Econ 301Intermediate MicroeconomicsProf. Marek WeretkaProblem set 7(due Thursday, March 18th, before class)Problem 1 (Production function)Consider the followin g three production fun ctionsf(K; L) = K2Lf(K; L) = K13L13f(K; L) = 2K + La) in the (K; L) space, sketch t he map of isoquants for each of them.b) gi ve some economic interpre tation for MPK and MPL (fo r an abstra ct production fun ction).c) …nd anali tically MPK for each production function above and pl ot it in the graph, with K on thehorizontal ax is and MP K on the vertical axis, assumingL = 1: Is MPK increasing, decreasing or constant?d) …nd MPL for each of the fo ur productions funct ions and plot it on the graph, with L on the horizontalax is and MP L on the vert ical axis, assumingK = 2: Is M P L increasing, decreasing or constant?e) Provide some economic intuition behind increasing, constant and decreasing returns to s cale. Give anexample of technology from real life that cap tur es each of the three cases (one ex am ple pe r technology).f) For each of the three production functions above, show formally whether it exhibits i ncreasing, decreasingor constant returns to scale.Problem 2 (Pro…t Maximization- Short run)GMC is producing cars using machine s (K) and labor (L) : The technology is capital intensive; theproduction funct ion is given byF (K; L) = K34L12The value of GMC physical capital (machines, real estate etc. ) is equal t oK = $16 billon (in calculationsignore billions). We analyze the behavior of the …rm in the sho rt run , i.e. in the period in whichK cannotbe changed. Suppose the price of a car is equal to p and the wage rate is w (parameters).a) write down the pro…t as a function of L:b) on a graph with L on the horizontal axis and $ on t he vertic al axis, plot two components of pro…tfunction: total revenue pF (K; L) and labor cost wL (when drawing, assume p = 1 and w = 2). On thegraph, mar k the le vel o f pro…t as the di¤erence between the two lines (for some L).c) We know that in order to …nd x that maximizes some function f(x), we take the …rst derivative of thefunction an d set it equal to zero (we call it a …rst order condition). Please explain intuitively why this methodallows us to …nd the optimum.d) S et the derivative of your pro…t fun ction with respect to L equal to zero and derive the secret of happine ss(the equation that tells that M PL =wp). Explain the economic intuition behind the latter conditi on.e) …nd analitically the optimal level of labor L that maximizes the pro…t as a function of the real wagew=p: (we call it the …rm’s labor demand). Find the values of L for the following values of parametersp 1 1 1w 8 4 2Plot your demand for labor on the graph with the real wage (wp) on the vertical axis and L on the horiz ontalone. Mark three points corresponding to the three values ofwpfrom the table.f) What is the maximal pro…t for each of the three values ofwpfrom the table?1Problem 3 (Labor Market)a) re view Problem 3 fro m PS4 (you will need it for the mid term). In that problem, we fo und that Kate’slabor sup ply is inelastic and e qual to 12h (pe r day). Plot her la bor supply in a graph with the real wage (wp)on the vertical ax is and L on the horizontal one .b) Su ppo se Kate works for GMC described in Problem 2 (in the present PS ). Add the labor demand curveof this com pany to your graph from ( a). Find the equilibri um real wage ratewpthat clears the labor mar ket(do i t analitically and using the graph)c) Suppose the wage rate is a bove the equilibrium level you found in (b). Do we observe unemploymentor excess demand of labor on the market? What market fo rces drive the wage rate down (give a newspaperstory; why will the wage go down?)d) Suppose Kate’s p references change so that now she is willing to supply only 8h: Show on the g raphhow th is a¤ects equilibriu m real wage and the level of labor. Find the two values analytically.e) Suppose the government passe s the law requiring that the minimal (real) wage rate is equal to w=p = 2:How doe this a¤ect the equilibrium on the labor market (assume Kate’s sup ply is 8h)? F ind the unemploymentrate associated with such poli cy.Problem 4 (Long Run)Jimmy produces milk using milk-making machines (read: cows) (K) and l abor (milkmen) (L). He h asaccess to the technology given byy = F (K; L) = K13L13The price of a gallon of m ilk is p = $1; the price of one machine (cow) is wk= $2 and the (milkmen) wagerate is wL= $1a) does this function exhibit increasing, c onstant or decreasing returns to scale?b) Write down the pro…t function - a function that de pends on K and L:c) Find the opt imal level of inputs and output and the maximal pro…t.(Hint: follow the steps shown in class - slides L14)d) Argue that the optimal choice of inputs also minimizes the cost o f production of the optimal level ofproduction (from (c))(show that t he cost minimiza tion condition