α 1 e α 0 P α 1 1 N P α K P α 3 W P α 4. So to increase the firm s use of N by 50 percent we have N 0 = α 1 e α 0 P α 1 1 N P α K P α 3 W P α 4 and 3 N 0 = α 1 e α 0 [(1 + λ) P N ] α1 1 P α K P α 3 W P α 4 and dividing the second line by the first obtain which can be solved for λ. 3 = (1 + λ)α 1 1, 6. (a) What is a profit function? Is it convex or concave in prices? Prove your answer in the context of one output and two inputs. (b) A profit-maximizing price-taking firm is asked to choose between two environments A and B. In A, output price is in odd-numbered years (like 1999) and p in even-numbered years (like 000), with p. In B, price is p 3 = ( + p )/ every year. Assume the real interest rate is zero. Which environment would the firm prefer? Justify your answer.answer both parts (a) and (b); they are of equal weight. (c) Suppose output q can be produced with inputs z 1 and z. When the output price is p and input prices are w 1 and w, the profit-maximizing levels of output and inputs are q, z 1 and z. Suppose in the short run z cannot be varied but z 1 can be, but in the long 6

7 run both inputs are variable. Let the price of output rise to p > p, and let input prices be constant. Prove that the increase in output in the long run is at least as large as the increase in output in the short run. S (a) A profit function is the solution to the following problem. π(w 1, w, p) = Max z 1, z pf(z 1, z ) w 1 z 1 w z, where w i are input prices, p is output price and f(z 1, z ) is the production function. π(w 1, w, p) is convex in prices. Proof: Let r 1 (w 1 1, w 1, ) (w 1, w, p ) r, and r 3 = αr 1 + (1 α)r, for some α [0, 1]. Suppose q, z 1 and z maximize profits at r 3. Then π(w1, 3 w, 3 p 3 ) = [ z1 z q ] w 3 1 w 3 = α [ z1 z q ] p 3 w 1 1 w 1 = [ z1 z q ] απ(w 1 1, w 1, ) + (1 α)π(w 1, w, p ), which means that the profit function is convex. αw1 1 + (1 α)w1 αw 1 + (1 α)w α + (1 α)p + (1 α) [ z1 z q ] (b) Without loss of generality, for the next part of the question, assume < p. Graph the firm s profits, π(p) against p. Average annual profit in environment A is (π( )+π(p ))/ (ignore discounting). Annual profit in environment B is π(( + p )/). Since π(p) is convex in p, (π( ) + π(p ))/ π(p). Thus the firm prefers A. (c) Write short-run total cost as C S (q) and long-run total cost as C L (q). In C S (q) z is fixed at z and so C S (q) C L (q) and we know the two are equal at q. So q is a minimizer of g(q) C S (q) C L (q). Thus w 1 w p g (q ) = C S(q ) C L(q ) = 0 g (q ) = C S(q ) C L(q ) > 0. So at q the short-run and long-run marginal costs are equal to each other and the shortrun MC is steeper than the long-run MC. Given p > p, then, the q at p = C L (q) will exceed the q at p = C S (q). 7. Textbooks introduce various kinds of systems of demand equations for consumers (e.g. Walrasian and Hicksian demands) and firms (e.g. inputs demands based on profit maximization or on cost minimization). How are the various consumer demand systems related to the 7

