Additional questions for Chapter 1 - solutions: 1. See Varian, p.20. Notice that Varian set a1=a2. 2. See Varian, p.20.

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1 Additional questions for Chapter 1 - solutions: 1. See Varian, p.0. Notice that Varian set a1=a.. See Varian, p.0.. See Varian, p.0. Idea: use the TRS 4.

2 Additional questions for Chapter - solutions: 1.

3 If B < 1, Ytx) = = t Bp/p *x 1 p = x p ). If B < 1 & we know 0<p <1, then BP < p, t Bp/p then < t, then DRS.

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7 Louisiana State University Economics 770 Handout This problem concerns a price-taking firm that produces two outputs, goods and, using a single input, good 1. Its implicit production function aka transformation function) is: F : R R : F y 1, y, y ) = y ) + y ) + y 1, so that any vector y 1, y, y ) satisfying y 1 0 and y ) + y ) + y 1 0 is technologically feasible. Moreover, its production set satisfies the free disposal assumption. 1 Does the technology display nonincreasing returns to scale? 1 Show your work and explain. Nonincreasing returns to scale requires that for any y Y and any t [0, 1], then ty Y. The production set Y is only partially defined by F y) 0 y Y, as there are other points in Y as a consequence of the free disposal assumption. However, we can limit ourselves to those vectors such that F y) 0. Therefore nonincreasing returns to scale is satisfied if whenever F y) 0 and t [0, 1], we then also have F ty) 0. Let y R such that F y) 0 and let t [0, 1]. In particular, we then have y ) + y ) + y 1 0. Then, F ty) = ty ) + ty ) + ty 1 = t y ) + y ) ) + ty 1 = t t y ) + y ) ) + y 1 ) t y ) + y ) ) + y 1 ) = t y ) + y ) + y 1 ). From y ) + y ) + y 1 0 and t [0, 1] it follows that t y ) + y ) + y 1 ) 0 and therefore F ty) 0. This shows that the technology displays nonincreasing returns to scale. Write the PROFITMAX Problem. The P ROF IT MAX[p] problem is max p y y 1,y,y R s.t. y 1 0, y + y + y Your answer must be based on the definition presented in class and in the textbook); alternative definitions of this term might not be equivalent. 1

8 Obtain the supply function and profit function. assuming price-taking in all input and output markets). Since y 0 and y 0, and hence y + y 0, it follows from F y) 0 that y 1 0. Therefore the y 1 0 constraint is redundant. Let ŷ Y such that F ŷ) < 0, or ŷ ) + ŷ ) + ŷ 1 < 0. The production vector ŷ is feasible but is not on the transformation function frontier.) Then for some positive value y 1 we have ŷ ) + ŷ ) + ŷ 1 + y 1 = 0. The increment y 1 reduces the negative value of y 1 so that we are using less of this input. Then profit is increased by lowering this input expenditure by p 1 y 1. It follows that ŷ cannot be profit maximizing. Therefore we can limit our attention to production vectors on the the transformation function frontier defined by y + y + y 1 = 0. With these two observations we know that we can instead work with this simpler equivalent profit maximization problem with only constraint, an equality constraint: The Lagrangian function is then max y 1,y,y R p 1 y 1 + p y + p y s.t. y + y + y 1 = 0. Ly 1, y, y, λ) = p 1 y 1 + p y + p y λ y + y + y 1 ) with first order conditions, A 1 ) : p 1 λ = 0, A ) : p 4λy = 0, A ) : p 6λy = 0, B) : y + y + y 1 = 0 From these, we can obtain the component supply functions in the following order, ỹ p 1, p, p ) = p 4p 1, ỹ p 1, p, p ) = p, 6p 1 ỹ 1 p 1, p, p ) = = 1 8 = 1 4p 1 p ) ) ) p + 4p 1 6p 1 ) + 1 ) ) p p 1 1 p 1 + p ) p p = p + p 4p 1,

