Consider a firm with a production function given by y = a ln x1 + ß lIn x2 with a, ß E (0,1) which can sell its product for price p and by its inputs at input costs w1 and w2. Find the conditional input demand functions and the cost function for some given level of output y.

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Given the following data on input and output levels. Suppose the output price is $5 and input price is $10. Find the values of APP and MPP when X = 6: X 0 2 4 6 8 10 12 Y 0 100 250 450 600 700 750 50 and 150 6 and 200 100 and 75 75 and 100

A firm produces basketballs with two variable inputs - labour (L) and plastic (K) - and has the following production function:   f (L , K) = 6L1/3K1/6   If the price of output is 4, the price of labour is $2 and the price of plastic is $3, calculate the a) profit-maximizing level of each input b) the total production and c) the profit.

BJO Ⓒ4 f(X₁, X₂) = (x₁² + X₂²) & 0

(a) For the cost function C(w1, w2, y) = 2y²w} w, calculate the Allen elasticity of substitution between the two inputs at the cost-minimizing input point (xf(w1, w2, y), a(w1, w2, y)). (b) Consider the production function f(r, y, z) = Vry + rz+ yz. Find the scale elasticity SE at (x, y, z) = (1,2, 3), (5, 1,6), (6, 6, 6) and determine if the pro- duction function is IRTS, CRTS, or DRTS locally at each point. (c) A profit maximizing firm in the market operates where the production exhibits decreasing return to scale (DRTS). Is this market in its long-run equilibrium? Justify your answer. (d) Suppose that there are the infinite number of potential firms that produce the identical output good y under the cost function C(y) = + 3. Assume free entry and exit. Find the long-run equilibrium output price p, the amount of the output that each firm in the market produces in the long-run equilibrium, and the value of profit that each firm earns in the equilibrium.

Assume that the firm’s operation is subject to the following production function and price data: Q=3X+5Y-XY Px= 3 $ Py= 6 $ where X and Y are two variable input factors employed in the production of Q. A. In the unconstrained case, what levels of X and Y will maximize Q? B. It is possible to express the cost function associated with the use of X and Y in the production of Q as TC = 3X + 6Y subject to the firm’s constraint of an operating budget of $25. Use the Lagrange multiplier technique to determine the optimal levels of X and Y. What is the firm’s total output at these levels of input usage? What is the value of Lagrangian multiplier? C. What will happen to the firm’s output from a marginal increase in the operating budget? (Hint: This is another way of asking you to interpret the Lagrangian multiplier you found in B above).

3. A firm has a production function given by 1 1 A) y = 4x³x² ; 1 1 B) y = 3x4x² ; a) What are the factor demand functions? b) What are the conditional factor demand functions? c) What is the cost function? d) What is the supply function? 1 1 C) y = 5x³x2; 11 D) y = 12xx².

A5

Suppose a firm producing wood burning stoves has the following production function Q(K, L) = 4K¹/2 [1/2 Where L, the labour, and K, the capital are the 2 inputs of production and Q the quantity of stoves. Assume the price of one unit of L is £1 and the price of one unit of K is £2. a) b) Assume that K=9 in the short run. Draw the production function and calculate the marginal products of L as L changes from L= 1 to L= 6. What does an isoquant curve show? Draw the graph of a production isoquant representing input combinations that will produce 8 units of output.