8 various producer demand systems? If a particular consumer (producer) demand system has no counterpart in producer (consumer) theory can you design a new demand system that would fill this gap? Explain. Assume goods or inputs. Marshallian demands derive from the problem Max x 1, x u(x 1, x ) such that w x 1 p x = 0 which yields x i (, p, w). (1) Hicksian demands derive from the problem Min h 1, h h 1 + p h such that u(h 1, h ) u 0 = 0 which yields h i (, p, u 0 ). () Input demands of a profit-maximizing price taking firm arise from the problem Max x 1, x pf(x 1, x ) w 1 x 1 w x (3) which yields x i (w 1, w, p). Input demands of a cost-minimizing firm derive from the problem Min x 1, x w 1 x 1 + w x such that f(x 1, x ) y 0 = 0 which yields x i (w 1, w, y 0 ) (4) Problems () and (4) are clearly the same and thus the associated demands functions h i (, p, u 0 ) and x i (w 1, w, y 0 ) have much in common. The only difference is that utility cannot be measured in the same sense as output. Problems (1) and (3) are different. The analogue of (1) in producer theory would be a firm that maximized sales subject to a constraint on the total costs of its inputs. The problem faced by this kind of firm would be Max x 1, x pf(x 1, x ) such that C 0 w 1 x 1 w x = 0 which would yield x i (w 1, w, C 0, p), which are much like the Marshallian demands of consumer theory. One could produce the analogue of (3) in consumer theory by using the concept of the marginal utility of money - the extra utility yielded by one more dollar of expenditure. Denote the marginal utility of money by λ. The inverse of λ is the price of utility - the number of dollars it takes to buy one unit of utility. 8

9 Max 1 x 1, x λ u(x 1, x ) x 1 p x (5) which yields x i (, p, λ). Note that the first-order conditions in (5) are the same as those in (3). 8. Consider a risk-averse individual with initial wealth w 0 and a Bernoulli utility function u(w) who must decide whether and for how much to insure his car. Assume the probability that he will not have an accident is π. In the event of an accident, he incurs a loss of $L < w 0 in damages. Suppose that insurance is available at an actuarially fair price, that is, one that yields insurance companies zero expected profits; denote the price of $1 worth of insurance coverage by p. Write down the individual s expected utility if he purchases x dollars of insurance. How much insurance will this individual purchase? Does this this individual s demand for insurance slope downward with respect to the price of insurance at the point where insurance is priced fairly? Defend your answer carefully. The individual s expected utility is f (x, p, π, w 0, L) πu (w 0 px) + (1 π) u (w 0 px L + x) If the insurance is priced fairly the firm s expected profit on each dollar of insurance is zero, so πp + (1 π) (p 1) = 0 or p = 1 π. To find the optimal value of x given the parameters of the problem set f (x, p, π, w 0, L) x and solve for x at p = 1 π. Thus = 0 ( p) πu (w 0 px) + (1 π) (1 p) u (w 0 px L + x) = 0 or u (w 0 px) = u (w 0 px L + x) and thus w 0 px = w 0 px L + x or x = L. We want to know 9

10 Sign dx dp p=1 π ******************************************************************** Aside on the comparative statics method: Suppose an agent is maximizing her objective function f (x, a), where x is the choice variable and a is a parameter. Assuming the maximization problem is well-behaved, and all we are interested in is the sign of dx/da then the method of comparative statics (the implicit function theorem) tells us that Sign dx da = Sign f (x, a) a x. Why? The first-order condition for this problem is f (x, a) = 0 x which implicitly defines the optimal level of x as a function of a. Taking a total differential of this equation obtain f (x, a) dx + f (x, a) da = 0 or x a x dx da = f (x, a) / a x f (x, a) / x The result follows immediately from this equation and that the second-order conditions for the maximization require f (x, a) / x < 0. ******************************************************************** In the context of this question then Sign dx dp = Sign f (x, p, π, w 0, L) p=1 π p x From above p=1 π f (x, p, π, w 0, L) x Checking second-order conditions = ( p) πu (w 0 px) + (1 π) (1 p) u (w 0 px L + x), f (x, p, π, w 0, L) x = ( p) πu (w 0 px) + (1 π) (1 p) u (w 0 px L + x) which must be negative if u (w) is strictly concave, that is, u (w) < 0. Then, applying the method of comparative statics, 10

11 f (x, p, π, w 0, L) p x = πu (w 0 px) + ( p) πu (w 0 px) ( x) (1 π) u (w 0 px L + x) + (1 π) (1 p) u (w 0 px L + x) ( x) But at p = 1 π we have proved x = L so f (x, p, π, w 0, L) p x = u (w 0 (1 π) L) < 0 p=1 π Thus, when insurance is priced fairly, an increase in the price of insurance lowers the demand for insurance. 11

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