Math Economics - Assignment Example

Math Economics Let . Compute , and . Also compute the total differential. SOLUTION 2. Let represent the demand for aparticular good at price p. Here I is income, is the price of a complement and is the price of a substitute. Discuss the economic interpretations of the signs of all partials of D.SOLUTIONIncome plays a major role in determining what and how much to purchase. People who suddenly inherit great wealth may purchase a Mercedes or other luxury items and would probably no longer buy do-it-yourself repair kitsThe prices of other goods also affect consumers’ purchase decisions.

Before deciding to buy Levi’s jeans, you might check the prices of other brands. If the price of a close substitute—a product that you view as similar or identical to the one you are considering purchasing—is much lower than the price of Levi’s jeans, you may buy that other brand instead.The price of a complement, , -a good that you like to consume at the same time as the product you are considering buying—may affect your decision. If you eat pie only with ice cream, the higher the price of ice cream, the less likely you are to buy pie3.

Consider a profit maximizing firm which rents capital K at rate R in order to produce goods q via the production function . Assume the firm takes the goods price as given. If determine the firm’s demand for capital.SOLUTIONThe idea is to find the derivative of 4. Same problem as above, but now the firm also has to hire labor at wage W and has production function where . Determine the firm’s demand for capital and labor.5. The idea is to find the partial derivative of Firm’s demand for capital is; Firm’s demand for labor is;6.

Consider a consumer with utility functionU(x, y) = α log(x) + (1 − α) log(y).Compute the marginal rate of substitution (MRS) between goods x and y. Compute an equation characterizing the agent’s indifference curve, and compute the slope of this curve. Compare this slope to the MRS. Use the MRS to compute the demand curves for x and y. Are the goods x and y complements? Substitutes?SOLUTIONNow that we have the marginal rate of substitution, we can find out what happens to the relative consumption of X and Y as we move along the consumer’s indifference curve.

To see what occurs as X increases, we take the partial derivative of MRS XY with respect to X:The consumer’s marginal rate of substitution — the slope of the indifference curve — remains the same as her consumption of X increases or decreases along the indifference curve. In other words, the consumption of commodity X has no any direct effect on consumption of commodity Y. thus the goods are not substitute but are compliments to each other.7. Consider the consumer choice problem given by(a) What is the total derivative of utility?

SOLUTIONUsing Lagrange multiplier we maximizeBy differentiating L with respect to , and , and setting the derivatives equal to zero, the resulting first order conditions are:and(b) If you sell a little of good x, say an amount dx, how much of good y can you buy?SOLUTIONYou can buy dy of good y(c) By selling a little x and buying a little y, how much does your utility change? (Hint: determined by the total differential.)SOLUTIONSetting the differential to zero yields