Microeconomics Fall 07 Problem set Possible answers Each answer resents only one way of solving the roblem. Other right answers are ossible and welcome. Exercise For each of the following roerties, draw a air of roduction sets that satisfy and violate this roerty a) free disosal; b) non-decreasing returns to scale; c) constant returns to scale; d) convexity. Solution shown in class Purose You have visualized your ideas about what the roerties of roduction set are. Exercise Consider the following single-inut single-outut roduction function f(x) = q The average roduct of x is de ned as AP (x) = q x Show that the roduction technology, associated with f(x), has non-decreasing returns to scale if, and only if, AP (x) is a non-decreasing function of x. Solution The su ciency of the condition AP (x) is a non-decreasing function of x, thus for any, AP (x) AP (x), which imlies f(x) = xap (x) xap (x) = f(x). We have established that f() has non-decreasing returns to scale. Next, we show that the associated roduction set Y exhibits non-decreasing returns to scale. Assume that ( x; q) Y, so q f(x). For, the following holds q f(x) f(x). Thus we have ( x; q) Y. The necessity of the condition can be established in a similar way. Purose You have learnt to deal with di erent reresentations of roerties of roduction set. Exercise 3 For any homothetic roduction function show that the conditional factor demand function is exressed in the form z(w; q) = z(w; )h(q); where h(q) is an increasing function of q. Find h(q) for the following Cobb-Douglas roduction function f(z ; z ) = z z
Solution 3 We can construct the required conditional factor demand using the roerty of homothetic functions f( x ) = f(x). Then a roblem of minimizing wx s.t f(x) q can h(q) q equivalently be written as that of mimimizing wx x s.t. f( ). solution to this roblem h(q) h(q) is x = z(w; ), so the solution to the initial roblem can be written as h(q) x = z(w; )h(q), QED. Conditional factor demand functions for the given Cobb-Douglas roduction technology are given by z (w ; ; q) = ( w ) + q + Thus, h(q) = q + z (w ; ; q) = ( w ) + q + Purose You have learnt a new roerty of conditional demand functions. Exercise 4 Consider a constant returns to scale roduction rocess in which there is only one outut q There are two (non-negative) inuts, z and z Outut can be made in either of two ways, each of which can be emloyed at any scale. Cometitive conditions determine whether only one of these methods is used, or whether they are both used. The two rocesses are described by and Let the rices of the inuts be w and q = minfz ; z g q = z + z a) What are the cost functions for these two methods? b) Suose that these roduction rocesses are available to all rms in an economy, and that cometitive conditions revail. What is the factor-rice frontier (set of rices at which the rm breaks even) for this economy? c) Assume that from erfect data you know recisely factor rice frontier. Can you use this knowledge to determine what technology is used? Solution 4 a) The cost function for the rst technology is C = q(w + ); for the second C = q min(w ; ).
b) A ro t maximizing rm always chooses technology that minimizes costs, hence the unit cost of roduction is c = min(c ; c ) = min(w + ; w ; ) = min(w ; ) Since there are no xed costs, the factor rice frontier includes all the oints at which rice is equal to unit costs where is given. F P F = f(w ; ) min(w ; ; w + ) = g; c) If we observe that w = = =; we can not say which technology is used, as they are equally e cient. If the inut rices di er, however, it means that the second (linear) technology is used. Purose You have thought about how to aly your knowledge of cost minimization to a real dataset. Exercise 5 State the rm s maximization roblem, derive the rm s factor demand functions, suly function, ro t function and verify Hotelling s lemma for q(w; ); z (w; ); and z (w; ); set u the rm s minimization roblem and derive conditional factor demand functions for the following single-outut technology with roduction function given by f(z) = z z ; > 0; > 0 Do you need additional restrictions on the arameters for the functions to be well de ned? Solution 5 The maximization roblem is max z ;z q w z z st q z z ; z 0; z 0 With > 0 the roduction function constraint is binding (otherwise we would lose some ro t from underroduced q). So for an interior solution we can rewrite the roblem as max z ;z z z w z z and the necessary conditions for an interior maximum are z z = w ; z z = This can be rewritten as z z = w 3
We see that this relation only determines a ratio of inuts, but not their absolute value. Indeed, as we said in the lecture, with non-decreasing returns to scale (here + ) we either have zero or in nite roduction. To see that, write down the ro t function using the otimal ratio The FOC is and the SOC is max z + z ( + ) z + w w ( + ) ( + ) z + w + z w + = 0 () w < 0; so for + the FOC describes a minimum of our function. By insecting the sign of the exression (), we can nd the solution for this case. Since this exression is non-decreasing in z, if it is ositive at z! +, we have unbounded ro t and no well-de ned solution; if it is negative, we have a corner solution z = z = 0. Thus, for + > there is no solution, and for + = there is no solution if > w w () and z = z = 0 in the oosite case. In case of equality in the above condition any roduction lan is otimal - we have a suly corresondence. Now, let us turn to the more normal case of decreasing returns to scale ( + < ). In this case the condition () is also su cient, so that we have z = w z = w ; ; which is of course a standard Cobb-Douglas case. With this knowledge at hand, we can restrict ourselves to the nice case of + < and write the factor demand functions as above, suly function as ro t function as () = q = + w w w To verify Hotelling s lemma, we di erentiate @ () = @ @ () = @w @ () = @ ; ( ) w w ; w w w w 4 ;
The cost minimization roblem is min z ;z w z + z st q z z ; z 0; z 0 Because w i > 0, the roduction function constraint is binding, and as long as q > 0 we must have z i > 0, so the roblem can be rewritten as FOC is then and SOC is min z w w z + q z q z = 0 + q z > 0; which is always satis ed. So we have a minimum indeed at Corresondingly, z = z = These are conditional factor demands. w + q + w w + q + Purose You have trained your technical skills related to ro t maximization. Exercise 6 Consider a rm with technology described by the roduction function y = (min fz ; z g) Find the outut suly function y(; w), the demand function of the two inuts z(; w) and the ro t function (; w). Which restriction do you need to imose on the values of for such functions to be well de ned? Solution 6 Solve max q w z z st q (min fz ; z g) We know that with erfect comlements we must have z = z, and with this roduction function also equal to q. Denote =. The ro t maximization can then be rewritten as max q (w + ) q F OC = (w + ) q ; q > 0 5
Thus, the suly function is and the demand functions are q = z = z = (w + ) (w + ) The ro t function is (w + ) (w + ) (w + ) Clearly, we need > 0, as otherwise roduction is not de ned (ro t increases in outut monotonically, so it is unbounded). Moreover, if <, FOC actually icks u the minimum of the ro t ( ( ) (w + ) q > 0), so the relevant solution is again unbounded the ro t q (w + ) q grows monotonically from the minimum on. Thus, we have to imose the restriction. Corresondingly, for the initial arameter = this imlies 0 <. Purose You have trained your technical skills even more ) 6