9 and the profit function, 1 p Πp) = p 1 4p 1 + p ) p p + p + p 4p 1 6p 1 = 1 p 4p 1 + p ) + 1 p p 1 + p ) = 1 p 4p 1 + p ) = p + p. 4p 1 4 Verfy that Hotelling s Lemma applies with all three commodities, input and outputs. Hotelling s lemma states that Πp) p l = ỹ l p), l = 1,..., L. In our case, Πp) = 1) 1 1 p p 1 4 p 1 + p ) = ỹ 1 p), Πp) = 1 p ) = ỹ p), p 4p 1 Πp) = 1 p ) = ỹ p),. p 4p 1 5 Show that the simple law of supply applies to both outputs and that the simple law of demand applies the single factor of production. Supply: ỹ p 1, p, p ) = 1 > 0, p 4p 1 ỹ p 1, p, p ) = 1 > 0. p 6p 1 Thus both outputs increase with their own price, satisfying the law of supply. Factor Demand: ỹ 1 p 1 p 1, p, p ) = ) 1 4p 1 + p ) p = 1 p p 1 + p ) > 0. This equation shows that y 1 increases in value on the real line with its own price. However, since y 1 0, this means that y 1 is moving towards zero from the left. Therefore the absolute value of y 1 is getting smaller so that in fact there is a smaller demand for the input with a higher input price. Thus, the the law of demand is satisfied.

10 Louisiana State University Economics 770 Fall 014 Answer Key General CES Production. For this problem set you are going to work with a more general version of CES production than presented in class. Let the production function f : R + R be defined by: fx) = a 1 x 1 ) ρ +a x ) ρ ) α/ρ, where a 1 > 0, a > 0, α > 0 and ρ,0) 0,1). 1. Carefully state this production technology as a transformation function. Include all appropriate restrictions on independent variable values. We have two inputs and a single output so that Y R. For each netput vector y R, let y 1 and y be the two respective inputs and let y be the output. 1 Recall that the convention with transformation functions requires all inputs be stated as negative values, so that y 1 = x 1 and y = x. Therefore, in creating the transformation function from the production function we need to substitute for x 1 and x with x 1 = y 1 and x = y. Then with technical efficiency we have y = a 1 y 1 ) ρ +a y ) ρ ) α/ρ, y a 1 y 1 ) ρ +a y ) ρ ) α/ρ = 0. 1) Note that with technical inefficiency, the left hand part of equation 1) is less than zero. Also, with combinations of inputs and output that are not feasible this same is expression is greater than zero. Thus the transformation function is F : R R + R, Fy 1,y,y ) = y a 1 y 1 ) ρ +a y ) ρ ) α/ρ.. Specify the general input requirement set as a function of output, y. The input requirement set specifies for any given output level, the feasible combinations of inputs. A combination of inputs is infeasible if the output level cannot be obtained, fx) < y. Thus with feasibility, we may have either technical efficiency with fx) = y or inefficiency with fx) > y. Therefore the input requirement set is Vy) = { x 1,x ) R + a 1 x 1 ) ρ +a x ) ρ ) α/ρ y }. 1 Alternatively, you could have let y 1 be the output, and y and y be the inputs. That is in the context of the production function, fx) < y. 1

