Final answer:
The production function is concave and does not exhibit decreasing returns to scale. The firm's long-run cost minimization problem involves finding the combination of inputs that minimizes total cost while producing a desired level of output. Contingent input demands must be strictly positive when output is positive. The contingent input demands are concave in q, the cost function is convex in q, and marginal costs are higher than average costs.
Step-by-step explanation:
The production function f(k,l)=4l¹/² + 6k²/³ represents a firm's relationship between the quantities of inputs (labor and capital) and the quantity of output produced. To determine if the production function is concave, we need to check if the second derivative is negative. Taking the second derivatives, we find that both d²f/dk² and d²f/dl² are positive, meaning the production function is concave.
To determine if the production function exhibits decreasing returns to scale, we can check if increasing all inputs by a constant proportion results in an output increase that is proportionately smaller. Mathematically, if f(λk,λl) < λf(k,l) for λ > 1, then the function exhibits decreasing returns to scale. In this case, it can be shown that f(λk,λl) = 4λl¹/² + 6λ²k²/³, and λf(k,l) = λ(4l¹/² + 6k²/³). Since the two expressions are equal, the production function does not exhibit decreasing returns to scale.
The firm's (long-run) cost minimization problem involves finding the combination of inputs (k and l) that minimizes total cost while producing the desired level of output (q). The firm will choose the inputs that yield the lowest cost for a given level of output. Contingent input demands refer to the amounts of inputs required at each level of output. In this case, we need to show that the contingent input demands must be strictly positive whenever q > 0, which means that the firm requires positive amounts of inputs to produce positive output.
Using the tangency condition, we can derive a relationship between k and l. We take the ratio of the marginal product of capital (MPK) to the marginal product of labor (MPL) and set it equal to the ratio of the input prices (v/w). This gives us (2/3)(k/l) = 6.2.1, which simplifies to k/l = 9.3. The firm's expansion path represents the combinations of k and l that minimize cost for each level of output. It can be depicted as a curve in a figure, with different points on the curve representing different levels of output.
Solving the firm's (long-run) cost minimization problem involves finding the values of k and l that minimize the cost function. Using the tangency condition as before, we have k/l = 9.3. Substituting this into the production function f(k,l), we can write it as a function of one variable, q. Solving for q in terms of k, we get q = 60k²/³. The contingent input demands can then be expressed in terms of q as k(q) = (3/10)q²/³ and l(q) = (10/9)q²/³. The contingent input demands are concave in q because the second derivatives of both k(q) and l(q) are negative. The cost function can also be expressed in terms of q, and it is convex in q because the second derivative of the cost function is positive. Comparing marginal costs and average costs, marginal costs are higher than average costs when the cost function is convex, which is the case here.
In the short-run cost minimization problem, where k = 1 is fixed, the firm needs to find the optimal level of labor that minimizes cost for a given level of output. Using the tangency condition, we set MPL = w, where w is the price of labor. Solving for l, we get l = (208w/9)²/³. The short-run cost function can be expressed as C(q) = 3(208w/9)²/³q²/³.
Comparing the long-run and short-run input demands and costs at q = 10, we can substitute q = 10 into the expressions for k(q) and l(q) to find the values of k and l. We can also substitute q = 10 into the cost function to find the cost at this level of output. When q > 10, both the input demands and costs will increase.