Kindly solve these

1. (15 points) Marginal rate of substitution and utility maximization For each of the following utility functions, do the following: 1. Find the marginal rate of substitution. 2. Then find the value for MRS at the bundle X = 1, Y = 2. Describe in one sentence the interpretation of this value for MRS. 3. Find the bundle that maximizes utility subject to the budget constraint if the price of good X is $2, the price of good Y is $2, and income is $20. (a) U = X1/2y1/4 (b) U = In(X) +2Y (c) U = X +4Y6. Now let's consider a Solow growth model where technology is constant and does not evolve over time. The production function is given as Y (t) = F(A, K(t), L(t) ) = AK(t) " L(t) 1 -a Where Y is output, A is technology, K is physical capital, and L is labor. Labor grows as a rate of n and technology is constant, i.e. g = 0 Suppose further that capital accumulation is given as K = SF(A, K(t), L(t) ) - 6K(t) a. Defining k = *, prove that y = = = Aka b. With the same definition of k, find the equation for k c. Find steady state levels of k (in terms of given parameters) d. Find steady state level of y e. Find steady state level of c4. A firm produces computers with two factors of production: labor L and capital K. It's pro- duction function is 10 Suppose the factor prices are wy = 10 and wx - 100. (a) Graph the isoquants for y equal to 1,2, and 3. Does this technology show increasing constant, or decreasing returns to scale? Why? (b) Derive the conditional factor demands. (c) Derive the long run cast function C'(y) (d) If the firm wants to produce one computer, how many units of labor and how many units of capital should it use? How much will it cost? What if the firm wants to produce two computers of Derive the firm's long run average cost function AC(y) and long run marginal cost He tion 1/Cty Graph AC(v) and NIGIvy () Label the firm's long run supply curse on the graph.1. Acme, Inc. produces widgets (w) and gadgets (g). Its profit function is given by: IT = 20w-w-+16g -2g a) How many widgets and gadgets should Acme sell to maximize profits? b) Suppose there is a constraint on production such that w + g $ 8. Compute the optimal quantity of widgets and gadgets.8. If Acme Manufacturing can produce 100 widgets for a linear production cost of $2,300 and they can produce 200 widgets for a linear production cost of $3,20 then:... a) What is the total production cost to produce 500 widgets? b) What is the fixed cost for Acme Manufacturing in producing widgets