max(x1,x2)R+2px2wx1 subject to x2x1, where p>0 and w>0 are parameters. The interpretation of this problem is that a firm produces its output with only a single kind of inputs, with x1 denoting the input quantity and x2 the output quantity, so that the firm needs to choose an input-output plan, (x1,x2) R+2, to maximize its profit px2wx1, with p being the market price for the output, and w the wage rate for the input. The constraint x2x1 says that, if it employs a quantity x1 of the input, the firm can produce up to x1 units of the output but nothing beyond. a. On a diagram of the x1x2 plane, draw the set of all (x1,x2)R+2 for which x2x1. This is the entire choice set in this decision problem. b. Given any x1>0, calculate the slope of the graph of the equation x2=x1. c. Pick any positive constant c. Draw the level surface of the objective function for the constant c. Calculate the vertical (x2) intercept of this level surface. Does the decision maker prefer to be on a level surface with higher vertical intercept or lower vertical intercept? If this level surface is a straight line, calculate its slope (in terms of w and p ). d. Draw the level surface of the objective function that is a supporting hyperplane of the choice set such that the choice set is contained in the lower side of the hyperplane. e. Denote (x1,x2) for the point at which the above-specified supporting hyperplane touches the boundary of the choice set (i.e., the curve of x2=x1). Why is (x1,x2) the solution of our decision problem? f. What is the relationship between the slope of the hyperplane and the slope of the choice set boundary at (x1,x2) ? Calculate (x1,x2) in terms of the parameters p and w. max(x1,x2)R+2px2wx1 subject to x2x1, where p>0 and w>0 are parameters. The interpretation of this problem is that a firm produces its output with only a single kind of inputs, with x1 denoting the input quantity and x2 the output quantity, so that the firm needs to choose an input-output plan, (x1,x2) R+2, to maximize its profit px2wx1, with p being the market price for the output, and w the wage rate for the input. The constraint x2x1 says that, if it employs a quantity x1 of the input, the firm can produce up to x1 units of the output but nothing beyond. a. On a diagram of the x1x2 plane, draw the set of all (x1,x2)R+2 for which x2x1. This is the entire choice set in this decision problem. b. Given any x1>0, calculate the slope of the graph of the equation x2=x1. c. Pick any positive constant c. Draw the level surface of the objective function for the constant c. Calculate the vertical (x2) intercept of this level surface. Does the decision maker prefer to be on a level surface with higher vertical intercept or lower vertical intercept? If this level surface is a straight line, calculate its slope (in terms of w and p ). d. Draw the level surface of the objective function that is a supporting hyperplane of the choice set such that the choice set is contained in the lower side of the hyperplane. e. Denote (x1,x2) for the point at which the above-specified supporting hyperplane touches the boundary of the choice set (i.e., the curve of x2=x1). Why is (x1,x2) the solution of our decision problem? f. What is the relationship between the slope of the hyperplane and the slope of the choice set boundary at (x1,x2) ? Calculate (x1,x2) in terms of the parameters p and w
