Provide solutions to the following attachment.

Problem 3 [24 marks] A competitive firm uses two inputs, capital (?) and labour (?), to produce one output, (?). The price of capital, jQuery22401822330914895809_1621382561482, is $1 per unit and the price of labor, ??, is $1 per unit. The firm operates in competitive markets for outputs and inputs, so takes the prices as given. The production function is ?(?,?) = 3?0.25?0.25. The maximum amount of output produced for a given amount of inputs is ? = ?(?,?) units.

a) Use the method of Lagrange to find the conditional factor demands for cost minimization. [8 marks]

b) Find the firm's cost function. [2 marks]

c) Would you call this a short-run cost function or a long-run cost function? Explain why. [1 mark]

d) Write the equations for the firm's average cost function and marginal cost function. [2 marks]

e) Draw the firm's total cost function, average cost function, and marginal cost function on a diagram. Clearly label the axes, the curves, and any key points on the graph (eg., axis intercepts, curve intersections, and minimums) with the numbers specifying the exact prices and quantities at these points. What are the coordinates of the points where the average cost curve and marginal cost curve intersect with the total cost curve? [6 marks]

f) Does your graph indicate increasing, decreasing, or constant returns to scale? Explain. [1 mark] Hint: Think about the relationship between the total cost function and returns to scale.

g) Show the firm's long-run supply function on your diagram and write a supply equation for the firm. [2 marks]

h) Using your supply function, find the profit maximising quantity if the price of output ? = 4. What price would be needed for the firm to supply 18 units of output? [2 marks]

3. (20 points) Consider a competitive market with 100 firms (producers) which produce and offer an identical good on the market. There are two types of firms, type A and type B, which differ in their cost functions. Each firm of type A has cost function CA(y) = where y 2 0 denotes an amount of individual firm's output. On the other hand, each firm of type B has cost function CB(y) = y. Suppose 50 firms are of type A, and other 50 firms are of type B. The market (industry) demand is given by X (p) = 110 - 10p for nonnegative price p 2 0. (a) Derive the individual supply function of each firm of type A. (b) Derive equilibrium price, quantity, and total surplus. (20 points) Consider a pure exchange economy with three goods, 21, 12, and 13. There a consumers who are either of type A or of type B.Question 3: The Closed-Economy IS-LM-FE Framework (25 Marks) Consider the following closed economy: Odd = 150 - 400r + 0.6(Y - T) 1= 200 - 200m Government purchases, G equal 100 and the government runs a balanced budget. The liquidity function is given by: L(Y, i) = 200 + 0.2Y - 5002 The nominal money supply, price level and expected inflation are: M = 100 P - 5 IT = 0.05 (a) Derive the /S curve with the real interest rate r as a function of output Y. (b) Derive the LM curve with the real interest rate r as a function of output Y for the given price level, P = 5. (c) Find the long-run equilibrium values of the real interest rate r, full employment output, Y', consump- tion C. and investment /. Illustrate the results using a IS - LM - FE diagram to depict the long-run equilibrium. (d) Imagine that a financial innovation reduces the demand for real money balances so that the new liquidity function is: L(Y. () = 100 + 0.2Y - 500 i) Find the short-run equilibrium values of Y and r (when the price level is fixed). ii) What will happen to r. Y', P as the economy moves towards a long run equilibrium? You do not need to solve for the new values, please just describe whether they will increase, decrease or stay the same relative to the short run equilibrium in part i). ifi) Illustrate the short and long-run equilibria using an IS - LM - FE diagram