11 . Carefully show that this technology is monotone. In class I presented four different equivalent characterizations of production monotonicity, including two based on the input requirement set and one based on the transformation function. I will work with the transformation function. In that context, monotonicity requires that whenever Fy) 0 the netput vector is feasible) and y y, then Fy ) 0 with both y,y R N ). Working directly with this definition, let y R with Fy) 0 so that y a 1 y 1 ) ρ +a y ) ρ ) α/ρ 0. Also let y y for y R. Then y 1 y 1, y y and y y. We now need to work separately with the two ranges for values of ρ. Starting with ρ 0,1), we have y 1) ρ y 1 ) ρ, y ) ρ y ) ρ, and hence a1 y 1) ρ +a y ) ρ) α/ρ y a 1 y 1) ρ +a y ) ρ) α/ρ a 1 y 1 ) ρ +a y ) ρ ) α/ρ, y a 1 y 1 ) ρ +a y ) ρ ) α/ρ. Thus Fy ) Fy) and hence Fy ) 0. Because negative powers are monotonically decreasing, the range ρ,0) is a little bit more tricky. First we have y k y k y k )ρ y k ) ρ for k = 1,, and hence a 1 y 1) ρ +a y ) ρ a 1 y 1 ) ρ +a y ) ρ. Then since also α/ρ < 0, a1 y 1) ρ +a y ) ρ) α/ρ a 1 y 1 ) ρ +a y ) ρ ) α/ρ. The rest follows as before. When you first work with a new concept such as monotone production, it is best to work directly with the definition to develop familiarity, as we just did. However there are sometimes other approaches. In this case with Fy) differentiable, the three first-order partial derivatives are useful. Beginning with y 1, F y 1 y) = α/ρ)a 1 y 1 ) ρ +a y ) ρ ) α/ρ) 1 ρa 1 y 1 ) ρ 1 1) = αa 1 y 1 ) ρ 1 a 1 y 1 ) ρ +a y ) ρ ) α/ρ) 1. Since each of these factors is positive with y 1 positive), F/ y 1 y) > 0. Similarly F/ y y) > 0, and F/ y y) > 0 is trivial. It then follows that whenever y y, we also have Fy ) Fy) and hence Fy ) 0. For x,y R + and a < 0, x y x a y a.

12 4. Provide an exact characterization of all parameter value combinations such that the consequent production technology has decreasing returns to scale. Show your work i.e., justify your characterization). One way of providing an exact characterization is with set notation. For example with the subset of real numbers R = ρ,0) 0,1), you could define this set with the form, { a 1,a,α,ρ) R ++ R } Again working directly with our definition for transformation functions, decreasing returns-to-scale requires that whenever Fy) = 0 for y R y 0) and t > 1, we also have Fty) > 0. In this case, Fty) = ty a 1 ty 1 ) ρ +a ty ) ρ ) α/ρ = ty a 1 t ρ y 1 ) ρ +a t ρ y ) ρ ) α/ρ = ty t ρ a 1 y 1 ) ρ +a y ) ρ )) α/ρ = ty t ρ ) α/ρ a 1 y 1 ) ρ +a y ) ρ ) α/ρ = ty t α a 1 y 1 ) ρ +a y ) ρ ) α/ρ. From Fy) = 0 we have y = a 1 y 1 ) ρ +a y ) ρ ) α/ρ > 0, so that Fty) = ty t α y = t t α )y. Then Fty) > 0 t t α > 0 t > t α α < 1. Thus whether or not this technology exhibits decreasing returns-to-scale depends only on the value of α and not at all on the other parameter values. With the suggested set notation this becomes { a 1,a,α,ρ) R ++ R α 0,1) }. 5. Is this production technology homothetic perhaps for only some parameter value combinations)? Fully justify your answer. Homothetic technology is defined in terms of the production function. It requires that f = T g where g : R + R is homogeneous of degree one HOD-1) and T : R R is an increasing monotonic function. For this we need to find out if f is homogeneous and of what degree), ftx) = a 1 tx 1 ) ρ +a tx ) ρ ) α/ρ = a 1 t ρ x 1 ) ρ +a t ρ x ) ρ ) α/ρ = t ρ a 1 x 1 ) ρ +a x ) ρ )) α/ρ = t ρ ) α/ρ a 1 x 1 ) ρ +a x ) ρ ) α/ρ = t α a 1 x 1 ) ρ +a x ) ρ ) α/ρ = t α fx). Thus f is HOD-α. It follows that with Th) = h α and gx) = a 1 x 1 ) ρ +a x ) ρ ) 1/ρ HOD-1, we have fx) = Tgx)), so that f is homothetic.

13 6. Provide the Technical Rate of Substitution of input 1 in terms of input for this technology as a function in the form TRS 1, =... Show your work, starting with the production function provided above. I shall work with the general equation that we found, Beginning with x 1, TRS 1, x) = f x) x 1. f x) x f x 1 x) = α/ρ)a 1 x 1 ) ρ +a x ) ρ ) α/ρ) 1 ρa 1 x 1 ) ρ 1 = αa 1 x 1 ) ρ 1 a 1 x 1 ) ρ +a x ) ρ ) α/ρ) 1. From the symmetry of the production function we also have so that f x x) = αa x ) ρ 1 a 1 x 1 ) ρ +a x ) ρ ) α/ρ) 1, TRS 1, x) = f x 1 x) f x x) = αa 1x 1 ) ρ 1 a 1 x 1 ) ρ +a x ) ρ ) α/ρ) 1 αa x ) ρ 1 a 1 x 1 ) ρ +a x ) ρ ) α/ρ) 1 = a 1x 1 ) ρ 1 a x ) ρ 1. Thus, TRS 1, x) = a ) ρ 1 1 x1. a x Note that this is the same as what we found in class with the regular CES production function without α). This is discussed in the context of problem 9 below. 7. Provide the gradient for this production function. Show your work. The gradient of the production function is simply the vector of first partials which we have already found above, f x) x 1 fx) = f x) x = αa 1 x 1 ) ρ 1 a 1 x 1 ) ρ +a x ) ρ ) α/ρ) 1. αa x ) ρ 1 a 1 x 1 ) ρ +a x ) ρ ) α/ρ) 1 4

14 Envelope theorem basic example: 1) Let fx;a) = -x + ax + 4a be a function in one variable x that depends on a parameter a. Use the Envelope theorems to show that df*/da = 10a

15 Part Cost minimization. The technology of another firm can be represented by the production function f : R + R, fx 1,x ) = x 1 + x ) /..1 pt.) If you can, please name this type of technology. Recall that with your other problem set, you worked with a general production function f : R + R defined by: fx) = a 1 x 1 ) ρ +a x ) ρ ) α/ρ, where a 1 > 0, a > 0, α > 0 and ρ,0) 0,1). The production function for this part of the exam is clearly an example of that general production function, which was a variation on CES.. 4pt.) Formally state the firm s cost minimization problem with all appropriate constraints in their appropriate form. The cost minimization problem can be stated either with the nonnegativity constraints implied with the min notation or explicitly listed: min x 1,x ) R + w 1 x 1 +w 1 x s.t. x 1 + x ) / y. min x 1,x ) R w 1 x 1 +w 1 x s.t. x 1 + x ) / y, x 1 0, x 0.. 1pt.) Please find the firms s two component conditional factor demand functions. These need to be explicitly written as functions. You need to be clear about how you deal with any inequality constraints. This time we have three inequality constraints, two nonnegativity constraints and production feasibility constraint. As with the nonpositivity constraint in part 1, I will defer these nonnegativity constraints until after the Lagrangian analysis. Suppose that x1 + x ) / > y. Then we can lower the x1 or x values at least some and still satisfy the constraint. Moreover, reduction in either value, by say x k, will reduce the cost of inputs by p k x k. We have shown that x 1 + x ) / > y is not cost minimizing, and can assume the equality constraint x 1 + x ) / = y. Therefore we will work with the simpler optimization problem, min x 1,x ) R w 1 x 1 +w 1 x s.t. x 1 + x ) / = y. 1

16 with Lagrangian, and first order conditions, Lx,λ) = w 1 x 1 +w x λ x1 + x ) / y ), A 1 ) : w 1 λ x 1 + x ) 1/ ) 1 1 = 0, x1 A ) : w λ x 1 + x ) 1/ ) 1 1 = 0, x B) : x1 + x ) / = y. From A 1 ) and A ) we have w 1 x yielding the non-binding) technical rate of substitution, TRS 1, x 1,x ) = x. x 1 =, 1) w x1 Going back to equation 1), at this point we would typically solve for either x 1 or x, and then substitute into B). However, in this case a quick look at B) suggests that it will be quicker to solve for either x 1 or x. Since I want one of these by itself on the right, I will solve for x 1, Substituting this into B) yields, Substituting equation ) into ) gives us x1 = w w 1 x, ) ) w x + ) / x = y w 1 ) 4 x + = y w / w 1 4w +9w 1 x = y / w 1 w 1 x = y / ) 4w +9w 1 ) w 1 x = y 4/. 4) 4w +9w 1 x1 = w 1 y w / w 1 4w +9w 1 w x1 = y / 4w +9w 1 ) w x 1 = y 4/ 5) 4w +9w 1

17 We have obtained equations4) and5) without imposing the nonnegativity constraints on both inputs. However both expressions are nonnegative for all values of the economic independent variables, p 1 > 0, p > 0 and y 0. Thus the nonnegativity constraints are always satisfied with this pair of expressions and we can use them as the requested component conditional factor demand functions, x 1 w 1,w,y) = w 9w 1 +4w ) y 4/ and x w 1,w,y) = w 1 9w 1 +4w.4 6pt.) Provide the cost function for this firm clearly stated as a function. The cost of production is the firm s expenditure on the two inputs, Summarizing, cw 1,w,y) = w 1 x 1 w 1,w,y)+w x w 1,w,y) ) ) w = w 1 y 4/ w 1 +w y 4/ 9w 1 +4w 9w 1 +4w = w 1 w ) +w w 1 ) ) ) 1 y 4/ 9w 1 +4w ) 1 = w 1 w 4w +9w 1 ) y 4/ 9w 1 +4w w 1 w = y 4/. 9w 1 +4w cw 1,w,y) = w 1 w 9w 1 +4w y 4/. ) y 4/. ) p9w 1 +4w ).5 8pt.) Suppose that this firm s supply function is ŷp,w 1,w ) =. 4 w 1 w Provide the regular) factor demand function for input one and the firm s profit function, both clearly stated as functions. For this problem full credit was given for correctly fully setting up each of these functions. I did not take off any points because of simplification. A regular) factor demand function can be found by substituting the supply function into the corresponding conditional factor demand function, x 1 p,w 1,w ) = x 1 w 1,w,ŷp,w 1,w )) ) w = ŷp,w 1,w )) 4/ 9w 1 +4w ) ) ) 4/ w p9w 1 +4w ) = 9w 1 +4w 4 w 1 w ) ) 4 w p9w 1 +4w ) = 9w 1 +4w 4 w 1 w ) 4 = ) p) 4 w ) 4 w 1 w ) 4 9w 1 +4w ) = 81 p) 4 64w 1 ) 4 w ) 9w 1 +4w ).

18 Profit is revenue less cost, and revenue is the price of output times the quantity supplied, πp,w 1,w ) = pŷp,w 1,w ) cw 1,w,ŷp,w 1,w )) w 1 w = pŷp,w 1,w ) ŷp,w 1,w )) 4/ 9w 1 +4w ) ) ) 4/ p9w 1 +4w ) w 1 w p9w 1 +4w ) = p 4 w 1 w 9w 1 +4w 4 w 1 w ) ) = p) 4 9w 1 +4w ) 4 w 1 w p) 4 9w 1 +4w ) 4 4 w 1 w ) 9w 1 +4w 4 w 1 w ) 4 ) ) = p) 4 9w 1 +4w ) 4 p) 4 9w 1 +4w ) 4 w 1 w ) 4 w 1 w ) ) = 1 )p) 9w 1 +4w ) 4 4 w 1 w ) ) 1 = )p) 9w 1 +4w ) 4 4 w 1 w ) = 7 9w 1 +4w ) 56 p) w 1 w ) 4

19 Additional Questions for chapter 5 - solutions 1- Answer: Same idea for inputs -